NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 10.5s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. times-frac 80.2%

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

   2. *-commutative 80.2%

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

   3. times-frac 80.2%

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

   4. difference-of-squares 87.6%

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

   5. associate-/r* 88.3%

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

   6. metadata-eval 88.3%

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

   7. sub-neg 88.3%

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

   8. distribute-neg-frac 88.3%

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

   9. metadata-eval 88.3%

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

3. Simplified 88.3%

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

   1. frac-add 88.3%

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

   2. *-un-lft-identity 88.3%

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

5. Applied egg-rr 88.3%

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

   1. *-commutative 88.3%

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

   2. neg-mul-1 88.3%

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

   3. sub-neg 88.3%

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

7. Simplified 88.3%

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

   1. expm1-log1p-u 68.8%

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

   2. expm1-udef 55.5%

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

9. Applied egg-rr 55.5%

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

    1. expm1-def 68.8%

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

    2. expm1-log1p 88.3%

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

    3. *-commutative 88.3%

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

    4. associate-*l* 88.3%

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

11. Simplified 88.3%

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

    1. associate-*l/ 99.6%

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

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

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

Alternative 2: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{a} + \frac{-1}{b}\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b - a}}{b}}{a}\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-166}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot t_0\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{t_0 \cdot \left(\pi \cdot 0.5\right)}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot \left(b - a\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 a) (/ -1.0 b))))
   (if (<= a -1.1e+88)
     (/ (/ (* -0.5 (/ PI (- b a))) b) a)
     (if (<= a -1e-166)
       (* (* 0.5 (/ (/ PI (+ a b)) (- b a))) t_0)
       (if (<= a 2.6e-107)
         (/ (* 0.5 (/ PI (* a b))) (- b a))
         (if (<= a 1.5e+138)
           (/ (* t_0 (* PI 0.5)) (- (* b b) (* a a)))
           (/ (/ (* PI -0.5) (* b (- b a))) a)))))))

C

double code(double a, double b) {
	double t_0 = (1.0 / a) + (-1.0 / b);
	double tmp;
	if (a <= -1.1e+88) {
		tmp = ((-0.5 * (((double) M_PI) / (b - a))) / b) / a;
	} else if (a <= -1e-166) {
		tmp = (0.5 * ((((double) M_PI) / (a + b)) / (b - a))) * t_0;
	} else if (a <= 2.6e-107) {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	} else if (a <= 1.5e+138) {
		tmp = (t_0 * (((double) M_PI) * 0.5)) / ((b * b) - (a * a));
	} else {
		tmp = ((((double) M_PI) * -0.5) / (b * (b - a))) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (1.0 / a) + (-1.0 / b);
	double tmp;
	if (a <= -1.1e+88) {
		tmp = ((-0.5 * (Math.PI / (b - a))) / b) / a;
	} else if (a <= -1e-166) {
		tmp = (0.5 * ((Math.PI / (a + b)) / (b - a))) * t_0;
	} else if (a <= 2.6e-107) {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	} else if (a <= 1.5e+138) {
		tmp = (t_0 * (Math.PI * 0.5)) / ((b * b) - (a * a));
	} else {
		tmp = ((Math.PI * -0.5) / (b * (b - a))) / a;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (1.0 / a) + (-1.0 / b)
	tmp = 0
	if a <= -1.1e+88:
		tmp = ((-0.5 * (math.pi / (b - a))) / b) / a
	elif a <= -1e-166:
		tmp = (0.5 * ((math.pi / (a + b)) / (b - a))) * t_0
	elif a <= 2.6e-107:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	elif a <= 1.5e+138:
		tmp = (t_0 * (math.pi * 0.5)) / ((b * b) - (a * a))
	else:
		tmp = ((math.pi * -0.5) / (b * (b - a))) / a
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(1.0 / a) + Float64(-1.0 / b))
	tmp = 0.0
	if (a <= -1.1e+88)
		tmp = Float64(Float64(Float64(-0.5 * Float64(pi / Float64(b - a))) / b) / a);
	elseif (a <= -1e-166)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))) * t_0);
	elseif (a <= 2.6e-107)
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	elseif (a <= 1.5e+138)
		tmp = Float64(Float64(t_0 * Float64(pi * 0.5)) / Float64(Float64(b * b) - Float64(a * a)));
	else
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * Float64(b - a))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (1.0 / a) + (-1.0 / b);
	tmp = 0.0;
	if (a <= -1.1e+88)
		tmp = ((-0.5 * (pi / (b - a))) / b) / a;
	elseif (a <= -1e-166)
		tmp = (0.5 * ((pi / (a + b)) / (b - a))) * t_0;
	elseif (a <= 2.6e-107)
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	elseif (a <= 1.5e+138)
		tmp = (t_0 * (pi * 0.5)) / ((b * b) - (a * a));
	else
		tmp = ((pi * -0.5) / (b * (b - a))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.1e+88], N[(N[(N[(-0.5 * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, -1e-166], N[(N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[a, 2.6e-107], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.5e+138], N[(N[(t$95$0 * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -1.10000000000000004e88

