NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.7%

Time: 10.7s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

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));
}

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));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

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

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
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]

Initial Program: 78.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

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));
}

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));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

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

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
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]

Alternative 1: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi}{\frac{b \cdot a}{0.5}}}{b + a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (/ PI (/ (* b a) 0.5)) (+ b a)))

C

double code(double a, double b) {
	return (((double) M_PI) / ((b * a) / 0.5)) / (b + a);
}

Java

public static double code(double a, double b) {
	return (Math.PI / ((b * a) / 0.5)) / (b + a);
}

Python

def code(a, b):
	return (math.pi / ((b * a) / 0.5)) / (b + a)

Julia

function code(a, b)
	return Float64(Float64(pi / Float64(Float64(b * a) / 0.5)) / Float64(b + a))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi / ((b * a) / 0.5)) / (b + a);
end

Wolfram

code[a_, b_] := N[(N[(Pi / N[(N[(b * a), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

   1. associate-*r/ 77.4%

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

   2. *-rgt-identity 77.4%

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

   3. associate-*l/ 77.4%

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

   4. difference-of-squares 85.6%

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

   5. *-commutative 85.6%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. frac-add 99.6%

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

   2. *-un-lft-identity 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\color{blue}{\frac{b + a \cdot -1}{a \cdot b}}}{b + a} \]
7. Step-by-step derivation

   1. *-commutative 99.6%

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

   2. neg-mul-1 99.6%

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

   3. sub-neg 99.6%

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

8. Simplified 99.6%

   \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\color{blue}{\frac{b - a}{a \cdot b}}}{b + a} \]
9. Step-by-step derivation

   1. associate-*r/ 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. *-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

11. Taylor expanded in b around 0 99.7%

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

    1. associate-*r/ 99.7%

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

    2. *-commutative 99.7%

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

    3. *-commutative 99.7%

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

    4. associate-/l* 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

    \[\leadsto \frac{\frac{\pi}{\frac{b \cdot a}{0.5}}}{b + a} \]
15. Add Preprocessing

Alternative 2: 60.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-131} \lor \neg \left(a \leq -5 \cdot 10^{-311}\right):\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.05e-131) (not (<= a -5e-311)))
   (* (/ PI a) (/ 0.5 (* b a)))
   (* (/ PI (* b a)) (/ -0.5 a))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.05e-131) || !(a <= -5e-311)) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * a));
	} else {
		tmp = (((double) M_PI) / (b * a)) * (-0.5 / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.05e-131) || !(a <= -5e-311)) {
		tmp = (Math.PI / a) * (0.5 / (b * a));
	} else {
		tmp = (Math.PI / (b * a)) * (-0.5 / a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.05e-131) or not (a <= -5e-311):
		tmp = (math.pi / a) * (0.5 / (b * a))
	else:
		tmp = (math.pi / (b * a)) * (-0.5 / a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.05e-131) || !(a <= -5e-311))
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)));
	else
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(-0.5 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.05e-131) || ~((a <= -5e-311)))
		tmp = (pi / a) * (0.5 / (b * a));
	else
		tmp = (pi / (b * a)) * (-0.5 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.05e-131], N[Not[LessEqual[a, -5e-311]], $MachinePrecision]], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.04999999999999999e-131 or -5.00000000000023e-311 < a

   1. Initial program 75.5%

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

      1. associate-*r/ 75.5%

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

      2. *-rgt-identity 75.5%

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

      3. associate-*l/ 75.5%

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

      4. difference-of-squares 83.8%

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

      5. *-commutative 83.8%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 73.9%

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

   6. Taylor expanded in b around 0 66.2%

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

      1. *-commutative 66.2%

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

      2. frac-2neg 66.2%

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

      3. metadata-eval 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-*r/ 66.2%

         \[\leadsto \frac{1}{-b \cdot a} \cdot \color{blue}{\frac{-0.5 \cdot \pi}{a}} \]

      6. *-commutative 66.2%

         \[\leadsto \frac{1}{-b \cdot a} \cdot \frac{\color{blue}{\pi \cdot -0.5}}{a} \]

