NMSE Section 6.1 mentioned, B

Average Error: 14.2 → 0.3

Time: 8.3s

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.7925240140726623 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \mathbf{elif}\;b \leq 1.0983780295633275 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot \left(\left(b + a\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{b}{\frac{\pi}{a}}}\\ \end{array} \]

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
 (if (<= b -3.7925240140726623e+120)
   (/ (* (/ PI a) (/ 0.5 b)) b)
   (if (<= b 1.0983780295633275e+95)
     (/ (/ PI a) (* b (* (+ b a) 2.0)))
     (/ (/ 0.5 b) (/ b (/ PI 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) {
	double tmp;
	if (b <= -3.7925240140726623e+120) {
		tmp = ((((double) M_PI) / a) * (0.5 / b)) / b;
	} else if (b <= 1.0983780295633275e+95) {
		tmp = (((double) M_PI) / a) / (b * ((b + a) * 2.0));
	} else {
		tmp = (0.5 / b) / (b / (((double) M_PI) / a));
	}
	return tmp;
}

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) {
	double tmp;
	if (b <= -3.7925240140726623e+120) {
		tmp = ((Math.PI / a) * (0.5 / b)) / b;
	} else if (b <= 1.0983780295633275e+95) {
		tmp = (Math.PI / a) / (b * ((b + a) * 2.0));
	} else {
		tmp = (0.5 / b) / (b / (Math.PI / a));
	}
	return tmp;
}

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):
	tmp = 0
	if b <= -3.7925240140726623e+120:
		tmp = ((math.pi / a) * (0.5 / b)) / b
	elif b <= 1.0983780295633275e+95:
		tmp = (math.pi / a) / (b * ((b + a) * 2.0))
	else:
		tmp = (0.5 / b) / (b / (math.pi / a))
	return tmp

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)
	tmp = 0.0
	if (b <= -3.7925240140726623e+120)
		tmp = Float64(Float64(Float64(pi / a) * Float64(0.5 / b)) / b);
	elseif (b <= 1.0983780295633275e+95)
		tmp = Float64(Float64(pi / a) / Float64(b * Float64(Float64(b + a) * 2.0)));
	else
		tmp = Float64(Float64(0.5 / b) / Float64(b / Float64(pi / a)));
	end
	return tmp
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_2 = code(a, b)
	tmp = 0.0;
	if (b <= -3.7925240140726623e+120)
		tmp = ((pi / a) * (0.5 / b)) / b;
	elseif (b <= 1.0983780295633275e+95)
		tmp = (pi / a) / (b * ((b + a) * 2.0));
	else
		tmp = (0.5 / b) / (b / (pi / a));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[b, -3.7925240140726623e+120], N[(N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[b, 1.0983780295633275e+95], N[(N[(Pi / a), $MachinePrecision] / N[(b * N[(N[(b + a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b), $MachinePrecision] / N[(b / N[(Pi / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79252401407266233e120

   1. Initial program 25.6

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

   2. Taylor expanded in b around inf 12.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]

   3. Simplified 12.3

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

   4. Applied egg-rr 0.2

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

   if -3.79252401407266233e120 < b < 1.0983780295633275e95

   1. Initial program 8.0

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

   2. Simplified 0.4

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

   3. Taylor expanded in b around 0 0.4

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

   4. Simplified 0.4

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

   5. Applied egg-rr 0.3

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

   if 1.0983780295633275e95 < b

   1. Initial program 23.1

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

   2. Simplified 0.2

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

   3. Taylor expanded in b around inf 11.8

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]

   4. Simplified 10.9

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

   5. Applied egg-rr 0.2

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

   6. Applied egg-rr 0.3

      \[\leadsto \frac{\frac{0.5}{b}}{\color{blue}{\frac{b}{\frac{\pi}{a}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7925240140726623 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \mathbf{elif}\;b \leq 1.0983780295633275 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot \left(\left(b + a\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{b}{\frac{\pi}{a}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.1
Cost	7176

\[\begin{array}{l} t_0 := \frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \mathbf{if}\;b \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.6465985719733 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot 0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.1
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \mathbf{elif}\;b \leq 5.6465985719733 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot 0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{b \cdot \frac{a}{\pi}}\\ \end{array} \]

Alternative 3
Error	7.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \mathbf{elif}\;b \leq 5.6465985719733 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot 0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{b}{\frac{\pi}{a}}}\\ \end{array} \]

Alternative 4
Error	7.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \mathbf{elif}\;b \leq 5.6465985719733 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\pi}{b}}{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{b}{\frac{\pi}{a}}}\\ \end{array} \]

Alternative 5
Error	7.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b}\\ \mathbf{elif}\;b \leq 5.6465985719733 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{\frac{b}{\frac{\pi}{a}}}\\ \end{array} \]

Alternative 6
Error	25.8
Cost	6912

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

Reproduce

herbie shell --seed 2022228 
(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))))