NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.6%

Time: 8.2s

Alternatives: 6

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: 78.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

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

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.9× 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 79.9%

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

   1. associate-*l* 79.9%

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

   2. associate-*l/ 79.9%

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

   3. *-lft-identity 79.9%

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

   4. difference-of-squares 87.4%

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

   5. associate-/l/ 99.2%

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

   6. sub-neg 99.2%

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

   7. distribute-neg-frac 99.2%

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

   8. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in a around 0 99.5%

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

   1. expm1-log1p-u 80.9%

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

   2. expm1-udef 47.8%

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

   3. div-inv 47.8%

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

   4. metadata-eval 47.8%

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

   5. associate-/l/ 47.8%

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

   6. +-commutative 47.8%

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

7. Applied egg-rr 47.8%

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

   1. expm1-def 79.1%

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

   2. expm1-log1p 97.8%

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

   3. associate-*r/ 97.8%

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

   4. *-rgt-identity 97.8%

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

   5. /-rgt-identity 97.8%

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

   6. times-frac 99.5%

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

   7. /-rgt-identity 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{a \cdot b}\\ \mathbf{if}\;b \leq 1.7 \cdot 10^{-39}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* a b))))
   (if (<= b 1.7e-39) (* t_0 (/ PI a)) (* t_0 (/ PI b)))))

C

double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (b <= 1.7e-39) {
		tmp = t_0 * (((double) M_PI) / a);
	} else {
		tmp = t_0 * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (b <= 1.7e-39) {
		tmp = t_0 * (Math.PI / a);
	} else {
		tmp = t_0 * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 / (a * b)
	tmp = 0
	if b <= 1.7e-39:
		tmp = t_0 * (math.pi / a)
	else:
		tmp = t_0 * (math.pi / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 / Float64(a * b))
	tmp = 0.0
	if (b <= 1.7e-39)
		tmp = Float64(t_0 * Float64(pi / a));
	else
		tmp = Float64(t_0 * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 / (a * b);
	tmp = 0.0;
	if (b <= 1.7e-39)
		tmp = t_0 * (pi / a);
	else
		tmp = t_0 * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.7e-39], N[(t$95$0 * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7e-39

   1. Initial program 77.5%

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

      1. associate-*l* 77.5%

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

      2. associate-*l/ 77.5%

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

      3. *-lft-identity 77.5%

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

      4. difference-of-squares 85.6%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around 0 99.5%

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

      1. expm1-log1p-u 76.4%

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

      2. expm1-udef 43.9%

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

      3. div-inv 43.9%

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

      4. metadata-eval 43.9%

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

      5. associate-/l/ 43.9%

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

      6. +-commutative 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. expm1-def 75.2%

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

      2. expm1-log1p 98.3%

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

      3. associate-*r/ 98.4%

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

      4. *-rgt-identity 98.4%

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

      5. /-rgt-identity 98.4%

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

      6. times-frac 99.5%

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

      7. /-rgt-identity 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in a around inf 73.4%

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

   if 1.7e-39 < b

   1. Initial program 85.0%

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

      1. associate-*l* 85.0%

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

      2. associate-*l/ 85.0%

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

      3. *-lft-identity 85.0%

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

      4. difference-of-squares 91.0%

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

      5. associate-/l/ 98.4%

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

      6. sub-neg 98.4%

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

      7. distribute-neg-frac 98.4%

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

      8. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{b - a}}{b + a}} \]
   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 90.1%

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

      2. expm1-udef 56.0%

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

      3. div-inv 56.0%

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

      4. metadata-eval 56.0%

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

      5. associate-/l/ 56.0%

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

      6. +-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. expm1-def 87.1%

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

      2. expm1-log1p 96.6%

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

      3. associate-*r/ 96.6%

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

      4. *-rgt-identity 96.6%

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

      5. /-rgt-identity 96.6%

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

      6. times-frac 99.6%

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

      7. /-rgt-identity 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 89.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.2e-39) (/ (* PI (/ 0.5 a)) (* a b)) (* (/ 0.5 (* a b)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.2e-39) {
		tmp = (((double) M_PI) * (0.5 / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.2e-39) {
		tmp = (Math.PI * (0.5 / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.2e-39:
		tmp = (math.pi * (0.5 / a)) / (a * b)
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.2e-39)
		tmp = Float64(Float64(pi * Float64(0.5 / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.2e-39)
		tmp = (pi * (0.5 / a)) / (a * b);
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.2e-39], N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.1999999999999998e-39