      7. metadata-eval 66.2%

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

      8. associate-*l* 66.2%

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

      9. frac-times 66.3%

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

      10. *-un-lft-identity 66.3%

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

      11. associate-*l* 66.3%

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

      12. metadata-eval 66.3%

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

   8. Applied egg-rr 66.3%

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

      1. *-commutative 66.3%

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

      2. times-frac 66.2%

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

      3. metadata-eval 66.2%

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

      4. frac-2neg 66.2%

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

   10. Applied egg-rr 66.2%

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

   if -1.04999999999999999e-131 < a < -5.00000000000023e-311

   1. Initial program 92.5%

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

      1. associate-*r/ 92.5%

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

      2. *-rgt-identity 92.5%

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

      3. associate-*l/ 92.8%

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

      4. difference-of-squares 99.9%

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

      5. *-commutative 99.9%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 18.6%

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

   6. Taylor expanded in b around 0 12.2%

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

      1. *-commutative 12.2%

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

      2. frac-2neg 12.2%

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

      3. metadata-eval 12.2%

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

      4. *-commutative 12.2%

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

      5. associate-*r/ 12.2%

         \[\leadsto \frac{1}{-b \cdot a} \cdot \color{blue}{\frac{-0.5 \cdot \pi}{a}} \]

      6. *-commutative 12.2%

         \[\leadsto \frac{1}{-b \cdot a} \cdot \frac{\color{blue}{\pi \cdot -0.5}}{a} \]

      7. metadata-eval 12.2%

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

      8. associate-*l* 12.2%

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

      9. frac-times 12.2%

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

      10. *-un-lft-identity 12.2%

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

      11. associate-*l* 12.2%

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

      12. metadata-eval 12.2%

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

   8. Applied egg-rr 12.2%

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

      1. times-frac 12.2%

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

      2. add-sqr-sqrt 0.5%

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

      3. sqrt-unprod 0.7%

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

      4. sqr-neg 0.7%

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

      5. sqrt-unprod 0.2%

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

      6. add-sqr-sqrt 26.8%

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

   10. Applied egg-rr 26.8%

       \[\leadsto \color{blue}{\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-131} \lor \neg \left(a \leq -5 \cdot 10^{-311}\right):\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{-\pi}{a}}{b \cdot \frac{a}{-0.5}}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.8e-132)
   (/ (/ (- PI) a) (* b (/ a -0.5)))
   (if (<= a 9e-309)
     (* (/ PI (* b a)) (/ -0.5 a))
     (* (/ PI a) (/ 0.5 (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.8e-132) {
		tmp = (-((double) M_PI) / a) / (b * (a / -0.5));
	} else if (a <= 9e-309) {
		tmp = (((double) M_PI) / (b * a)) * (-0.5 / a);
	} else {
		tmp = (((double) M_PI) / a) * (0.5 / (b * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9.8e-132) {
		tmp = (-Math.PI / a) / (b * (a / -0.5));
	} else if (a <= 9e-309) {
		tmp = (Math.PI / (b * a)) * (-0.5 / a);
	} else {
		tmp = (Math.PI / a) * (0.5 / (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9.8e-132:
		tmp = (-math.pi / a) / (b * (a / -0.5))
	elif a <= 9e-309:
		tmp = (math.pi / (b * a)) * (-0.5 / a)
	else:
		tmp = (math.pi / a) * (0.5 / (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.8e-132)
		tmp = Float64(Float64(Float64(-pi) / a) / Float64(b * Float64(a / -0.5)));
	elseif (a <= 9e-309)
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9.8e-132)
		tmp = (-pi / a) / (b * (a / -0.5));
	elseif (a <= 9e-309)
		tmp = (pi / (b * a)) * (-0.5 / a);
	else
		tmp = (pi / a) * (0.5 / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.8e-132], N[(N[((-Pi) / a), $MachinePrecision] / N[(b * N[(a / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9e-309], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.79999999999999961e-132

   1. Initial program 73.2%

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

      1. associate-*r/ 73.3%

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

      2. *-rgt-identity 73.3%

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

      3. associate-*l/ 73.3%

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

      4. difference-of-squares 84.0%

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

      5. *-commutative 84.0%

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

      6. times-frac 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 84.1%

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

   6. Taylor expanded in b around 0 75.3%

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

      1. associate-*r/ 75.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \pi}{a}} \cdot \frac{-1}{a \cdot b} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\color{blue}{\pi \cdot -0.5}}{a} \cdot \frac{-1}{a \cdot b} \]

