NMSE Section 6.1 mentioned, B

Average Error: 14.2 → 0.9

Time: 12.8s

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;a \leq 2.65 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{a}\\ \end{array} \]

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 (<= a -4.2e+167)
   (/ (* PI 0.5) (* a (* b a)))
   (if (<= a 2.65e+119)
     (* 0.5 (/ PI (* b (* a (+ b a)))))
     (/ (* 0.5 (/ PI (* b a))) 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 (a <= -4.2e+167) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else if (a <= 2.65e+119) {
		tmp = 0.5 * (((double) M_PI) / (b * (a * (b + a))));
	} else {
		tmp = (0.5 * (((double) M_PI) / (b * a))) / 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 (a <= -4.2e+167) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else if (a <= 2.65e+119) {
		tmp = 0.5 * (Math.PI / (b * (a * (b + a))));
	} else {
		tmp = (0.5 * (Math.PI / (b * a))) / 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 a <= -4.2e+167:
		tmp = (math.pi * 0.5) / (a * (b * a))
	elif a <= 2.65e+119:
		tmp = 0.5 * (math.pi / (b * (a * (b + a))))
	else:
		tmp = (0.5 * (math.pi / (b * a))) / 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 (a <= -4.2e+167)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	elseif (a <= 2.65e+119)
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * Float64(b + a)))));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(b * a))) / 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 (a <= -4.2e+167)
		tmp = (pi * 0.5) / (a * (b * a));
	elseif (a <= 2.65e+119)
		tmp = 0.5 * (pi / (b * (a * (b + a))));
	else
		tmp = (0.5 * (pi / (b * a))) / 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[a, -4.2e+167], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.65e+119], N[(0.5 * N[(Pi / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.1999999999999998e167

   1. Initial program 31.0

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

   2. Simplified 31.0

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

      [Start] 31.0	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*l* [=>] 31.0	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

   3. Taylor expanded in b around 0 13.8

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

   4. Simplified 0.7

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

      [Start] 13.8	\[ 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]
      associate-*r/ [=>] 13.8	\[ \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]
      unpow2 [=>] 13.8	\[ \frac{0.5 \cdot \pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]
      associate-*l* [=>] 0.7	\[ \frac{0.5 \cdot \pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \]

   if -4.1999999999999998e167 < a < 2.64999999999999986e119

   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 7.9

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

      [Start] 8.0	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*l* [=>] 8.0	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]
      associate-*l/ [=>] 7.9	\[ \frac{\pi}{2} \cdot \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b \cdot b - a \cdot a}} \]
      *-lft-identity [=>] 7.9	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} - \frac{1}{b}}}{b \cdot b - a \cdot a} \]
      sub-neg [=>] 7.9	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} + \left(-\frac{1}{b}\right)}}{b \cdot b - a \cdot a} \]
      distribute-neg-frac [=>] 7.9	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b \cdot b - a \cdot a} \]
      metadata-eval [=>] 7.9	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b \cdot b - a \cdot a} \]

   3. Applied egg-rr 36.1

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \cdot 0.5\right)} - 1} \]

   4. Simplified 1.2

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

      [Start] 36.1	\[ e^{\mathsf{log1p}\left(\frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \cdot 0.5\right)} - 1 \]
      expm1-def [=>] 13.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \cdot 0.5\right)\right)} \]
      expm1-log1p [=>] 0.7	\[ \color{blue}{\frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)} \cdot 0.5} \]
      *-commutative [=>] 0.7	\[ \color{blue}{0.5 \cdot \frac{\pi}{\left(a + b\right) \cdot \left(a \cdot b\right)}} \]
      associate-*r* [=>] 1.2	\[ 0.5 \cdot \frac{\pi}{\color{blue}{\left(\left(a + b\right) \cdot a\right) \cdot b}} \]
      *-commutative [=>] 1.2	\[ 0.5 \cdot \frac{\pi}{\color{blue}{b \cdot \left(\left(a + b\right) \cdot a\right)}} \]
      *-commutative [=>] 1.2	\[ 0.5 \cdot \frac{\pi}{b \cdot \color{blue}{\left(a \cdot \left(a + b\right)\right)}} \]

   if 2.64999999999999986e119 < a

   1. Initial program 25.7

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

   2. Taylor expanded in b around 0 13.2

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

   3. Simplified 13.2

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

      [Start] 13.2	\[ 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]
      unpow2 [=>] 13.2	\[ 0.5 \cdot \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]

   4. Applied egg-rr 0.1

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;a \leq 2.65 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.8
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-28} \lor \neg \left(a \leq 2.6 \cdot 10^{+64}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 2
Error	8.1
Cost	7177

\[\begin{array}{l} \mathbf{if}\;a \leq -6.1 \cdot 10^{-31} \lor \neg \left(a \leq 4.9 \cdot 10^{+70}\right):\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{b}\\ \end{array} \]

Alternative 3
Error	17.1
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-28}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \end{array} \]

Alternative 4
Error	12.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \end{array} \]

Alternative 5
Error	8.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;\pi \cdot \frac{\frac{\frac{0.5}{b}}{a}}{a}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \end{array} \]

Alternative 6
Error	8.1
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+73}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{\frac{0.5}{b \cdot a}}{a}\\ \end{array} \]

Alternative 7
Error	8.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.66 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{a}\\ \end{array} \]

Alternative 8
Error	8.1
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;a \leq -3.1 \cdot 10^{-28}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot t_0}{a}\\ \end{array} \]

Alternative 9
Error	8.2
Cost	7176

\[\begin{array}{l} t_0 := \frac{\pi}{b \cdot a}\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{\pi}{2}}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot t_0}{a}\\ \end{array} \]

Alternative 10
Error	0.2
Cost	7040

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

Alternative 11
Error	0.3
Cost	7040

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

Alternative 12
Error	0.3
Cost	7040

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

Alternative 13
Error	29.7
Cost	6912

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

Reproduce

herbie shell --seed 2023187 
(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))))