NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.7%

Time: 8.3s

Alternatives: 8

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: 78.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

   1. div-inv 82.3%

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

   2. clear-num 82.3%

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

   3. frac-sub 82.3%

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

   4. frac-times 77.1%

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

   5. *-un-lft-identity 77.1%

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

   6. *-un-lft-identity 77.1%

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

   7. div-inv 77.1%

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

   8. metadata-eval 77.1%

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

3. Applied egg-rr 77.1%

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

   1. *-rgt-identity 77.1%

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

   2. *-commutative 77.1%

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

   3. associate-*r/ 77.0%

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

   4. associate-/l* 76.7%

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

   5. *-commutative 76.7%

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

   6. associate-*l* 76.7%

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

   7. *-commutative 76.7%

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

5. Simplified 76.7%

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

   1. *-un-lft-identity 76.7%

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

   2. times-frac 82.3%

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

   3. *-commutative 82.3%

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

7. Applied egg-rr 82.3%

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

   1. *-lft-identity 82.3%

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

   2. associate-*l/ 82.3%

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

   3. difference-of-squares 89.7%

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

   4. +-commutative 89.7%

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

   5. times-frac 99.6%

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

   6. *-commutative 99.6%

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

   7. times-frac 89.3%

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

9. Simplified 89.3%

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

    1. frac-times 82.5%

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

    2. *-commutative 82.5%

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

11. Applied egg-rr 82.5%

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

    1. *-commutative 82.5%

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

    2. times-frac 89.3%

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

    3. associate-*l/ 88.5%

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

    4. associate-/r/ 89.3%

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

    5. associate-*l/ 89.3%

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

    6. associate-*r/ 89.3%

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

    7. *-commutative 89.3%

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

    8. associate-*l/ 89.4%

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

    9. associate-/r/ 99.7%

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

    10. *-inverses 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 66.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-119}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e-119)
   (* (/ 0.5 a) (/ PI (* a b)))
   (* (/ 0.5 a) (/ PI (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.3e-119) {
		tmp = (0.5 / a) * (((double) M_PI) / (a * b));
	} else {
		tmp = (0.5 / a) * (((double) M_PI) / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.3e-119) {
		tmp = (0.5 / a) * (Math.PI / (a * b));
	} else {
		tmp = (0.5 / a) * (Math.PI / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.3e-119:
		tmp = (0.5 / a) * (math.pi / (a * b))
	else:
		tmp = (0.5 / a) * (math.pi / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.3e-119)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.3e-119)
		tmp = (0.5 / a) * (pi / (a * b));
	else
		tmp = (0.5 / a) * (pi / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.3e-119], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.30000000000000008e-119

   1. Initial program 84.0%

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

   2. Taylor expanded in b around 0 74.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 74.7%

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

      2. unpow2 74.7%

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

   4. Simplified 74.7%

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

      1. associate-*l* 80.7%

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

      2. times-frac 80.8%

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

   6. Applied egg-rr 80.8%

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

   if -3.30000000000000008e-119 < a

   1. Initial program 81.3%

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

   2. Taylor expanded in b around inf 61.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 61.2%

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

      2. unpow2 61.2%

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

   4. Simplified 61.2%

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

      1. times-frac 60.8%

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

   6. Applied egg-rr 60.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-119}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \]

Alternative 3: 72.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.5e-116)
   (* (/ 0.5 a) (/ PI (* a b)))
   (* (/ 0.5 (* a b)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.5e-116) {
		tmp = (0.5 / a) * (((double) M_PI) / (a * b));
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.5e-116) {
		tmp = (0.5 / a) * (Math.PI / (a * b));
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.5e-116:
		tmp = (0.5 / a) * (math.pi / (a * b))
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.5e-116)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.5e-116)
		tmp = (0.5 / a) * (pi / (a * b));
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.5e-116], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5000000000000001e-116

