NMSE Section 6.1 mentioned, B

Average Error: 14.4 → 0.7

Time: 35.8s

Precision: binary64

Cost: 7552

Math

\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

↓

\[\frac{0.5 \cdot \left(a - b\right)}{a - b} \cdot \frac{\pi}{\left(b \cdot a\right) \cdot \left(b + a\right)} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

↓

(FPCore (a b)
 :precision binary64
 (* (/ (* 0.5 (- a b)) (- a b)) (/ PI (* (* b a) (+ b a)))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

↓

double code(double a, double b) {
	return ((0.5 * (a - b)) / (a - b)) * (((double) M_PI) / ((b * a) * (b + a)));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

↓

public static double code(double a, double b) {
	return ((0.5 * (a - b)) / (a - b)) * (Math.PI / ((b * a) * (b + a)));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

↓

def code(a, b):
	return ((0.5 * (a - b)) / (a - b)) * (math.pi / ((b * a) * (b + a)))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

↓

function code(a, b)
	return Float64(Float64(Float64(0.5 * Float64(a - b)) / Float64(a - b)) * Float64(pi / Float64(Float64(b * a) * Float64(b + a))))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

↓

function tmp = code(a, b)
	tmp = ((0.5 * (a - b)) / (a - b)) * (pi / ((b * a) * (b + a)));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[(N[(N[(0.5 * N[(a - b), $MachinePrecision]), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(N[(b * a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.4

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

2. Simplified 14.3

   \[\leadsto \color{blue}{\left(0.5 \cdot \frac{a - b}{\left(\left(b + a\right) \cdot \left(a - b\right)\right) \cdot \left(b \cdot a\right)}\right) \cdot \pi} \]
   Proof

3. Applied egg-rr 0.7

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(a - b\right)}{a - b} \cdot \frac{\pi}{\left(b \cdot a\right) \cdot \left(b + a\right)}} \]

Alternatives

Alternative 1
Error	12.0
Cost	7176

\[\begin{array}{l} t_0 := \pi \cdot \frac{\frac{0.5}{a}}{a \cdot b}\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(b \cdot b\right)} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	12.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{a \cdot b}\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(b \cdot b\right)} \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]

Alternative 3
Error	12.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{a}}{a \cdot b}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]

Alternative 4
Error	12.0
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	7.4
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	7.6
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(b \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	7.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(b \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a}}{b \cdot a}\\ \end{array} \]

Alternative 8
Error	7.6
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(b \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{a}\\ \end{array} \]

Alternative 9
Error	7.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(b \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\pi}{a}}{2}}{b}}{a}\\ \end{array} \]

Alternative 10
Error	7.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\pi}{a}}{b}}{b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\pi}{a}}{2}}{b}}{a}\\ \end{array} \]

Alternative 11
Error	0.3
Cost	7168

\[\frac{\frac{\frac{1}{b}}{a}}{b + a} \cdot \left(\pi \cdot 0.5\right) \]

Alternative 12
Error	0.2
Cost	7040

\[\frac{\frac{\pi}{a \cdot b}}{\frac{b + a}{0.5}} \]

Alternative 13
Error	25.4
Cost	6912

\[\pi \cdot \frac{\frac{0.5}{a}}{a \cdot b} \]

Reproduce

herbie shell --seed 2023010 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))