NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.7%

Time: 9.9s

Alternatives: 5

Speedup: 1.9×

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: 78.8% 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, 1.9× speedup

Math

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

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(0.5 * Float64(pi / Float64(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[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

   1. associate-*l* 76.9%

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

   2. associate-*l/ 76.9%

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

   3. *-lft-identity 76.9%

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

   4. difference-of-squares 85.8%

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

   5. associate-/l/ 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. associate-*r/ 99.7%

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

   2. div-inv 99.7%

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

   3. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 71.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.2e-22)
   (/ (* (/ 0.5 a) (/ PI a)) b)
   (* (/ PI b) (/ 0.5 (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.2e-22) {
		tmp = ((0.5 / a) * (((double) M_PI) / a)) / b;
	} else {
		tmp = (((double) M_PI) / b) * (0.5 / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.2e-22) {
		tmp = ((0.5 / a) * (Math.PI / a)) / b;
	} else {
		tmp = (Math.PI / b) * (0.5 / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.2e-22:
		tmp = ((0.5 / a) * (math.pi / a)) / b
	else:
		tmp = (math.pi / b) * (0.5 / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.2e-22)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(pi / a)) / b);
	else
		tmp = Float64(Float64(pi / b) * Float64(0.5 / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.2e-22)
		tmp = ((0.5 / a) * (pi / a)) / b;
	else
		tmp = (pi / b) * (0.5 / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.2e-22], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.1999999999999999e-22

   1. Initial program 76.2%

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

      1. associate-*l* 76.1%

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

      2. associate-*l/ 76.2%

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

      3. *-lft-identity 76.2%

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

      4. difference-of-squares 85.9%

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

      5. associate-/l/ 99.6%

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

      6. sub-neg 99.6%

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

      7. distribute-neg-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l/ 97.9%

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

      3. frac-times 97.9%

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

      4. *-un-lft-identity 97.9%

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

      5. +-commutative 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. associate-/l/ 97.9%

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

      2. associate-*r* 86.0%

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

      3. *-un-lft-identity 86.0%

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

      4. *-commutative 86.0%

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

      5. times-frac 85.8%

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

      6. div-inv 85.8%

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

      7. metadata-eval 85.8%

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

      8. +-commutative 85.8%

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

      9. times-frac 85.8%

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

      10. +-commutative 85.8%

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

   9. Applied egg-rr 85.8%

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

       1. associate-*l/ 85.9%

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

       2. *-lft-identity 85.9%

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

       3. *-commutative 85.9%

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

   11. Simplified 85.9%

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

   12. Taylor expanded in a around inf 73.0%

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

   if -8.1999999999999999e-22 < a

   1. Initial program 77.1%

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

      1. associate-*l* 77.1%

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

      2. associate-*l/ 77.1%

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

      3. *-lft-identity 77.1%

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

      4. difference-of-squares 85.8%

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

      5. associate-/l/ 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around 0 99.7%

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

      1. expm1-log1p-u 81.2%

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

      2. expm1-udef 52.1%

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

      3. *-commutative 52.1%

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

      4. associate-/l/ 52.1%

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

      5. frac-times 52.1%

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

      6. *-un-lft-identity 52.1%

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

      7. +-commutative 52.1%

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

   7. Applied egg-rr 52.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(\left(a + b\right) \cdot \left(a \cdot b\right)\right) \cdot 2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 79.7%

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

      2. expm1-log1p 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-/r* 98.2%

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

      5. associate-*r* 93.8%

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

   9. Simplified 93.8%

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

   10. Taylor expanded in a around 0 69.2%

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

       1. div-inv 69.2%

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

       2. metadata-eval 69.2%

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

       3. *-commutative 69.2%

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

       4. times-frac 69.8%

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

   12. Applied egg-rr 69.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \frac{\pi}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.5e-22)
   (/ (* (/ 0.5 a) (/ PI a)) b)
   (/ (* (/ 0.5 a) (/ PI b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.5e-22) {
		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 <= -7.5e-22) {
		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 <= -7.5e-22:
		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 <= -7.5e-22)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(pi / a)) / b);
	else
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(pi / b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.5e-22)
		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, -7.5e-22], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.49999999999999978e-22