      3. associate-/l* 75.3%

         \[\leadsto \color{blue}{\frac{\pi}{\frac{a}{-0.5}}} \cdot \frac{-1}{a \cdot b} \]

      4. associate-/r* 75.4%

         \[\leadsto \frac{\pi}{\frac{a}{-0.5}} \cdot \color{blue}{\frac{\frac{-1}{a}}{b}} \]

      5. frac-times 75.3%

         \[\leadsto \color{blue}{\frac{\pi \cdot \frac{-1}{a}}{\frac{a}{-0.5} \cdot b}} \]

   8. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\frac{\pi \cdot \frac{-1}{a}}{\frac{a}{-0.5} \cdot b}} \]
   9. Step-by-step derivation

      1. associate-*r/ 75.3%

         \[\leadsto \frac{\color{blue}{\frac{\pi \cdot -1}{a}}}{\frac{a}{-0.5} \cdot b} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \pi}}{a}}{\frac{a}{-0.5} \cdot b} \]

      3. mul-1-neg 75.3%

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

      4. *-commutative 75.3%

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

   10. Simplified 75.3%

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

   if -9.79999999999999961e-132 < a < 9.0000000000000021e-309

   1. Initial program 92.5%

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

      1. associate-*r/ 92.5%

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

      2. *-rgt-identity 92.5%

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

      3. associate-*l/ 92.8%

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

      4. difference-of-squares 99.9%

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

      5. *-commutative 99.9%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 18.6%

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

   6. Taylor expanded in b around 0 12.2%

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

      1. *-commutative 12.2%

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

      2. frac-2neg 12.2%

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

      3. metadata-eval 12.2%

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

      4. *-commutative 12.2%

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

      5. associate-*r/ 12.2%

         \[\leadsto \frac{1}{-b \cdot a} \cdot \color{blue}{\frac{-0.5 \cdot \pi}{a}} \]

      6. *-commutative 12.2%

         \[\leadsto \frac{1}{-b \cdot a} \cdot \frac{\color{blue}{\pi \cdot -0.5}}{a} \]

      7. metadata-eval 12.2%

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

      8. associate-*l* 12.2%

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

      9. frac-times 12.2%

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

      10. *-un-lft-identity 12.2%

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

      11. associate-*l* 12.2%

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

      12. metadata-eval 12.2%

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

   8. Applied egg-rr 12.2%

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

      1. times-frac 12.2%

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

      2. add-sqr-sqrt 0.5%

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

      3. sqrt-unprod 0.7%

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

      4. sqr-neg 0.7%

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

      5. sqrt-unprod 0.2%

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

      6. add-sqr-sqrt 26.8%

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

   10. Applied egg-rr 26.8%

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

   if 9.0000000000000021e-309 < a

   1. Initial program 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-rgt-identity 77.0%

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

      3. associate-*l/ 77.0%

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

      4. difference-of-squares 83.7%

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

      5. *-commutative 83.7%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 67.0%

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

   6. Taylor expanded in b around 0 60.0%

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

      1. *-commutative 60.0%

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

      2. frac-2neg 60.0%

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

      3. metadata-eval 60.0%

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

      4. *-commutative 60.0%

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

      5. associate-*r/ 60.0%

         \[\leadsto \frac{1}{-b \cdot a} \cdot \color{blue}{\frac{-0.5 \cdot \pi}{a}} \]

      6. *-commutative 60.0%

         \[\leadsto \frac{1}{-b \cdot a} \cdot \frac{\color{blue}{\pi \cdot -0.5}}{a} \]

