NMSE Section 6.1 mentioned, B

Average Error: 14.5 → 0.4

Time: 12.5s

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b \cdot a}}{a}}{2}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b \cdot a}}{a}\\ \end{array} \]

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 -3.1e+127)
   (/ (/ (/ PI (* b a)) a) 2.0)
   (if (<= a 3.8e+86)
     (* (/ 0.5 b) (/ PI (* a (+ b a))))
     (/ (* PI (/ 0.5 (* 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 <= -3.1e+127) {
		tmp = ((((double) M_PI) / (b * a)) / a) / 2.0;
	} else if (a <= 3.8e+86) {
		tmp = (0.5 / b) * (((double) M_PI) / (a * (b + a)));
	} else {
		tmp = (((double) M_PI) * (0.5 / (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 <= -3.1e+127) {
		tmp = ((Math.PI / (b * a)) / a) / 2.0;
	} else if (a <= 3.8e+86) {
		tmp = (0.5 / b) * (Math.PI / (a * (b + a)));
	} else {
		tmp = (Math.PI * (0.5 / (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 <= -3.1e+127:
		tmp = ((math.pi / (b * a)) / a) / 2.0
	elif a <= 3.8e+86:
		tmp = (0.5 / b) * (math.pi / (a * (b + a)))
	else:
		tmp = (math.pi * (0.5 / (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 <= -3.1e+127)
		tmp = Float64(Float64(Float64(pi / Float64(b * a)) / a) / 2.0);
	elseif (a <= 3.8e+86)
		tmp = Float64(Float64(0.5 / b) * Float64(pi / Float64(a * Float64(b + a))));
	else
		tmp = Float64(Float64(pi * Float64(0.5 / 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 <= -3.1e+127)
		tmp = ((pi / (b * a)) / a) / 2.0;
	elseif (a <= 3.8e+86)
		tmp = (0.5 / b) * (pi / (a * (b + a)));
	else
		tmp = (pi * (0.5 / (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, -3.1e+127], N[(N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[a, 3.8e+86], N[(N[(0.5 / b), $MachinePrecision] * N[(Pi / N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.1000000000000002e127

   1. Initial program 27.4

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

   2. Simplified 27.4

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

      [Start] 27.4	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*l* [=>] 27.4	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]
      associate-*l/ [=>] 27.4	\[ \frac{\pi}{2} \cdot \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b \cdot b - a \cdot a}} \]
      *-lft-identity [=>] 27.4	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} - \frac{1}{b}}}{b \cdot b - a \cdot a} \]
      sub-neg [=>] 27.4	\[ \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a} + \left(-\frac{1}{b}\right)}}{b \cdot b - a \cdot a} \]
      distribute-neg-frac [=>] 27.4	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \color{blue}{\frac{-1}{b}}}{b \cdot b - a \cdot a} \]
      metadata-eval [=>] 27.4	\[ \frac{\pi}{2} \cdot \frac{\frac{1}{a} + \frac{\color{blue}{-1}}{b}}{b \cdot b - a \cdot a} \]

   3. Applied egg-rr 0.6

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

   4. Taylor expanded in a around inf 13.9

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

   5. Simplified 0.1

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

      [Start] 13.9	\[ \frac{\frac{\pi}{{a}^{2} \cdot b}}{2} \]
      unpow2 [=>] 13.9	\[ \frac{\frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b}}{2} \]
      associate-*r* [<=] 0.6	\[ \frac{\frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}}}{2} \]
      *-commutative [=>] 0.6	\[ \frac{\frac{\pi}{\color{blue}{\left(a \cdot b\right) \cdot a}}}{2} \]
      associate-/r* [=>] 0.1	\[ \frac{\color{blue}{\frac{\frac{\pi}{a \cdot b}}{a}}}{2} \]

   if -3.1000000000000002e127 < a < 3.79999999999999978e86

   1. Initial program 8.1

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

   2. Simplified 8.1

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

      [Start] 8.1	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*r/ [=>] 8.1	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-rgt-identity [=>] 8.1	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      sub-neg [=>] 8.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
      distribute-neg-frac [=>] 8.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
      metadata-eval [=>] 8.1	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.4

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

   5. Applied egg-rr 0.4

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

   6. Applied egg-rr 0.4

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

   if 3.79999999999999978e86 < a

   1. Initial program 21.6

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

   2. Simplified 21.7

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

      [Start] 21.6	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*l* [=>] 21.7	\[ \color{blue}{\frac{\pi}{2} \cdot \left(\frac{1}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)} \]

   3. Taylor expanded in b around 0 11.1

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

   4. Simplified 1.1

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

      [Start] 11.1	\[ 0.5 \cdot \frac{\pi}{{a}^{2} \cdot b} \]
      associate-*r/ [=>] 11.1	\[ \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]
      unpow2 [=>] 11.1	\[ \frac{0.5 \cdot \pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]
      associate-*l* [=>] 1.1	\[ \frac{0.5 \cdot \pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \]

   5. Applied egg-rr 1.1

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

   6. Applied egg-rr 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b \cdot a}}{a}}{2}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\pi}{a \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b \cdot a}}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	20160

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

Alternative 2
Error	16.4
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-26} \lor \neg \left(b \leq 6.6 \cdot 10^{+33}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot a\right)}\\ \end{array} \]

Alternative 3
Error	16.4
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-26} \lor \neg \left(b \leq 3.9 \cdot 10^{+34}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \end{array} \]

Alternative 4
Error	11.9
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-29} \lor \neg \left(b \leq 2 \cdot 10^{+38}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \end{array} \]

Alternative 5
Error	7.2
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-26} \lor \neg \left(b \leq 7.2 \cdot 10^{+33}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 6
Error	7.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -2.45 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 7
Error	7.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-28}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+33}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{b \cdot a}\\ \end{array} \]

Alternative 8
Error	0.3
Cost	7040

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

Alternative 9
Error	0.3
Cost	7040

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

Alternative 10
Error	29.9
Cost	6912

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

Reproduce

herbie shell --seed 2023018 
(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))))