NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.1%

Time: 7.0s

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: 78.7% 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: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. associate-*r/ 79.6%

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

   2. *-rgt-identity 79.6%

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

   3. sub-neg 79.6%

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

   4. distribute-neg-frac 79.6%

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

   5. metadata-eval 79.6%

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

3. Simplified 79.6%

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

   1. frac-add 79.6%

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

   2. *-un-lft-identity 79.6%

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

5. Applied egg-rr 79.6%

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

   1. *-commutative 79.6%

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

   2. neg-mul-1 79.6%

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

   3. sub-neg 79.6%

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

7. Simplified 79.6%

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

   1. frac-times 75.1%

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

   2. div-inv 75.1%

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

   3. metadata-eval 75.1%

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

   4. *-commutative 75.1%

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

9. Applied egg-rr 75.1%

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

    1. times-frac 79.6%

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

    2. associate-*l/ 79.6%

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

    3. associate-*l* 79.6%

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

    4. difference-of-squares 89.7%

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

    5. times-frac 99.6%

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

    6. +-commutative 99.6%

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

    7. associate-*r/ 99.6%

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

    8. *-commutative 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in b around 0 99.6%

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

    1. associate-/l/ 99.7%

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

14. Simplified 99.7%

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

    1. associate-/l/ 99.6%

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

    2. frac-times 99.7%

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

16. Applied egg-rr 99.7%

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

17. Final simplification 99.7%

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

Alternative 2: 80.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-53} \lor \neg \left(b \leq 4.5 \cdot 10^{-40}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -4.2e-53) (not (<= b 4.5e-40)))
   (* 0.5 (/ (/ PI a) (* b b)))
   (* (/ (/ 0.5 b) a) (/ PI a))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -4.2e-53) || !(b <= 4.5e-40)) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * b));
	} else {
		tmp = ((0.5 / b) / a) * (((double) M_PI) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -4.2e-53) || !(b <= 4.5e-40)) {
		tmp = 0.5 * ((Math.PI / a) / (b * b));
	} else {
		tmp = ((0.5 / b) / a) * (Math.PI / a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -4.2e-53) or not (b <= 4.5e-40):
		tmp = 0.5 * ((math.pi / a) / (b * b))
	else:
		tmp = ((0.5 / b) / a) * (math.pi / a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -4.2e-53) || !(b <= 4.5e-40))
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * b)));
	else
		tmp = Float64(Float64(Float64(0.5 / b) / a) * Float64(pi / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -4.2e-53) || ~((b <= 4.5e-40)))
		tmp = 0.5 * ((pi / a) / (b * b));
	else
		tmp = ((0.5 / b) / a) * (pi / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -4.2e-53], N[Not[LessEqual[b, 4.5e-40]], $MachinePrecision]], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.19999999999999955e-53 or 4.5000000000000001e-40 < b

   1. Initial program 77.9%

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

      1. associate-*r/ 77.9%

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

      2. *-rgt-identity 77.9%

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

      3. sub-neg 77.9%

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

      4. distribute-neg-frac 77.9%

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

      5. metadata-eval 77.9%

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

   3. Simplified 77.9%

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

      1. frac-add 77.8%

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

      2. *-un-lft-identity 77.8%

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

   5. Applied egg-rr 77.8%

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

      1. *-commutative 77.8%

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

      2. neg-mul-1 77.8%

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

      3. sub-neg 77.8%

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

   7. Simplified 77.8%

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

      1. frac-times 72.4%

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

      2. div-inv 72.4%

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

      3. metadata-eval 72.4%

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

      4. *-commutative 72.4%

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

   9. Applied egg-rr 72.4%

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

       1. times-frac 77.8%

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

       2. associate-*l/ 77.8%

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

       3. associate-*l* 77.8%

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

       4. difference-of-squares 89.0%

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

       5. times-frac 99.6%

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

       6. +-commutative 99.6%

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

       7. associate-*r/ 99.6%

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

       8. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in a around 0 80.4%

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

       1. associate-/r* 80.1%

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

       2. unpow2 80.1%

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

   14. Simplified 80.1%

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

   if -4.19999999999999955e-53 < b < 4.5000000000000001e-40

   1. Initial program 82.2%

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

      1. associate-*r/ 82.2%

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

      2. *-rgt-identity 82.2%

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

      3. sub-neg 82.2%

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

      4. distribute-neg-frac 82.2%

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

      5. metadata-eval 82.2%

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

   3. Simplified 82.2%

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

      1. frac-add 82.2%

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

      2. *-un-lft-identity 82.2%

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

   5. Applied egg-rr 82.2%

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

      1. *-commutative 82.2%

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

      2. neg-mul-1 82.2%

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

      3. sub-neg 82.2%

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

   7. Simplified 82.2%

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

      1. frac-times 78.9%

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

      2. div-inv 78.9%

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

      3. metadata-eval 78.9%

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

      4. *-commutative 78.9%

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

   9. Applied egg-rr 78.9%

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

       1. times-frac 82.2%

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

       2. associate-*l/ 82.1%

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

       3. associate-*l* 82.1%

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

       4. difference-of-squares 90.7%

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

       5. times-frac 99.6%

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

       6. +-commutative 99.6%

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

       7. associate-*r/ 99.6%

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

       8. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in b around 0 99.7%

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

       1. associate-/l/ 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in a around inf 89.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-53} \lor \neg \left(b \leq 4.5 \cdot 10^{-40}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \end{array} \]

