NMSE Section 6.1 mentioned, B

Average Error: 14.6 → 0.3

Time: 11.8s

Precision: binary64

Cost: 7040

Math

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

↓

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

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

Derivation

1. Initial program 14.6

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

2. Applied egg-rr 9.1

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.3

   \[\leadsto \frac{\color{blue}{\frac{\pi}{1} \cdot \frac{0.5}{b + a}}}{b \cdot a} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	7.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4329783210836582 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{a}}{\frac{b}{0.5}}\\ \mathbf{elif}\;b \leq 1.2402147671858271 \cdot 10^{-29}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(\left(b \cdot a\right) \cdot 2\right)}\\ \end{array} \]

Alternative 2
Error	7.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4329783210836582 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{a}}{\frac{b}{0.5}}\\ \mathbf{elif}\;b \leq 1.2402147671858271 \cdot 10^{-29}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(\left(b \cdot a\right) \cdot 2\right)}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7040

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

Alternative 4
Error	0.3
Cost	7040

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

Alternative 5
Error	0.2
Cost	7040

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

Alternative 6
Error	25.2
Cost	6912

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

Alternative 7
Error	25.1
Cost	6912

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

Alternative 8
Error	25.1
Cost	6912

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

Reproduce

herbie shell --seed 2022294 
(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))))