   1. Initial program 84.0%

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

   2. Taylor expanded in b around 0 74.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 74.7%

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

      2. unpow2 74.7%

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

   4. Simplified 74.7%

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

      1. associate-*l* 80.7%

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

      2. times-frac 80.8%

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

   6. Applied egg-rr 80.8%

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

   if -2.5000000000000001e-116 < a

   1. Initial program 81.3%

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

      1. div-inv 81.3%

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

      2. clear-num 81.3%

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

      3. frac-sub 81.3%

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

      4. frac-times 75.9%

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

      5. *-un-lft-identity 75.9%

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

      6. *-un-lft-identity 75.9%

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

      7. div-inv 75.9%

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

      8. metadata-eval 75.9%

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

   3. Applied egg-rr 75.9%

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

      1. *-rgt-identity 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r/ 75.9%

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

      4. associate-/l* 76.0%

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

      5. *-commutative 76.0%

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

      6. associate-*l* 76.0%

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

      7. *-commutative 76.0%

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

   5. Simplified 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 81.3%

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

      3. *-commutative 81.3%

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

   7. Applied egg-rr 81.3%

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

      1. *-lft-identity 81.3%

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

      2. associate-*l/ 81.3%

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

      3. difference-of-squares 87.5%

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

      4. +-commutative 87.5%

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

      5. times-frac 99.5%

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

      6. *-commutative 99.5%

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

      7. times-frac 92.4%

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

   9. Simplified 92.4%

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

   10. Taylor expanded in a around 0 61.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 61.2%

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

       2. unpow2 61.2%

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

       3. associate-*r* 70.4%

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

       4. times-frac 70.1%

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

   12. Simplified 70.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 4: 72.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.55 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{b}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.55e-116)
   (* (/ 0.5 a) (/ PI (* a b)))
   (/ 0.5 (* (* a b) (/ b PI)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.55e-116) {
		tmp = (0.5 / a) * (((double) M_PI) / (a * b));
	} else {
		tmp = 0.5 / ((a * b) * (b / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.55e-116) {
		tmp = (0.5 / a) * (Math.PI / (a * b));
	} else {
		tmp = 0.5 / ((a * b) * (b / Math.PI));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.55e-116:
		tmp = (0.5 / a) * (math.pi / (a * b))
	else:
		tmp = 0.5 / ((a * b) * (b / math.pi))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.55e-116)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(a * b)));
	else
		tmp = Float64(0.5 / Float64(Float64(a * b) * Float64(b / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.55e-116)
		tmp = (0.5 / a) * (pi / (a * b));
	else
		tmp = 0.5 / ((a * b) * (b / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.55e-116], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a * b), $MachinePrecision] * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5500000000000001e-116

   1. Initial program 84.0%

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

   2. Taylor expanded in b around 0 74.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 74.7%

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

      2. unpow2 74.7%

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

   4. Simplified 74.7%

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

      1. associate-*l* 80.7%

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

      2. times-frac 80.8%

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

   6. Applied egg-rr 80.8%

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

   if -2.5500000000000001e-116 < a

   1. Initial program 81.3%

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

      1. div-inv 81.3%

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

      2. clear-num 81.3%

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

      3. frac-sub 81.3%

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

      4. frac-times 75.9%

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

      5. *-un-lft-identity 75.9%

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

      6. *-un-lft-identity 75.9%

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

      7. div-inv 75.9%

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

      8. metadata-eval 75.9%

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

   3. Applied egg-rr 75.9%

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

      1. *-rgt-identity 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r/ 75.9%

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

      4. associate-/l* 76.0%

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

      5. *-commutative 76.0%

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

      6. associate-*l* 76.0%

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

      7. *-commutative 76.0%

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

   5. Simplified 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 81.3%

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

      3. *-commutative 81.3%

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

   7. Applied egg-rr 81.3%

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

      1. *-lft-identity 81.3%

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

      2. associate-*l/ 81.3%

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

      3. difference-of-squares 87.5%

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

      4. +-commutative 87.5%

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

      5. times-frac 99.5%

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

      6. *-commutative 99.5%

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

      7. times-frac 92.4%

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

   9. Simplified 92.4%

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

   10. Taylor expanded in a around 0 61.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 61.2%

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

       2. unpow2 61.2%

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

       3. associate-*r* 70.4%

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

       4. times-frac 70.1%

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

   12. Simplified 70.1%

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

       1. *-commutative 70.1%

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

       2. clear-num 70.1%

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

       3. frac-times 70.4%

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

       4. metadata-eval 70.4%

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

   14. Applied egg-rr 70.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.55 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot \frac{b}{\pi}}\\ \end{array} \]

Alternative 5: 72.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.2e-116)
   (* (/ 0.5 a) (/ PI (* a b)))
   (/ (* 0.5 PI) (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.2e-116) {
		tmp = (0.5 / a) * (((double) M_PI) / (a * b));
	} else {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.2e-116) {
		tmp = (0.5 / a) * (Math.PI / (a * b));
	} else {
		tmp = (0.5 * Math.PI) / (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.2e-116:
		tmp = (0.5 / a) * (math.pi / (a * b))
	else:
		tmp = (0.5 * math.pi) / (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.2e-116)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.2e-116)
		tmp = (0.5 / a) * (pi / (a * b));
	else
		tmp = (0.5 * pi) / (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.2e-116], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2000000000000001e-116