Alternative 3: 86.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.5}{b}}{a}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-52} \lor \neg \left(b \leq 2.4 \cdot 10^{-40}\right):\\ \;\;\;\;t_0 \cdot \frac{\pi}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\pi}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ 0.5 b) a)))
   (if (or (<= b -2e-52) (not (<= b 2.4e-40)))
     (* t_0 (/ PI b))
     (* t_0 (/ PI a)))))

C

double code(double a, double b) {
	double t_0 = (0.5 / b) / a;
	double tmp;
	if ((b <= -2e-52) || !(b <= 2.4e-40)) {
		tmp = t_0 * (((double) M_PI) / b);
	} else {
		tmp = t_0 * (((double) M_PI) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (0.5 / b) / a;
	double tmp;
	if ((b <= -2e-52) || !(b <= 2.4e-40)) {
		tmp = t_0 * (Math.PI / b);
	} else {
		tmp = t_0 * (Math.PI / a);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (0.5 / b) / a
	tmp = 0
	if (b <= -2e-52) or not (b <= 2.4e-40):
		tmp = t_0 * (math.pi / b)
	else:
		tmp = t_0 * (math.pi / a)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(0.5 / b) / a)
	tmp = 0.0
	if ((b <= -2e-52) || !(b <= 2.4e-40))
		tmp = Float64(t_0 * Float64(pi / b));
	else
		tmp = Float64(t_0 * Float64(pi / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (0.5 / b) / a;
	tmp = 0.0;
	if ((b <= -2e-52) || ~((b <= 2.4e-40)))
		tmp = t_0 * (pi / b);
	else
		tmp = t_0 * (pi / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]}, If[Or[LessEqual[b, -2e-52], N[Not[LessEqual[b, 2.4e-40]], $MachinePrecision]], N[(t$95$0 * N[(Pi / b), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(Pi / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -2e-52 or 2.39999999999999991e-40 < b

   1. Initial program 77.9%

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

      1. associate-*r/ 77.9%

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

      2. *-rgt-identity 77.9%

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

      3. sub-neg 77.9%

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

      4. distribute-neg-frac 77.9%

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

      5. metadata-eval 77.9%

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

   3. Simplified 77.9%

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

      1. frac-add 77.8%

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

      2. *-un-lft-identity 77.8%

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

   5. Applied egg-rr 77.8%

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

      1. *-commutative 77.8%

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

      2. neg-mul-1 77.8%

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

      3. sub-neg 77.8%

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

   7. Simplified 77.8%

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

      1. frac-times 72.4%

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

      2. div-inv 72.4%

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

      3. metadata-eval 72.4%

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

      4. *-commutative 72.4%

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

   9. Applied egg-rr 72.4%

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

       1. times-frac 77.8%

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

       2. associate-*l/ 77.8%

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

       3. associate-*l* 77.8%

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

       4. difference-of-squares 89.0%

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

       5. times-frac 99.6%

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

       6. +-commutative 99.6%

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

       7. associate-*r/ 99.6%

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

       8. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in b around 0 99.6%

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

       1. associate-/l/ 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in a around 0 90.7%

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

   if -2e-52 < b < 2.39999999999999991e-40

   1. Initial program 82.2%

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

      1. associate-*r/ 82.2%

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

      2. *-rgt-identity 82.2%

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

      3. sub-neg 82.2%

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

      4. distribute-neg-frac 82.2%

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

      5. metadata-eval 82.2%

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

   3. Simplified 82.2%

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

      1. frac-add 82.2%

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

      2. *-un-lft-identity 82.2%

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

   5. Applied egg-rr 82.2%

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

      1. *-commutative 82.2%

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

      2. neg-mul-1 82.2%

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

      3. sub-neg 82.2%

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

   7. Simplified 82.2%

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

      1. frac-times 78.9%

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

      2. div-inv 78.9%

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

      3. metadata-eval 78.9%

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

      4. *-commutative 78.9%

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

   9. Applied egg-rr 78.9%

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

       1. times-frac 82.2%

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

       2. associate-*l/ 82.1%

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

       3. associate-*l* 82.1%

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

       4. difference-of-squares 90.7%

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

       5. times-frac 99.6%

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

       6. +-commutative 99.6%

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

       7. associate-*r/ 99.6%

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

       8. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in b around 0 99.7%

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

       1. associate-/l/ 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in a around inf 89.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-52} \lor \neg \left(b \leq 2.4 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b}}{a} \cdot \frac{\pi}{a}\\ \end{array} \]