   1. Initial program 76.2%

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

      1. associate-*l* 76.1%

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

      2. associate-*l/ 76.2%

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

      3. *-lft-identity 76.2%

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

      4. difference-of-squares 85.9%

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

      5. associate-/l/ 99.6%

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

      6. sub-neg 99.6%

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

      7. distribute-neg-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l/ 97.9%

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

      3. frac-times 97.9%

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

      4. *-un-lft-identity 97.9%

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

      5. +-commutative 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. associate-/l/ 97.9%

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

      2. associate-*r* 86.0%

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

      3. *-un-lft-identity 86.0%

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

      4. *-commutative 86.0%

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

      5. times-frac 85.8%

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

      6. div-inv 85.8%

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

      7. metadata-eval 85.8%

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

      8. +-commutative 85.8%

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

      9. times-frac 85.8%

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

      10. +-commutative 85.8%

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

   9. Applied egg-rr 85.8%

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

       1. associate-*l/ 85.9%

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

       2. *-lft-identity 85.9%

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

       3. *-commutative 85.9%

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

   11. Simplified 85.9%

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

   12. Taylor expanded in a around inf 73.0%

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

   if -7.49999999999999978e-22 < a

   1. Initial program 77.1%

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

      1. associate-*l* 77.1%

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

      2. associate-*l/ 77.1%

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

      3. *-lft-identity 77.1%

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

      4. difference-of-squares 85.8%

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

      5. associate-/l/ 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l/ 98.2%

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

      3. frac-times 98.2%

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

      4. *-un-lft-identity 98.2%

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

      5. +-commutative 98.2%

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

   7. Applied egg-rr 98.2%

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

      1. associate-/l/ 98.2%

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

      2. associate-*r* 93.8%

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

      3. *-un-lft-identity 93.8%

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

      4. *-commutative 93.8%

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

      5. times-frac 94.4%

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

      6. div-inv 94.4%

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

      7. metadata-eval 94.4%

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

      8. +-commutative 94.4%

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

      9. times-frac 94.4%

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

      10. +-commutative 94.4%

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

   9. Applied egg-rr 94.4%

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

       1. associate-*l/ 94.5%

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

       2. *-lft-identity 94.5%

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

       3. *-commutative 94.5%

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

   11. Simplified 94.5%

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

   12. Taylor expanded in a around 0 69.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \frac{\pi}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \frac{\pi}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

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(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 76.9%

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

   1. associate-*l* 76.9%

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

   2. associate-*l/ 76.9%

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

   3. *-lft-identity 76.9%

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

   4. difference-of-squares 85.8%

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

   5. associate-/l/ 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. associate-*r/ 99.7%

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

   2. div-inv 99.7%

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

   3. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.7%

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

   1. expm1-log1p-u 83.0%

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

   2. expm1-udef 52.7%

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

   3. *-un-lft-identity 52.7%

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

   4. times-frac 52.7%

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

   5. metadata-eval 52.7%

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

   6. +-commutative 52.7%

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

9. Applied egg-rr 52.7%

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

    1. expm1-def 82.9%

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

    2. expm1-log1p 99.7%

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

    3. associate-*r/ 99.7%

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

    4. *-commutative 99.7%

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

    5. associate-*r/ 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 5: 62.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. associate-*l* 76.9%

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

   2. associate-*l/ 76.9%

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

   3. *-lft-identity 76.9%

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

   4. difference-of-squares 85.8%

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

   5. associate-/l/ 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a around 0 99.6%

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

   1. expm1-log1p-u 82.9%

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

   2. expm1-udef 52.7%

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

   3. *-commutative 52.7%

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

   4. associate-/l/ 52.7%

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

   5. frac-times 52.7%

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

   6. *-un-lft-identity 52.7%

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

   7. +-commutative 52.7%

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

7. Applied egg-rr 52.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(\left(a + b\right) \cdot \left(a \cdot b\right)\right) \cdot 2}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 81.4%

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

   2. expm1-log1p 98.1%

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

   3. *-commutative 98.1%

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

   4. associate-/r* 98.1%

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

   5. associate-*r* 91.9%

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

9. Simplified 91.9%

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

10. Taylor expanded in a around 0 63.1%

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

    1. div-inv 63.1%

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

    2. metadata-eval 63.1%

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

    3. *-commutative 63.1%

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

    4. times-frac 63.6%

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

12. Applied egg-rr 63.6%

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

13. Final simplification 63.6%

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

Reproduce

herbie shell --seed 2024026 
(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))))