NMSE Section 6.1 mentioned, B

Average Error: 14.2 → 0.3

Time: 11.6s

Precision: binary64

Cost: 7040

Math

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

↓

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

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 (* a b)) PI) (+ a b)))

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

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

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

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

Derivation

1. Initial program 14.2

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

2. Simplified 14.0

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

3. Applied egg-rr 0.7

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

4. Applied egg-rr 0.3

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

Alternatives

Alternative 1
Error	6.9
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi}{\frac{b \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{if}\;b \leq -5967.678746111406:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.3541544081358155 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot b} \cdot \pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.9
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi}{\frac{b \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{if}\;b \leq -5967.678746111406:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.3541544081358155 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	6.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -5967.678746111406:\\ \;\;\;\;\frac{\pi}{\frac{b \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;b \leq 3.3541544081358155 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 \cdot \pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 4
Error	6.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -5967.678746111406:\\ \;\;\;\;\frac{\pi}{\frac{b \cdot \left(a \cdot b\right)}{0.5}}\\ \mathbf{elif}\;b \leq 3.3541544081358155 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{b}}{a \cdot b}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	7040

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

Alternative 6
Error	25.5
Cost	6912

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

Alternative 7
Error	25.5
Cost	6912

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

Reproduce

herbie shell --seed 2022325 
(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))))