Alternative 4: 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 79.7%

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

   1. associate-*r/ 79.6%

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

   2. *-rgt-identity 79.6%

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

   3. sub-neg 79.6%

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

   4. distribute-neg-frac 79.6%

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

   5. metadata-eval 79.6%

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

3. Simplified 79.6%

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

   1. frac-add 79.6%

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

   2. *-un-lft-identity 79.6%

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

5. Applied egg-rr 79.6%

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

   1. *-commutative 79.6%

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

   2. neg-mul-1 79.6%

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

   3. sub-neg 79.6%

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

7. Simplified 79.6%

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

   1. frac-times 75.1%

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

   2. div-inv 75.1%

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

   3. metadata-eval 75.1%

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

   4. *-commutative 75.1%

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

9. Applied egg-rr 75.1%

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

    1. times-frac 79.6%

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

    2. associate-*l/ 79.6%

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

    3. associate-*l* 79.6%

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

    4. difference-of-squares 89.7%

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

    5. times-frac 99.6%

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

    6. +-commutative 99.6%

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

    7. associate-*r/ 99.6%

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

    8. *-commutative 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in b around 0 99.6%

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

    1. *-commutative 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

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(Float64(0.5 / 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[(N[(0.5 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.7%

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

   1. associate-*r/ 79.6%

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

   2. *-rgt-identity 79.6%

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

   3. sub-neg 79.6%

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

   4. distribute-neg-frac 79.6%

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

   5. metadata-eval 79.6%

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

3. Simplified 79.6%

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

   1. frac-add 79.6%

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

   2. *-un-lft-identity 79.6%

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

5. Applied egg-rr 79.6%

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

   1. *-commutative 79.6%

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

   2. neg-mul-1 79.6%

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

   3. sub-neg 79.6%

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

7. Simplified 79.6%

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

   1. frac-times 75.1%

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

   2. div-inv 75.1%

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

   3. metadata-eval 75.1%

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

   4. *-commutative 75.1%

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

9. Applied egg-rr 75.1%

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

    1. times-frac 79.6%

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

    2. associate-*l/ 79.6%

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

    3. associate-*l* 79.6%

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

    4. difference-of-squares 89.7%

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

    5. times-frac 99.6%

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

    6. +-commutative 99.6%

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

    7. associate-*r/ 99.6%

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

    8. *-commutative 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in b around 0 99.6%

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

    1. associate-/l/ 99.7%

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

14. Simplified 99.7%

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

15. Taylor expanded in b around 0 99.6%

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

    1. associate-/r* 99.7%

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

17. Simplified 99.7%

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

18. Final simplification 99.7%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

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(Float64(pi / Float64(a + b)) * Float64(Float64(0.5 / b) / a))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. associate-*r/ 79.6%

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

   2. *-rgt-identity 79.6%

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

   3. sub-neg 79.6%

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

   4. distribute-neg-frac 79.6%

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

   5. metadata-eval 79.6%

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

3. Simplified 79.6%

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

   1. frac-add 79.6%

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

   2. *-un-lft-identity 79.6%

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

5. Applied egg-rr 79.6%

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

   1. *-commutative 79.6%

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

   2. neg-mul-1 79.6%

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

   3. sub-neg 79.6%

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

7. Simplified 79.6%

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

   1. frac-times 75.1%

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

   2. div-inv 75.1%

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

   3. metadata-eval 75.1%

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

   4. *-commutative 75.1%

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

9. Applied egg-rr 75.1%

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

    1. times-frac 79.6%

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

    2. associate-*l/ 79.6%

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

    3. associate-*l* 79.6%

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

    4. difference-of-squares 89.7%

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

    5. times-frac 99.6%

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

    6. +-commutative 99.6%

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

    7. associate-*r/ 99.6%

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

    8. *-commutative 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in b around 0 99.6%

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

    1. associate-/l/ 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 7: 56.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. associate-*r/ 79.6%

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

   2. *-rgt-identity 79.6%

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

   3. sub-neg 79.6%

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

   4. distribute-neg-frac 79.6%

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

   5. metadata-eval 79.6%

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

3. Simplified 79.6%

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

   1. frac-add 79.6%

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

   2. *-un-lft-identity 79.6%

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

5. Applied egg-rr 79.6%

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

   1. *-commutative 79.6%

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

   2. neg-mul-1 79.6%

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

   3. sub-neg 79.6%

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

7. Simplified 79.6%

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

   1. frac-times 75.1%

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

   2. div-inv 75.1%

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

   3. metadata-eval 75.1%

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

   4. *-commutative 75.1%

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

9. Applied egg-rr 75.1%

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

    1. times-frac 79.6%

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

    2. associate-*l/ 79.6%

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

    3. associate-*l* 79.6%

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

    4. difference-of-squares 89.7%

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

    5. times-frac 99.6%

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

    6. +-commutative 99.6%

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

    7. associate-*r/ 99.6%

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

    8. *-commutative 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in a around 0 62.2%

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

    1. associate-/r* 62.1%

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

    2. unpow2 62.1%

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

14. Simplified 62.1%

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

15. Final simplification 62.1%

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

Reproduce

herbie shell --seed 2023178 
(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))))