      7. metadata-eval 60.0%

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

      8. associate-*l* 60.0%

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

      9. frac-times 60.0%

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

      10. *-un-lft-identity 60.0%

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

      11. associate-*l* 60.0%

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

      12. metadata-eval 60.0%

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

   8. Applied egg-rr 60.0%

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

      1. *-commutative 60.0%

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

      2. times-frac 60.0%

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

      3. metadata-eval 60.0%

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

      4. frac-2neg 60.0%

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

   10. Applied egg-rr 60.0%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{-\pi}{a}}{b \cdot \frac{a}{-0.5}}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{-\pi}{a}}{b \cdot \frac{a}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{b}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.1e-64)
   (/ (/ (- PI) a) (* b (/ a -0.5)))
   (* (* 0.5 (/ PI b)) (/ (/ 1.0 b) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.1e-64) {
		tmp = (-((double) M_PI) / a) / (b * (a / -0.5));
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) * ((1.0 / b) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.1e-64) {
		tmp = (-Math.PI / a) / (b * (a / -0.5));
	} else {
		tmp = (0.5 * (Math.PI / b)) * ((1.0 / b) / a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.1e-64:
		tmp = (-math.pi / a) / (b * (a / -0.5))
	else:
		tmp = (0.5 * (math.pi / b)) * ((1.0 / b) / a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.1e-64)
		tmp = Float64(Float64(Float64(-pi) / a) / Float64(b * Float64(a / -0.5)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) * Float64(Float64(1.0 / b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.1e-64)
		tmp = (-pi / a) / (b * (a / -0.5));
	else
		tmp = (0.5 * (pi / b)) * ((1.0 / b) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.1e-64], N[(N[((-Pi) / a), $MachinePrecision] / N[(b * N[(a / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1e-64

   1. Initial program 70.8%

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

      1. associate-*r/ 70.9%

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

      2. *-rgt-identity 70.9%

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

      3. associate-*l/ 70.8%

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

      4. difference-of-squares 83.0%

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

      5. *-commutative 83.0%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 90.3%

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

   6. Taylor expanded in b around 0 81.3%

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

      1. associate-*r/ 81.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \pi}{a}} \cdot \frac{-1}{a \cdot b} \]

      2. *-commutative 81.3%

         \[\leadsto \frac{\color{blue}{\pi \cdot -0.5}}{a} \cdot \frac{-1}{a \cdot b} \]

      3. associate-/l* 81.3%

         \[\leadsto \color{blue}{\frac{\pi}{\frac{a}{-0.5}}} \cdot \frac{-1}{a \cdot b} \]

      4. associate-/r* 81.4%

         \[\leadsto \frac{\pi}{\frac{a}{-0.5}} \cdot \color{blue}{\frac{\frac{-1}{a}}{b}} \]

      5. frac-times 81.4%

         \[\leadsto \color{blue}{\frac{\pi \cdot \frac{-1}{a}}{\frac{a}{-0.5} \cdot b}} \]

   8. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\frac{\pi \cdot \frac{-1}{a}}{\frac{a}{-0.5} \cdot b}} \]
   9. Step-by-step derivation

      1. associate-*r/ 81.4%

         \[\leadsto \frac{\color{blue}{\frac{\pi \cdot -1}{a}}}{\frac{a}{-0.5} \cdot b} \]

      2. *-commutative 81.4%

         \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \pi}}{a}}{\frac{a}{-0.5} \cdot b} \]

      3. mul-1-neg 81.4%

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

      4. *-commutative 81.4%

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

   10. Simplified 81.4%

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

   if -1.1e-64 < a

   1. Initial program 80.4%

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

      1. associate-*r/ 80.4%

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

      2. *-rgt-identity 80.4%

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

      3. associate-*l/ 80.4%

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

      4. difference-of-squares 86.8%

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

      5. *-commutative 86.8%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-add 99.6%

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

      2. *-un-lft-identity 99.6%

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

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\color{blue}{\frac{b + a \cdot -1}{a \cdot b}}}{b + a} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. neg-mul-1 99.6%

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

      3. sub-neg 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in b around inf 69.3%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
   10. Step-by-step derivation

