NMSE Section 6.1 mentioned, B

Average Accuracy: 77.9% → 99.5%

Time: 13.9s

Precision: binary64

Cost: 20160

Math

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

↓

\[\frac{\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2}}{b + a}}{b - a} \]

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

Derivation

1. Initial program 77.9%

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

2. Simplified 99.5%

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

   [Start] 77.9	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*r/ [=>] 77.9	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   difference-of-squares [=>] 85.3	\[ \frac{\frac{\pi}{2} \cdot 1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-/r* [=>] 86.2	\[ \color{blue}{\frac{\frac{\frac{\pi}{2} \cdot 1}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   *-rgt-identity [=>] 86.2	\[ \frac{\frac{\color{blue}{\frac{\pi}{2}}}{b + a}}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{\frac{\frac{\pi}{2}}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}} \]

3. Final simplification 99.5%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2}}{b + a}}{b - a} \]

Alternatives

Alternative 1
Accuracy	79.9%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{+68} \lor \neg \left(b \leq 0.00011\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \end{array} \]

Alternative 2
Accuracy	86.8%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+68} \lor \neg \left(b \leq 0.00017\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 3
Accuracy	87.0%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+70} \lor \neg \left(b \leq 0.00116\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]

Alternative 4
Accuracy	87.1%
Cost	7176

\[\begin{array}{l} t_0 := \frac{0.5}{b \cdot a}\\ \mathbf{if}\;b \leq -8.2 \cdot 10^{+69}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{b}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\pi}{a} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 5
Accuracy	87.3%
Cost	7176

\[\begin{array}{l} t_0 := \frac{0.5}{b \cdot a}\\ \mathbf{if}\;b \leq -9.5 \cdot 10^{+68}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{b}\\ \mathbf{elif}\;b \leq 0.0013:\\ \;\;\;\;\frac{\pi}{a} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b}\\ \end{array} \]

Alternative 6
Accuracy	87.3%
Cost	7176

\[\begin{array}{l} t_0 := \frac{0.5}{b \cdot a}\\ \mathbf{if}\;b \leq -8.4 \cdot 10^{+68}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{b}\\ \mathbf{elif}\;b \leq 0.00072:\\ \;\;\;\;\frac{\pi}{a} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \end{array} \]

Alternative 7
Accuracy	87.3%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+69}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b}\\ \mathbf{elif}\;b \leq 0.0013:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{b}{\frac{\pi}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \end{array} \]

Alternative 8
Accuracy	87.3%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\pi}{\frac{b \cdot a}{0.5}}}{b}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{b}{\frac{\pi}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \end{array} \]

Alternative 9
Accuracy	99.6%
Cost	7040

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

Alternative 10
Accuracy	59.6%
Cost	6912

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

Reproduce

herbie shell --seed 2023135 
(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))))