   1. Initial program 84.0%

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

   2. Taylor expanded in b around 0 74.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 74.7%

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

      2. unpow2 74.7%

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

   4. Simplified 74.7%

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

      1. associate-*l* 80.7%

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

      2. times-frac 80.8%

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

   6. Applied egg-rr 80.8%

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

   if -2.2000000000000001e-116 < a

   1. Initial program 81.3%

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

      1. div-inv 81.3%

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

      2. clear-num 81.3%

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

      3. frac-sub 81.3%

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

      4. frac-times 75.9%

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

      5. *-un-lft-identity 75.9%

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

      6. *-un-lft-identity 75.9%

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

      7. div-inv 75.9%

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

      8. metadata-eval 75.9%

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

   3. Applied egg-rr 75.9%

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

      1. *-rgt-identity 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*r/ 75.9%

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

      4. associate-/l* 76.0%

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

      5. *-commutative 76.0%

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

      6. associate-*l* 76.0%

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

      7. *-commutative 76.0%

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

   5. Simplified 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. times-frac 81.3%

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

      3. *-commutative 81.3%

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

   7. Applied egg-rr 81.3%

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

      1. *-lft-identity 81.3%

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

      2. associate-*l/ 81.3%

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

      3. difference-of-squares 87.5%

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

      4. +-commutative 87.5%

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

      5. times-frac 99.5%

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

      6. *-commutative 99.5%

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

      7. times-frac 92.4%

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

   9. Simplified 92.4%

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

   10. Taylor expanded in a around 0 61.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 61.2%

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

       2. unpow2 61.2%

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

       3. associate-*r* 70.4%

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

       4. times-frac 70.1%

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

   12. Simplified 70.1%

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

       1. frac-times 70.4%

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

   14. Applied egg-rr 70.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

   1. div-inv 82.3%

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

   2. clear-num 82.3%

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

   3. frac-sub 82.3%

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

   4. frac-times 77.1%

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

   5. *-un-lft-identity 77.1%

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

   6. *-un-lft-identity 77.1%

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

   7. div-inv 77.1%

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

   8. metadata-eval 77.1%

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

3. Applied egg-rr 77.1%

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

   1. *-rgt-identity 77.1%

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

   2. *-commutative 77.1%

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

   3. associate-*r/ 77.0%

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

   4. associate-/l* 76.7%

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

   5. *-commutative 76.7%

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

   6. associate-*l* 76.7%

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

   7. *-commutative 76.7%

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

5. Simplified 76.7%

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

   1. *-un-lft-identity 76.7%

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

   2. times-frac 82.3%

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

   3. *-commutative 82.3%

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

7. Applied egg-rr 82.3%

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

   1. *-lft-identity 82.3%

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

   2. associate-*l/ 82.3%

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

   3. difference-of-squares 89.7%

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

   4. +-commutative 89.7%

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

   5. times-frac 99.6%

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

   6. *-commutative 99.6%

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

   7. times-frac 89.3%

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

9. Simplified 89.3%

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

10. Taylor expanded in a around 0 99.6%

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

11. Final simplification 99.6%

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

Alternative 7: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

   1. div-inv 82.3%

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

   2. clear-num 82.3%

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

   3. frac-sub 82.3%

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

   4. frac-times 77.1%

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

   5. *-un-lft-identity 77.1%

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

   6. *-un-lft-identity 77.1%

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

   7. div-inv 77.1%

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

   8. metadata-eval 77.1%

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

3. Applied egg-rr 77.1%

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

   1. *-rgt-identity 77.1%

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

   2. *-commutative 77.1%

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

   3. associate-*r/ 77.0%

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

   4. associate-/l* 76.7%

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

   5. *-commutative 76.7%

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

   6. associate-*l* 76.7%

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

   7. *-commutative 76.7%

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

5. Simplified 76.7%

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

   1. *-un-lft-identity 76.7%

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

   2. times-frac 82.3%

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

   3. *-commutative 82.3%

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

7. Applied egg-rr 82.3%

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

   1. *-lft-identity 82.3%

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

   2. associate-*l/ 82.3%

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

   3. difference-of-squares 89.7%

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

   4. +-commutative 89.7%

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

   5. times-frac 99.6%

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

   6. *-commutative 99.6%

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

   7. times-frac 89.3%

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

9. Simplified 89.3%

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

    1. frac-times 82.5%

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

    2. *-commutative 82.5%

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

11. Applied egg-rr 82.5%

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

    1. *-commutative 82.5%

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

    2. times-frac 89.3%

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

    3. associate-*l/ 88.5%

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

    4. associate-/r/ 89.3%

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

    5. associate-*l/ 89.3%

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

    6. associate-*r/ 89.3%

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

    7. *-commutative 89.3%

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

    8. associate-*l/ 89.4%

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

    9. associate-/r/ 99.7%

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

    10. *-inverses 99.7%

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

13. Simplified 99.7%

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

    1. clear-num 99.7%

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

    2. inv-pow 99.7%

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

    3. *-un-lft-identity 99.7%

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

    4. times-frac 99.7%

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

    5. metadata-eval 99.7%

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

15. Applied egg-rr 99.7%

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

    1. unpow-1 99.7%

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

    2. associate-/r* 99.7%

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

    3. metadata-eval 99.7%

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

17. Simplified 99.7%

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

18. Final simplification 99.7%

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

Alternative 8: 62.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

2. Taylor expanded in b around 0 58.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
3. Step-by-step derivation

   1. associate-*r/ 58.8%

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

   2. unpow2 58.8%

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

4. Simplified 58.8%

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

   1. associate-*l* 62.8%

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

   2. times-frac 62.8%

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

6. Applied egg-rr 62.8%

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

7. Final simplification 62.8%

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

Reproduce

herbie shell --seed 2023272 
(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))))