   1. Initial program 77.5%

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

      1. associate-*l* 77.5%

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

      2. associate-*l/ 77.5%

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

      3. *-lft-identity 77.5%

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

      4. difference-of-squares 85.6%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around 0 99.5%

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

      1. expm1-log1p-u 76.4%

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

      2. expm1-udef 43.9%

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

      3. div-inv 43.9%

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

      4. metadata-eval 43.9%

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

      5. associate-/l/ 43.9%

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

      6. +-commutative 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. expm1-def 75.2%

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

      2. expm1-log1p 98.3%

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

      3. associate-*r/ 98.4%

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

      4. *-rgt-identity 98.4%

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

      5. /-rgt-identity 98.4%

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

      6. times-frac 99.5%

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

      7. /-rgt-identity 99.5%

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

   9. Simplified 99.5%

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

       1. *-commutative 99.5%

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

       2. associate-/r* 99.5%

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

       3. frac-times 97.8%

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

   11. Applied egg-rr 97.8%

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

   12. Taylor expanded in b around 0 73.5%

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

   if 3.1999999999999998e-39 < b

   1. Initial program 85.0%

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

      1. associate-*l* 85.0%

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

      2. associate-*l/ 85.0%

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

      3. *-lft-identity 85.0%

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

      4. difference-of-squares 91.0%

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

      5. associate-/l/ 98.4%

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

      6. sub-neg 98.4%

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

      7. distribute-neg-frac 98.4%

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

      8. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{b - a}}{b + a}} \]
   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 90.1%

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

      2. expm1-udef 56.0%

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

      3. div-inv 56.0%

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

      4. metadata-eval 56.0%

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

      5. associate-/l/ 56.0%

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

      6. +-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. expm1-def 87.1%

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

      2. expm1-log1p 96.6%

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

      3. associate-*r/ 96.6%

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

      4. *-rgt-identity 96.6%

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

      5. /-rgt-identity 96.6%

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

      6. times-frac 99.6%

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

      7. /-rgt-identity 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 89.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.8e-39) (/ (/ 0.5 a) (* b (/ a PI))) (* (/ 0.5 (* a b)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.8e-39) {
		tmp = (0.5 / a) / (b * (a / ((double) M_PI)));
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.8e-39) {
		tmp = (0.5 / a) / (b * (a / Math.PI));
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.8e-39:
		tmp = (0.5 / a) / (b * (a / math.pi))
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.8e-39)
		tmp = Float64(Float64(0.5 / a) / Float64(b * Float64(a / pi)));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.8e-39)
		tmp = (0.5 / a) / (b * (a / pi));
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.8e-39], N[(N[(0.5 / a), $MachinePrecision] / N[(b * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8000000000000002e-39

   1. Initial program 77.5%

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

      1. associate-*l* 77.5%

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

      2. associate-*l/ 77.5%

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

      3. *-lft-identity 77.5%

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

      4. difference-of-squares 85.6%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around 0 99.5%

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

      1. expm1-log1p-u 76.4%

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

      2. expm1-udef 43.9%

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

      3. div-inv 43.9%

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

      4. metadata-eval 43.9%

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

      5. associate-/l/ 43.9%

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

      6. +-commutative 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. expm1-def 75.2%

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

      2. expm1-log1p 98.3%

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

      3. associate-*r/ 98.4%

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

      4. *-rgt-identity 98.4%

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

      5. /-rgt-identity 98.4%

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

      6. times-frac 99.5%

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

      7. /-rgt-identity 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in a around inf 73.4%

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

       1. clear-num 73.4%

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

       2. associate-/r* 73.4%

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

       3. frac-times 73.5%

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

       4. *-un-lft-identity 73.5%

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

   12. Applied egg-rr 73.5%

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

   if 3.8000000000000002e-39 < b

   1. Initial program 85.0%

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

      1. associate-*l* 85.0%

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

      2. associate-*l/ 85.0%

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

      3. *-lft-identity 85.0%

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

      4. difference-of-squares 91.0%

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

      5. associate-/l/ 98.4%

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

      6. sub-neg 98.4%

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

      7. distribute-neg-frac 98.4%

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

      8. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{b - a}}{b + a}} \]
   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 90.1%

