NMSE Section 6.1 mentioned, B

Average Accuracy: 77.4% → 99.5%

Time: 13.4s

Precision: binary64

Cost: 7552

Math

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

↓

\[\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{a + b}}{b - a} \]

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 2.0) (/ (/ (+ (/ 1.0 a) (/ -1.0 b)) (+ a b)) (- 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) / 2.0) * ((((1.0 / a) + (-1.0 / b)) / (a + b)) / (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 / 2.0) * ((((1.0 / a) + (-1.0 / b)) / (a + b)) / (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 / 2.0) * ((((1.0 / a) + (-1.0 / b)) / (a + b)) / (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(pi / 2.0) * Float64(Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(a + b)) / 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 / 2.0) * ((((1.0 / a) + (-1.0 / b)) / (a + b)) / (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[(Pi / 2.0), $MachinePrecision] * N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

2. Simplified 77.4%

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

   [Start] 77.4	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*l* [=>] 77.4	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

3. Applied egg-rr 99.5%

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

   [Start] 77.4	\[ \frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right) \]
   associate-*l/ [=>] 77.5	\[ \frac{\pi}{2} \cdot \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b \cdot b - a \cdot a}} \]
   *-un-lft-identity [<=] 77.5	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} - \frac{1}{b}}}{b \cdot b - a \cdot a} \]
   difference-of-squares [=>] 85.0	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \]
   associate-/r* [=>] 99.5	\[ \frac{\pi}{2} \cdot \color{blue}{\frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}} \]
   sub-neg [=>] 99.5	\[ \frac{\pi}{2} \cdot \frac{\frac{\color{blue}{\frac{1}{a} + \left(-\frac{1}{b}\right)}}{b + a}}{b - a} \]
   distribute-neg-frac [=>] 99.5	\[ \frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b + a}}{b - a} \]
   metadata-eval [=>] 99.5	\[ \frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b + a}}{b - a} \]

4. Final simplification 99.5%

   \[\leadsto \frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{a + b}}{b - a} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{+105} \lor \neg \left(a \leq 2.7 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	74.3%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+25} \lor \neg \left(a \leq 1.8 \cdot 10^{-52}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 3
Accuracy	81.0%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+30} \lor \neg \left(a \leq 1.8 \cdot 10^{-52}\right):\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 4
Accuracy	81.1%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{+27} \lor \neg \left(a \leq 1.8 \cdot 10^{-52}\right):\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \]

Alternative 5
Accuracy	88.4%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{+27} \lor \neg \left(a \leq 1.8 \cdot 10^{-52}\right):\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 6
Accuracy	74.4%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{a}}{b}\\ \end{array} \]

Alternative 7
Accuracy	88.4%
Cost	7176

\[\begin{array}{l} t_0 := a \cdot \left(a \cdot b\right)\\ \mathbf{if}\;a \leq -8.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{t_0}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b} \cdot \frac{\pi}{b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{t_0}\\ \end{array} \]

Alternative 8
Accuracy	99.6%
Cost	7040

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

Alternative 9
Accuracy	53.4%
Cost	6912

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

Reproduce

herbie shell --seed 2023137 
(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))))