       1. *-commutative 69.3%

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

       2. associate-/r* 69.3%

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

   11. Simplified 69.3%

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

   12. Taylor expanded in b around inf 67.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\pi}{b}\right)} \cdot \frac{\frac{1}{b}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{-\pi}{a}}{b \cdot \frac{a}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{b}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{-0.5}{b - a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{b}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e-66)
   (* (/ -0.5 (- b a)) (/ PI (* b a)))
   (* (* 0.5 (/ PI b)) (/ (/ 1.0 b) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-66) {
		tmp = (-0.5 / (b - a)) * (((double) M_PI) / (b * a));
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) * ((1.0 / b) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-66) {
		tmp = (-0.5 / (b - a)) * (Math.PI / (b * a));
	} else {
		tmp = (0.5 * (Math.PI / b)) * ((1.0 / b) / a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.7e-66:
		tmp = (-0.5 / (b - a)) * (math.pi / (b * a))
	else:
		tmp = (0.5 * (math.pi / b)) * ((1.0 / b) / a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e-66)
		tmp = Float64(Float64(-0.5 / Float64(b - a)) * Float64(pi / Float64(b * a)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) * Float64(Float64(1.0 / b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.7e-66)
		tmp = (-0.5 / (b - a)) * (pi / (b * a));
	else
		tmp = (0.5 * (pi / b)) * ((1.0 / b) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e-66], N[(N[(-0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.69999999999999996e-66

   1. Initial program 70.8%

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

      1. associate-*r/ 70.9%

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

      2. *-rgt-identity 70.9%

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

      3. associate-*l/ 70.8%

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

      4. difference-of-squares 83.0%

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

      5. *-commutative 83.0%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 90.3%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \color{blue}{\frac{-1}{a \cdot b}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 82.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)\right)} \]

      2. expm1-udef 56.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{2}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1} \]

      3. div-inv 56.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\pi \cdot \frac{1}{2}}}{b - a} \cdot \frac{-1}{a \cdot b}\right)} - 1 \]

      4. metadata-eval 56.1%

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

      5. associate-/r* 56.1%

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

   7. Applied egg-rr 56.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi \cdot 0.5}{b - a} \cdot \frac{\frac{-1}{a}}{b}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 82.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi \cdot 0.5}{b - a} \cdot \frac{\frac{-1}{a}}{b}\right)\right)} \]

      2. expm1-log1p 90.4%

         \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b - a} \cdot \frac{\frac{-1}{a}}{b}} \]

      3. associate-/l/ 90.3%

         \[\leadsto \frac{\pi \cdot 0.5}{b - a} \cdot \color{blue}{\frac{-1}{b \cdot a}} \]

      4. times-frac 90.3%

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

      5. associate-*l* 90.3%

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

      6. metadata-eval 90.3%

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

      7. *-commutative 90.3%

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

      8. times-frac 90.4%

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

   9. Simplified 90.4%

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

   if -2.69999999999999996e-66 < a

   1. Initial program 80.4%

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

      1. associate-*r/ 80.4%

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

      2. *-rgt-identity 80.4%

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

      3. associate-*l/ 80.4%

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

      4. difference-of-squares 86.8%

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

      5. *-commutative 86.8%

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

      6. times-frac 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-add 99.6%

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

      2. *-un-lft-identity 99.6%

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

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\color{blue}{\frac{b + a \cdot -1}{a \cdot b}}}{b + a} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. neg-mul-1 99.6%

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

      3. sub-neg 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in b around inf 69.3%

      \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \color{blue}{\frac{1}{a \cdot b}} \]
   10. Step-by-step derivation

       1. *-commutative 69.3%

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

       2. associate-/r* 69.3%

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

   11. Simplified 69.3%

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

   12. Taylor expanded in b around inf 67.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\pi}{b}\right)} \cdot \frac{\frac{1}{b}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{-0.5}{b - a} \cdot \frac{\pi}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{b}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{\left(a \cdot \frac{b}{0.5}\right) \cdot \left(b + a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ PI (* (* a (/ b 0.5)) (+ b a))))

C

double code(double a, double b) {
	return ((double) M_PI) / ((a * (b / 0.5)) * (b + a));
}

Java

public static double code(double a, double b) {
	return Math.PI / ((a * (b / 0.5)) * (b + a));
}

Python

def code(a, b):
	return math.pi / ((a * (b / 0.5)) * (b + a))

Julia

function code(a, b)
	return Float64(pi / Float64(Float64(a * Float64(b / 0.5)) * Float64(b + a)))
end

MATLAB

function tmp = code(a, b)
	tmp = pi / ((a * (b / 0.5)) * (b + a));
end

Wolfram

code[a_, b_] := N[(Pi / N[(N[(a * N[(b / 0.5), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

