NMSE Section 6.1 mentioned, B

Average Accuracy: 77.1% → 99.6%

Time: 16.5s

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 77.1%

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

2. Simplified 77.1%

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

   [Start] 77.1	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*r/ [=>] 77.1	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-rgt-identity [=>] 77.1	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   sub-neg [=>] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
   distribute-neg-frac [=>] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
   metadata-eval [=>] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Applied egg-rr 99.6%

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

   [Start] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   frac-add [=>] 77.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]
   associate-*r/ [=>] 77.1	\[ \color{blue}{\frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(1 \cdot b + a \cdot -1\right)}{a \cdot b}} \]
   *-un-lft-identity [<=] 77.1	\[ \frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\color{blue}{b} + a \cdot -1\right)}{a \cdot b} \]
   *-commutative [=>] 77.1	\[ \frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(b + \color{blue}{-1 \cdot a}\right)}{a \cdot b} \]
   metadata-eval [<=] 77.1	\[ \frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(b + \color{blue}{\left(-1\right)} \cdot a\right)}{a \cdot b} \]
   cancel-sign-sub-inv [<=] 77.1	\[ \frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(b - 1 \cdot a\right)}}{a \cdot b} \]
   *-un-lft-identity [<=] 77.1	\[ \frac{\frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(b - \color{blue}{a}\right)}{a \cdot b} \]
   associate-/r/ [<=] 77.1	\[ \frac{\color{blue}{\frac{\frac{\pi}{2}}{\frac{b \cdot b - a \cdot a}{b - a}}}}{a \cdot b} \]
   flip-+ [<=] 99.6	\[ \frac{\frac{\frac{\pi}{2}}{\color{blue}{b + a}}}{a \cdot b} \]
   div-inv [=>] 99.6	\[ \frac{\frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b + a}}{a \cdot b} \]
   metadata-eval [=>] 99.6	\[ \frac{\frac{\pi \cdot \color{blue}{0.5}}{b + a}}{a \cdot b} \]
   *-commutative [=>] 99.6	\[ \frac{\frac{\pi \cdot 0.5}{b + a}}{\color{blue}{b \cdot a}} \]

4. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	74.2%
Cost	7442

\[\begin{array}{l} \mathbf{if}\;b \leq -72000000000 \lor \neg \left(b \leq -6.6 \cdot 10^{-45}\right) \land \left(b \leq -1.12 \cdot 10^{-55} \lor \neg \left(b \leq 1.6 \cdot 10^{-48}\right)\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \end{array} \]

Alternative 2
Accuracy	81.5%
Cost	7442

\[\begin{array}{l} \mathbf{if}\;b \leq -0.07 \lor \neg \left(b \leq -6.6 \cdot 10^{-45} \lor \neg \left(b \leq -1.12 \cdot 10^{-55}\right) \land b \leq 1.4 \cdot 10^{-48}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]

Alternative 3
Accuracy	88.4%
Cost	7441

\[\begin{array}{l} \mathbf{if}\;b \leq -27000000000:\\ \;\;\;\;\frac{0.5}{\frac{b}{\frac{\pi}{b \cdot a}}}\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-45} \lor \neg \left(b \leq -1.12 \cdot 10^{-55}\right) \land b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

Alternative 4
Accuracy	88.3%
Cost	7440

\[\begin{array}{l} t_0 := \frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ t_1 := \pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;b \leq -36000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ t_1 := \frac{0.5}{a} \cdot t_0\\ \mathbf{if}\;b \leq -1.5:\\ \;\;\;\;\frac{0.5}{\frac{b}{t_0}}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 6
Accuracy	88.1%
Cost	7440

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;b \leq -135000000000:\\ \;\;\;\;\frac{0.5}{\frac{b}{t_0}}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{0.5}{a} \cdot t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 7
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} t_0 := \frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{if}\;b \leq -15500000000:\\ \;\;\;\;\frac{0.5}{\frac{b}{\frac{\pi}{b \cdot a}}}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 8
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} t_0 := \frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{if}\;b \leq -520000000:\\ \;\;\;\;\frac{0.5}{\frac{b}{\frac{\pi}{b \cdot a}}}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{b \cdot a}\\ \end{array} \]

Alternative 9
Accuracy	88.5%
Cost	7440

\[\begin{array}{l} t_0 := \frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{if}\;b \leq -9500:\\ \;\;\;\;\frac{0.5}{\frac{b}{\frac{\pi}{b \cdot a}}}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b}}{b \cdot a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{b \cdot a}{\pi}}\\ \end{array} \]

Alternative 10
Accuracy	88.4%
Cost	7440

\[\begin{array}{l} t_0 := \frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{if}\;b \leq -16500000:\\ \;\;\;\;\frac{0.5}{\frac{b}{\frac{\pi}{b \cdot a}}}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b \cdot b}{0.5}}}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{b \cdot a}{\pi}}\\ \end{array} \]

Alternative 11
Accuracy	99.0%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]

Alternative 12
Accuracy	99.5%
Cost	7040

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

Alternative 13
Accuracy	53.3%
Cost	6912

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

Reproduce

herbie shell --seed 2023136 
(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))))