NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.1% → 99.6%

Time: 12.2s

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

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 -2e+118)
   (/ (/ (* PI (/ 0.5 a)) b) b)
   (if (<= b 5e+93)
     (* 0.5 (/ (/ PI a) (* b (+ b a))))
     (* 0.5 (/ (/ PI b) (* 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) {
	double tmp;
	if (b <= -2e+118) {
		tmp = ((((double) M_PI) * (0.5 / a)) / b) / b;
	} else if (b <= 5e+93) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * (b + a)));
	} else {
		tmp = 0.5 * ((((double) M_PI) / b) / (b * 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 <= -2e+118) {
		tmp = ((Math.PI * (0.5 / a)) / b) / b;
	} else if (b <= 5e+93) {
		tmp = 0.5 * ((Math.PI / a) / (b * (b + a)));
	} else {
		tmp = 0.5 * ((Math.PI / b) / (b * 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 <= -2e+118:
		tmp = ((math.pi * (0.5 / a)) / b) / b
	elif b <= 5e+93:
		tmp = 0.5 * ((math.pi / a) / (b * (b + a)))
	else:
		tmp = 0.5 * ((math.pi / b) / (b * 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 <= -2e+118)
		tmp = Float64(Float64(Float64(pi * Float64(0.5 / a)) / b) / b);
	elseif (b <= 5e+93)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * Float64(b + a))));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / b) / Float64(b * 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 <= -2e+118)
		tmp = ((pi * (0.5 / a)) / b) / b;
	elseif (b <= 5e+93)
		tmp = 0.5 * ((pi / a) / (b * (b + a)));
	else
		tmp = 0.5 * ((pi / b) / (b * 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, -2e+118], N[(N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[b, 5e+93], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999993e118

   1. Initial program 59.9%

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

   2. Simplified 59.9%

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

      [Start] 59.9	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*r/ [=>] 59.9	\[ \color{blue}{\frac{\frac{\pi}{2} \cdot 1}{b \cdot b - a \cdot a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-rgt-identity [=>] 59.9	\[ \frac{\color{blue}{\frac{\pi}{2}}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      sub-neg [=>] 59.9	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]
      distribute-neg-frac [=>] 59.9	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]
      metadata-eval [=>] 59.9	\[ \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Taylor expanded in b around inf 87.1%

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

   4. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{\pi}{b \cdot b} \cdot \frac{0.5}{a}} \]
      Step-by-step derivation

      [Start] 87.1	\[ 0.5 \cdot \frac{\pi}{a \cdot {b}^{2}} \]
      associate-*r/ [=>] 87.1	\[ \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]
      *-commutative [<=] 87.1	\[ \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]
      *-commutative [=>] 87.1	\[ \frac{\pi \cdot 0.5}{\color{blue}{{b}^{2} \cdot a}} \]
      times-frac [=>] 87.1	\[ \color{blue}{\frac{\pi}{{b}^{2}} \cdot \frac{0.5}{a}} \]
      unpow2 [=>] 87.1	\[ \frac{\pi}{\color{blue}{b \cdot b}} \cdot \frac{0.5}{a} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{b}} \]
      Step-by-step derivation

      [Start] 87.1	\[ \frac{\pi}{b \cdot b} \cdot \frac{0.5}{a} \]
      associate-*l/ [=>] 87.2	\[ \color{blue}{\frac{\pi \cdot \frac{0.5}{a}}{b \cdot b}} \]
      associate-/r* [=>] 99.9	\[ \color{blue}{\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{b}} \]

   if -1.99999999999999993e118 < b < 5.0000000000000001e93

   1. Initial program 90.0%

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

   2. Simplified 90.0%

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

      [Start] 90.0	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-commutative [=>] 90.0	\[ \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-/r/ [<=] 89.9	\[ \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*l/ [=>] 90.0	\[ \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]
      *-commutative [=>] 90.0	\[ \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]
      associate-/r/ [=>] 90.0	\[ \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]
      times-frac [=>] 90.0	\[ \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Applied egg-rr 99.5%

