NMSE Section 6.1 mentioned, B

Average Error: 14.4 → 0.7

Time: 11.6s

Precision: binary64

Cost: 7040

Math

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

↓

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

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 (* (+ a b) (* a b))) 2.0))

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) / ((a + b) * (a * b))) / 2.0;
}

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 / ((a + b) * (a * b))) / 2.0;
}

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 / ((a + b) * (a * b))) / 2.0

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(Float64(a + b) * Float64(a * b))) / 2.0)
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 / ((a + b) * (a * b))) / 2.0;
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[(N[(a + b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 14.4

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

2. Simplified 14.3

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

   [Start] 14.4	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*l* [=>] 14.4	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]
   associate-*l/ [=>] 14.3	\[ \frac{\pi}{2} \cdot \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b \cdot b - a \cdot a}} \]
   *-lft-identity [=>] 14.3	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} - \frac{1}{b}}}{b \cdot b - a \cdot a} \]
   sub-neg [=>] 14.3	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} + \left(-\frac{1}{b}\right)}}{b \cdot b - a \cdot a} \]
   distribute-neg-frac [=>] 14.3	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b \cdot b - a \cdot a} \]
   metadata-eval [=>] 14.3	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b \cdot b - a \cdot a} \]

3. Applied egg-rr 0.7

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

4. Final simplification 0.7

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

Alternatives

Alternative 1
Error	16.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44} \lor \neg \left(a \leq 6.5 \cdot 10^{-40}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 2
Error	12.0
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44} \lor \neg \left(a \leq 6.5 \cdot 10^{-40}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 3
Error	12.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \end{array} \]

Alternative 4
Error	7.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \end{array} \]

Alternative 5
Error	7.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\frac{\pi}{a}}}}{a}\\ \end{array} \]

Alternative 6
Error	7.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\frac{b}{\frac{\pi}{a}}}}{a}\\ \end{array} \]

Alternative 7
Error	0.8
Cost	7040

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

Alternative 8
Error	0.3
Cost	7040

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

Alternative 9
Error	30.2
Cost	6912

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

Reproduce

herbie shell --seed 2023010 
(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))))