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

      2. expm1-udef 56.0%

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

      3. div-inv 56.0%

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

      4. metadata-eval 56.0%

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

      5. associate-/l/ 56.0%

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

      6. +-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. expm1-def 87.1%

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

      2. expm1-log1p 96.6%

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

      3. associate-*r/ 96.6%

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

      4. *-rgt-identity 96.6%

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

      5. /-rgt-identity 96.6%

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

      6. times-frac 99.6%

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

      7. /-rgt-identity 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 89.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{b \cdot \frac{a}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(a \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4e-39) (/ (/ PI (* b (* a 2.0))) a) (* (/ 0.5 (* a b)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4e-39) {
		tmp = (((double) M_PI) / (b * (a * 2.0))) / a;
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4e-39) {
		tmp = (Math.PI / (b * (a * 2.0))) / a;
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4e-39:
		tmp = (math.pi / (b * (a * 2.0))) / a
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4e-39)
		tmp = Float64(Float64(pi / Float64(b * Float64(a * 2.0))) / a);
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4e-39)
		tmp = (pi / (b * (a * 2.0))) / a;
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4e-39], N[(N[(Pi / N[(b * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.99999999999999972e-39

   1. Initial program 77.5%

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

      1. associate-*l* 77.5%

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

      2. associate-*l/ 77.5%

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

      3. *-lft-identity 77.5%

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

      4. difference-of-squares 85.6%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around 0 99.5%

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

      1. expm1-log1p-u 76.4%

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

      2. expm1-udef 43.9%

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

      3. div-inv 43.9%

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

      4. metadata-eval 43.9%

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

      5. associate-/l/ 43.9%

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

      6. +-commutative 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. expm1-def 75.2%

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

      2. expm1-log1p 98.3%

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

      3. associate-*r/ 98.4%

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

      4. *-rgt-identity 98.4%

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

      5. /-rgt-identity 98.4%

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

      6. times-frac 99.5%

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

      7. /-rgt-identity 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in a around inf 73.4%

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

       1. associate-*l/ 73.5%

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

       2. associate-/r* 73.5%

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

       3. clear-num 73.4%

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

       4. un-div-inv 73.5%

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

       5. div-inv 73.5%

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

       6. clear-num 73.5%

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

       7. div-inv 73.5%

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

       8. metadata-eval 73.5%

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

   12. Applied egg-rr 73.5%

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

   if 3.99999999999999972e-39 < b

   1. Initial program 85.0%

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

      1. associate-*l* 85.0%

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

      2. associate-*l/ 85.0%

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

      3. *-lft-identity 85.0%

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

      4. difference-of-squares 91.0%

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

      5. associate-/l/ 98.4%

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

      6. sub-neg 98.4%

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

      7. distribute-neg-frac 98.4%

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

      8. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{b - a}}{b + a}} \]
   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 90.1%

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

      2. expm1-udef 56.0%

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

      3. div-inv 56.0%

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

      4. metadata-eval 56.0%

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

      5. associate-/l/ 56.0%

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

      6. +-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. expm1-def 87.1%

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

      2. expm1-log1p 96.6%

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

      3. associate-*r/ 96.6%

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

      4. *-rgt-identity 96.6%

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

      5. /-rgt-identity 96.6%

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

      6. times-frac 99.6%

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

      7. /-rgt-identity 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 89.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\pi}{b \cdot \left(a \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.9%

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

   1. associate-*l* 79.9%

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

   2. associate-*l/ 79.9%

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

   3. *-lft-identity 79.9%

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

   4. difference-of-squares 87.4%

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

   5. associate-/l/ 99.2%

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

   6. sub-neg 99.2%

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

   7. distribute-neg-frac 99.2%

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

   8. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in a around 0 99.5%

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

   1. expm1-log1p-u 80.9%

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

   2. expm1-udef 47.8%

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

   3. div-inv 47.8%

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

   4. metadata-eval 47.8%

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

   5. associate-/l/ 47.8%

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

   6. +-commutative 47.8%

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

7. Applied egg-rr 47.8%

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

   1. expm1-def 79.1%

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

   2. expm1-log1p 97.8%

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

   3. associate-*r/ 97.8%

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

   4. *-rgt-identity 97.8%

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

   5. /-rgt-identity 97.8%

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

   6. times-frac 99.5%

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

   7. /-rgt-identity 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in a around inf 60.1%

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

11. Final simplification 60.1%

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

Reproduce

herbie shell --seed 2024027 
(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))))