NMSE Section 6.1 mentioned, B

Average Accuracy: 78.6% → 99.6%

Time: 13.0s

Precision: binary64

Cost: 7040

Math

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

↓

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

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 (/ (/ (* PI 0.5) (+ 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 ((((double) M_PI) * 0.5) / (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 ((Math.PI * 0.5) / (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 ((math.pi * 0.5) / (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(pi * 0.5) / 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 = ((pi * 0.5) / (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[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

2. Simplified 78.6%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   Proof

   [Start] 78.6	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*r/ [=>] 78.6	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-rgt-identity [=>] 78.6	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   sub-neg [=>] 78.6	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
   distribute-neg-frac [=>] 78.6	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
   metadata-eval [=>] 78.6	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	89.6%
Cost	7440

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{a}\\ \mathbf{if}\;a \leq -0.155:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -3.9 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(b \cdot \frac{b}{\pi}\right)}\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{elif}\;a \leq 0.00054:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	89.6%
Cost	7440

\[\begin{array}{l} \mathbf{if}\;a \leq -230000:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(b \cdot \frac{b}{\pi}\right)}\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-39}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \end{array} \]

Alternative 3
Accuracy	89.6%
Cost	7440

\[\begin{array}{l} \mathbf{if}\;a \leq -3000:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(b \cdot \frac{b}{\pi}\right)}\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{elif}\;a \leq 0.000114:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \frac{0.5}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+98} \lor \neg \left(a \leq 9.8 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	75.7%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0285 \lor \neg \left(a \leq 2.15 \cdot 10^{-6}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 6
Accuracy	75.7%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \lor \neg \left(a \leq 3.3 \cdot 10^{-5}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 7
Accuracy	82.6%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0038 \lor \neg \left(a \leq 2.6 \cdot 10^{-10}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 8
Accuracy	89.8%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -0.216 \lor \neg \left(a \leq 6.5 \cdot 10^{-5}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 9
Accuracy	82.6%
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi}{a \cdot a}\\ \mathbf{if}\;a \leq -3.2:\\ \;\;\;\;0.5 \cdot \frac{t_0}{b}\\ \mathbf{elif}\;a \leq 0.00125:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 10
Accuracy	99.5%
Cost	7040

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

Alternative 11
Accuracy	53.1%
Cost	6912

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

Reproduce

herbie shell --seed 2023129 
(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))))