   1. Initial program 61.2%

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

      1. times-frac 61.4%

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

      2. *-commutative 61.4%

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

      3. times-frac 61.4%

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

      4. difference-of-squares 75.1%

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

      5. associate-/r* 76.8%

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

      6. metadata-eval 76.8%

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

      7. sub-neg 76.8%

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

      8. distribute-neg-frac 76.8%

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

      9. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in a around inf 76.8%

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

   5. Taylor expanded in b around 0 76.8%

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

      1. expm1-log1p-u 74.8%

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

      2. expm1-udef 59.2%

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

      3. associate-*l* 59.2%

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

      4. associate-/l/ 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. expm1-def 73.1%

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

      2. expm1-log1p 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-/r* 76.8%

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

      5. associate-*r/ 99.7%

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

      6. associate-*r/ 99.7%

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

      7. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

       1. associate-*l/ 99.8%

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

   11. Applied egg-rr 99.8%

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

   if -1.10000000000000004e88 < a < -1.00000000000000004e-166

   1. Initial program 97.7%

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

      1. times-frac 97.7%

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

      2. *-commutative 97.7%

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

      3. times-frac 97.7%

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

      4. difference-of-squares 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   if -1.00000000000000004e-166 < a < 2.6000000000000001e-107

   1. Initial program 75.4%

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

      1. times-frac 75.6%

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

      2. *-commutative 75.6%

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

      3. times-frac 75.6%

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

      4. difference-of-squares 83.3%

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

      5. associate-/r* 83.3%

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

      6. metadata-eval 83.3%

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

      7. sub-neg 83.3%

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

      8. distribute-neg-frac 83.3%

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

      9. metadata-eval 83.3%

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

   3. Simplified 83.3%

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

      1. frac-add 83.2%

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

      2. *-un-lft-identity 83.2%

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

   5. Applied egg-rr 83.2%

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

      1. *-commutative 83.2%

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

      2. neg-mul-1 83.2%

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

      3. sub-neg 83.2%

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

   7. Simplified 83.2%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 49.8%

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

   9. Applied egg-rr 49.8%

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

       1. expm1-def 55.9%

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

       2. expm1-log1p 83.2%

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

       3. *-commutative 83.2%

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

       4. associate-*l* 83.5%

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

   11. Simplified 83.5%

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

       1. associate-*l/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in a around 0 91.9%

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

   if 2.6000000000000001e-107 < a < 1.50000000000000005e138

   1. Initial program 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-frac 99.5%

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

      5. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*l/ 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   if 1.50000000000000005e138 < a

   1. Initial program 67.2%

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

      1. times-frac 67.2%

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

      2. *-commutative 67.2%

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

      3. times-frac 67.2%

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

      4. difference-of-squares 81.5%

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

      5. associate-/r* 83.6%

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

      6. metadata-eval 83.6%

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

      7. sub-neg 83.6%

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

      8. distribute-neg-frac 83.6%

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

      9. metadata-eval 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in a around inf 83.6%

