NMSE Section 6.1 mentioned, B

Average Accuracy: 76.9% → 99.5%

Time: 11.9s

Precision: binary64

Cost: 7552

Math

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

↓

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

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 (- b a)) (/ (/ (- b a) (* b a)) (/ (+ b a) PI))))

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 / (b - a)) * (((b - a) / (b * a)) / ((b + a) / ((double) M_PI)));
}

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 / (b - a)) * (((b - a) / (b * a)) / ((b + a) / Math.PI));
}

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 / (b - a)) * (((b - a) / (b * a)) / ((b + a) / math.pi))

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(0.5 / Float64(b - a)) * Float64(Float64(Float64(b - a) / Float64(b * a)) / Float64(Float64(b + a) / pi)))
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 / (b - a)) * (((b - a) / (b * a)) / ((b + a) / pi));
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[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b - a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b + a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

2. Simplified 85.2%

   \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   Proof

   [Start] 76.9	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   times-frac [<=] 76.9	\[ \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-commutative [<=] 76.9	\[ \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   times-frac [=>] 76.9	\[ \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   difference-of-squares [=>] 84.4	\[ \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-/r* [=>] 85.2	\[ \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   metadata-eval [=>] 85.2	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   sub-neg [=>] 85.2	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
   distribute-neg-frac [=>] 85.2	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
   metadata-eval [=>] 85.2	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Applied egg-rr 84.4%

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

   [Start] 85.2	\[ \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   associate-*l* [=>] 85.2	\[ \color{blue}{\frac{\frac{\pi}{b + a}}{b - a} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)} \]
   clear-num [=>] 84.4	\[ \color{blue}{\frac{1}{\frac{b - a}{\frac{\pi}{b + a}}}} \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right) \]
   associate-*l/ [=>] 84.5	\[ \color{blue}{\frac{1 \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)}{\frac{b - a}{\frac{\pi}{b + a}}}} \]
   *-un-lft-identity [<=] 84.5	\[ \frac{\color{blue}{0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}}{\frac{b - a}{\frac{\pi}{b + a}}} \]
   +-commutative [=>] 84.5	\[ \frac{0.5 \cdot \color{blue}{\left(\frac{-1}{b} + \frac{1}{a}\right)}}{\frac{b - a}{\frac{\pi}{b + a}}} \]
   frac-add [=>] 84.4	\[ \frac{0.5 \cdot \color{blue}{\frac{-1 \cdot a + b \cdot 1}{b \cdot a}}}{\frac{b - a}{\frac{\pi}{b + a}}} \]
   neg-mul-1 [<=] 84.4	\[ \frac{0.5 \cdot \frac{\color{blue}{\left(-a\right)} + b \cdot 1}{b \cdot a}}{\frac{b - a}{\frac{\pi}{b + a}}} \]
   *-commutative [<=] 84.4	\[ \frac{0.5 \cdot \frac{\left(-a\right) + \color{blue}{1 \cdot b}}{b \cdot a}}{\frac{b - a}{\frac{\pi}{b + a}}} \]
   *-un-lft-identity [<=] 84.4	\[ \frac{0.5 \cdot \frac{\left(-a\right) + \color{blue}{b}}{b \cdot a}}{\frac{b - a}{\frac{\pi}{b + a}}} \]
   +-commutative [<=] 84.4	\[ \frac{0.5 \cdot \frac{\color{blue}{b + \left(-a\right)}}{b \cdot a}}{\frac{b - a}{\frac{\pi}{b + a}}} \]
   sub-neg [<=] 84.4	\[ \frac{0.5 \cdot \frac{\color{blue}{b - a}}{b \cdot a}}{\frac{b - a}{\frac{\pi}{b + a}}} \]
   div-inv [=>] 84.4	\[ \frac{0.5 \cdot \frac{b - a}{b \cdot a}}{\color{blue}{\left(b - a\right) \cdot \frac{1}{\frac{\pi}{b + a}}}} \]
   clear-num [<=] 84.4	\[ \frac{0.5 \cdot \frac{b - a}{b \cdot a}}{\left(b - a\right) \cdot \color{blue}{\frac{b + a}{\pi}}} \]

4. Simplified 99.5%

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

   [Start] 84.4	\[ \frac{0.5 \cdot \frac{b - a}{b \cdot a}}{\left(b - a\right) \cdot \frac{b + a}{\pi}} \]
   times-frac [=>] 99.5	\[ \color{blue}{\frac{0.5}{b - a} \cdot \frac{\frac{b - a}{b \cdot a}}{\frac{b + a}{\pi}}} \]
   *-commutative [=>] 99.5	\[ \frac{0.5}{b - a} \cdot \frac{\frac{b - a}{\color{blue}{a \cdot b}}}{\frac{b + a}{\pi}} \]
   +-commutative [=>] 99.5	\[ \frac{0.5}{b - a} \cdot \frac{\frac{b - a}{a \cdot b}}{\frac{\color{blue}{a + b}}{\pi}} \]

5. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	89.0%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-12} \lor \neg \left(a \leq 9 \cdot 10^{-23}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]

Alternative 2
Accuracy	89.2%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-12} \lor \neg \left(a \leq 7.5 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \frac{\pi}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]

Alternative 3
Accuracy	89.2%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-11} \lor \neg \left(a \leq 3.7 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a}}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{b \cdot a}\\ \end{array} \]

Alternative 4
Accuracy	89.2%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{b \cdot a}}{a}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a}}{b \cdot a}\\ \end{array} \]

Alternative 5
Accuracy	99.6%
Cost	7040

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

Alternative 6
Accuracy	60.3%
Cost	6912

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

Alternative 7
Accuracy	60.3%
Cost	6912

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

Reproduce

herbie shell --seed 2023151 
(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))))