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

      [Start] 90.0	\[ \frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5 \]
      *-un-lft-identity [=>] 90.0	\[ \frac{\color{blue}{1 \cdot \mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}{b \cdot b - a \cdot a} \cdot 0.5 \]
      difference-of-squares [=>] 92.4	\[ \frac{1 \cdot \mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot 0.5 \]
      times-frac [=>] 99.5	\[ \color{blue}{\left(\frac{1}{b + a} \cdot \frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b - a}\right)} \cdot 0.5 \]

   4. Taylor expanded in b around 0 99.6%

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

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{a}}{\left(a + b\right) \cdot b}} \cdot 0.5 \]
      Step-by-step derivation

      [Start] 99.6	\[ \left(\frac{1}{b + a} \cdot \frac{\pi}{a \cdot b}\right) \cdot 0.5 \]
      associate-/r* [=>] 99.5	\[ \left(\frac{1}{b + a} \cdot \color{blue}{\frac{\frac{\pi}{a}}{b}}\right) \cdot 0.5 \]
      frac-times [=>] 99.6	\[ \color{blue}{\frac{1 \cdot \frac{\pi}{a}}{\left(b + a\right) \cdot b}} \cdot 0.5 \]
      *-un-lft-identity [<=] 99.6	\[ \frac{\color{blue}{\frac{\pi}{a}}}{\left(b + a\right) \cdot b} \cdot 0.5 \]
      +-commutative [=>] 99.6	\[ \frac{\frac{\pi}{a}}{\color{blue}{\left(a + b\right)} \cdot b} \cdot 0.5 \]

   if 5.0000000000000001e93 < b

   1. Initial program 72.0%

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

   2. Simplified 72.0%

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

      [Start] 72.0	\[ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      *-commutative [=>] 72.0	\[ \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-/r/ [<=] 72.0	\[ \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
      associate-*l/ [=>] 72.0	\[ \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]
      *-commutative [=>] 72.0	\[ \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]
      associate-/r/ [=>] 72.0	\[ \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]
      times-frac [=>] 72.0	\[ \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Taylor expanded in b around inf 87.7%

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

   4. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{\pi}{\left(a \cdot b\right) \cdot b}} \cdot 0.5 \]
      Step-by-step derivation

      [Start] 87.7	\[ \frac{\pi}{a \cdot {b}^{2}} \cdot 0.5 \]
      unpow2 [=>] 87.7	\[ \frac{\pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \cdot 0.5 \]
      associate-*r* [=>] 98.8	\[ \frac{\pi}{\color{blue}{\left(a \cdot b\right) \cdot b}} \cdot 0.5 \]

   5. Taylor expanded in a around 0 87.7%

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

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{b}}{a \cdot b}} \cdot 0.5 \]
      Step-by-step derivation

      [Start] 87.7	\[ \frac{\pi}{a \cdot {b}^{2}} \cdot 0.5 \]
      unpow2 [=>] 87.7	\[ \frac{\pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \cdot 0.5 \]
      associate-*r* [=>] 98.8	\[ \frac{\pi}{\color{blue}{\left(a \cdot b\right) \cdot b}} \cdot 0.5 \]
      associate-/l/ [<=] 99.8	\[ \color{blue}{\frac{\frac{\pi}{b}}{a \cdot b}} \cdot 0.5 \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{\pi \cdot \frac{0.5}{a}}{b}}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.1%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -1050000:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.5}{a}}{\frac{-b}{\pi}}}{b}\\ \end{array} \]

Alternative 2
Accuracy	81.2%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -1000000 \lor \neg \left(b \leq 1.7 \cdot 10^{-20}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 3
Accuracy	87.0%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -3100000 \lor \neg \left(b \leq 1.08 \cdot 10^{-19}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 4
Accuracy	81.2%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -23000000:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \]

Alternative 5
Accuracy	87.1%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -920000:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{b \cdot a}\\ \end{array} \]

Alternative 6
Accuracy	99.6%
Cost	7168

\[\left(\frac{1}{b + a} \cdot \frac{\pi}{b \cdot a}\right) \cdot 0.5 \]

Alternative 7
Accuracy	57.0%
Cost	6912

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

Alternative 8
Accuracy	62.7%
Cost	6912

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

Alternative 9
Accuracy	62.6%
Cost	6912

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

Reproduce

herbie shell --seed 2023161 
(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))))