   1. associate-*r/ 77.4%

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

   2. *-rgt-identity 77.4%

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

   3. associate-*l/ 77.4%

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

   4. difference-of-squares 85.6%

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

   5. *-commutative 85.6%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. frac-add 99.6%

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

   2. *-un-lft-identity 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\color{blue}{\frac{b + a \cdot -1}{a \cdot b}}}{b + a} \]
7. Step-by-step derivation

   1. *-commutative 99.6%

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

   2. neg-mul-1 99.6%

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

   3. sub-neg 99.6%

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

8. Simplified 99.6%

   \[\leadsto \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\color{blue}{\frac{b - a}{a \cdot b}}}{b + a} \]
9. Step-by-step derivation

   1. associate-*r/ 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. *-commutative 99.6%

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

10. Applied egg-rr 99.6%

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

11. Taylor expanded in b around 0 99.7%

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

    1. associate-*r/ 99.7%

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

    2. *-commutative 99.7%

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

    3. *-commutative 99.7%

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

    4. associate-/l* 99.7%

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

13. Simplified 99.7%

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

    1. expm1-log1p-u 79.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{\frac{b \cdot a}{0.5}}}{b + a}\right)\right)} \]

    2. expm1-udef 51.5%

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

    3. associate-/l/ 51.5%

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

    4. associate-/l* 51.5%

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

15. Applied egg-rr 51.5%

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

    1. expm1-def 78.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{\left(b + a\right) \cdot \frac{b}{\frac{0.5}{a}}}\right)\right)} \]

    2. expm1-log1p 99.2%

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

    3. *-commutative 99.2%

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

    4. associate-/r/ 99.3%

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

    5. +-commutative 99.3%

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

17. Simplified 99.3%

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

18. Final simplification 99.3%

    \[\leadsto \frac{\pi}{\left(a \cdot \frac{b}{0.5}\right) \cdot \left(b + a\right)} \]
19. Add Preprocessing

Alternative 7: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{a} \cdot \frac{0.5}{b \cdot a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ PI a) (/ 0.5 (* b a))))

C

double code(double a, double b) {
	return (((double) M_PI) / a) * (0.5 / (b * a));
}

Java

public static double code(double a, double b) {
	return (Math.PI / a) * (0.5 / (b * a));
}

Python

def code(a, b):
	return (math.pi / a) * (0.5 / (b * a))

Julia

function code(a, b)
	return Float64(Float64(pi / a) * Float64(0.5 / Float64(b * a)))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi / a) * (0.5 / (b * a));
end

Wolfram

code[a_, b_] := N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

   1. associate-*r/ 77.4%

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

   2. *-rgt-identity 77.4%

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

   3. associate-*l/ 77.4%

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

   4. difference-of-squares 85.6%

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

   5. *-commutative 85.6%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a}} \]
4. Add Preprocessing

5. Taylor expanded in a around inf 67.9%

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

6. Taylor expanded in b around 0 60.3%

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

   1. *-commutative 60.3%

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

   2. frac-2neg 60.3%

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

   3. metadata-eval 60.3%

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

   4. *-commutative 60.3%

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

   5. associate-*r/ 60.3%

      \[\leadsto \frac{1}{-b \cdot a} \cdot \color{blue}{\frac{-0.5 \cdot \pi}{a}} \]

   6. *-commutative 60.3%

      \[\leadsto \frac{1}{-b \cdot a} \cdot \frac{\color{blue}{\pi \cdot -0.5}}{a} \]

   7. metadata-eval 60.3%

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

   8. associate-*l* 60.3%

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

   9. frac-times 60.4%

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

   10. *-un-lft-identity 60.4%

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

   11. associate-*l* 60.4%

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

   12. metadata-eval 60.4%

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

8. Applied egg-rr 60.4%

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

   1. *-commutative 60.4%

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

   2. times-frac 60.3%

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

   3. metadata-eval 60.3%

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

   4. frac-2neg 60.3%

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

10. Applied egg-rr 60.3%

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

11. Final simplification 60.3%

    \[\leadsto \frac{\pi}{a} \cdot \frac{0.5}{b \cdot a} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))