NMSE Section 6.1 mentioned, B

Average Error: 22.62% → 0.38%

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 22.62

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

2. Simplified 22.58

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

   [Start] 22.62	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*r/ [=>] 22.58	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-rgt-identity [=>] 22.58	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   sub-neg [=>] 22.58	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
   distribute-neg-frac [=>] 22.58	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
   metadata-eval [=>] 22.58	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Applied egg-rr 0.38

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

4. Final simplification 0.38

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

Alternatives

Alternative 1
Error	24.93%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-53} \lor \neg \left(a \leq 0.0015\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 2
Error	10.41%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-52} \lor \neg \left(a \leq 0.00025\right):\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 3
Error	24.96%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq 0.0096:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \end{array} \]

Alternative 4
Error	17.86%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq 0.0049:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \end{array} \]

Alternative 5
Error	10.55%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{0.5}{a \cdot \left(a \cdot \frac{b}{\pi}\right)}\\ \mathbf{elif}\;a \leq 0.0095:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \end{array} \]

Alternative 6
Error	10.54%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;a \leq 0.0105:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \end{array} \]

Alternative 7
Error	10.51%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;a \leq 0.0028:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot a}\\ \end{array} \]

Alternative 8
Error	0.42%
Cost	7040

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

Alternative 9
Error	46.49%
Cost	6912

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

Reproduce

herbie shell --seed 2023102 
(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))))