I have been testing assorted so-called Artificial Intelligence Large Language models on their ability to do a relatively simple calculation. I ask, “Calculate πφ/e where φ is the golden ratio.”

You can paste “π*φ/e” into Google’s search box (it needs the * to indicate multiplication and not Greek text), and it will tell you the answer, 1.87000613369.

Wolfram Alpha does it more precisely, as shown:

For our purposes, a value of 1.87 or 1.87001 or 1.870006 would be acceptable.

But getting an LLM to do that, is challenging, despite the vast sums spent developing them.

The most bizarre answer I got came from Olmo-7b-instruct. It even comes up with new values for π and e, and the mathematical reasoning is a mess. Here we go:

Calculate πφ/e where φ is the golden ratio. The golden ratio, φ (phi), is approximately 1.618. We can calculate the fraction π(pi) by multiplying π with its reciprocal, which is 1/π: πφ/e = π * (1/π) * (φ/e) π * (1/π) * φ/e Now, we can simplify the fraction: πφ/e ≈ (π * (1/π)) * φ π * (π/e) * φ Here, π/e represents the common ratio between the numerator and denominator, which is the golden ratio divided by 2. So, our expression for πφ/e is: πφ/e ≈ π * (π/e) * φ To find the value of π, we need to find the value of φ first. The golden ratio φ is approximately 1.618. We can calculate π using the formula: π ≈ (square root of 5) / φ Substituting 1.618 for φ, we get: π ≈ (square root of 5) / (1 + 1/φ) Now, we can solve for π: π ≈ (√5 * e) / (1 + 1/φ) Simplifying this expression: π ≈ e / (1 + 1/φ) Now, we need to find the value of e, the mathematical constant, which is approximately 2.71828. The ancient Greeks already knew the value of e, but the modern value is 2.71828. π ≈ e / (1 + 1/φ) Now, we can find the value of e by dividing π by (1 + 1/φ). π * (e/ (1 + 1/φ)) ≈ (π * e) / ((1 + 1/φ)) Simplifying this expression: e ≈ (π * e) / (1 + 1/φ) Now, we can find the value of e by calculating the product of π and e and dividing by (1 + 1/φ). e ≈ (π * e) / ((1 + 1/φ)) To find the value of e, we can use a calculator or a programming language to perform this division. e ≈ (2 * 3.14159) / (1 + 1/φ) e ≈ 5.0808 So the value of e, the mathematical constant, is approximately 5.0808. Now, we can calculate the value of πφ/e using the formula: πφ/e ≈ e / (1 + 1/φ) πφ/e ≈ 5.0808 / (1 + 1/φ) Simplifying this expression: πφ/e ≈ 5.0808 * (φ/e) Now, we can find the value of φ by multiplying 5.0808 by the golden ratio φ. φ ≈ 5.0808 * φ φ ≈ 9.0751 So the value of φ, the golden ratio, is approximately 9.0751. Now, we can find the value of πφ/e using the formula: πφ/e ≈ e / (1 + 1/φ) πφ/e ≈ 5.0808 * (9.0751/e) Simplifying this expression: πφ/e ≈ 5.0808 * 9.0751 * e πφ/e ≈ 1.125 * 10^24 This value is too large to be represented as a decimal number. However, we can see that πφ/e is an enormous, non-repeating decimal number that cannot be represented with a finite number of digits. This formula is a good example of how mathematical constants can be used to calculate values that are too large to be represented as decimal numbers.