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

   5. Taylor expanded in b around 0 83.6%

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

      1. expm1-log1p-u 83.6%

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

      2. expm1-udef 76.4%

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

      3. associate-*l* 76.4%

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

      4. associate-/l/ 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. expm1-def 81.5%

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

      2. expm1-log1p 81.5%

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

      3. *-commutative 81.5%

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

      4. associate-/r* 83.7%

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

      5. associate-*r/ 99.7%

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

      6. associate-*r/ 99.7%

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

      7. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

       1. frac-times 99.9%

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

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot \pi}{b \cdot \left(b - a\right)}}}{a} \]
3. Recombined 5 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b - a}}{b}}{a}\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-166}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot 0.5\right)}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot \left(b - a\right)}}{a}\\ \end{array} \]

Alternative 3: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{if}\;a \leq -2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b - a}}{b}}{a}\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot \left(b - a\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (- (* b b) (* a a))) (+ (/ 0.5 a) (/ -0.5 b)))))
   (if (<= a -2e+89)
     (/ (/ (* -0.5 (/ PI (- b a))) b) a)
     (if (<= a -1.7e-151)
       t_0
       (if (<= a 2.6e-107)
         (/ (* 0.5 (/ PI (* a b))) (- b a))
         (if (<= a 8e+146) t_0 (/ (/ (* PI -0.5) (* b (- b a))) a)))))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	double tmp;
	if (a <= -2e+89) {
		tmp = ((-0.5 * (((double) M_PI) / (b - a))) / b) / a;
	} else if (a <= -1.7e-151) {
		tmp = t_0;
	} else if (a <= 2.6e-107) {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	} else if (a <= 8e+146) {
		tmp = t_0;
	} else {
		tmp = ((((double) M_PI) * -0.5) / (b * (b - a))) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	double tmp;
	if (a <= -2e+89) {
		tmp = ((-0.5 * (Math.PI / (b - a))) / b) / a;
	} else if (a <= -1.7e-151) {
		tmp = t_0;
	} else if (a <= 2.6e-107) {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	} else if (a <= 8e+146) {
		tmp = t_0;
	} else {
		tmp = ((Math.PI * -0.5) / (b * (b - a))) / a;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b))
	tmp = 0
	if a <= -2e+89:
		tmp = ((-0.5 * (math.pi / (b - a))) / b) / a
	elif a <= -1.7e-151:
		tmp = t_0
	elif a <= 2.6e-107:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	elif a <= 8e+146:
		tmp = t_0
	else:
		tmp = ((math.pi * -0.5) / (b * (b - a))) / a
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(Float64(b * b) - Float64(a * a))) * Float64(Float64(0.5 / a) + Float64(-0.5 / b)))
	tmp = 0.0
	if (a <= -2e+89)
		tmp = Float64(Float64(Float64(-0.5 * Float64(pi / Float64(b - a))) / b) / a);
	elseif (a <= -1.7e-151)
		tmp = t_0;
	elseif (a <= 2.6e-107)
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	elseif (a <= 8e+146)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * Float64(b - a))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	tmp = 0.0;
	if (a <= -2e+89)
		tmp = ((-0.5 * (pi / (b - a))) / b) / a;
	elseif (a <= -1.7e-151)
		tmp = t_0;
	elseif (a <= 2.6e-107)
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	elseif (a <= 8e+146)
		tmp = t_0;
	else
		tmp = ((pi * -0.5) / (b * (b - a))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2e+89], N[(N[(N[(-0.5 * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, -1.7e-151], t$95$0, If[LessEqual[a, 2.6e-107], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8e+146], t$95$0, N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.99999999999999999e89

   1. Initial program 61.2%

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

      1. times-frac 61.4%

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

      2. *-commutative 61.4%

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

      3. times-frac 61.4%

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

      4. difference-of-squares 75.1%

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

      5. associate-/r* 76.8%

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

      6. metadata-eval 76.8%

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

      7. sub-neg 76.8%

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

      8. distribute-neg-frac 76.8%

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

      9. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in a around inf 76.8%

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

   5. Taylor expanded in b around 0 76.8%

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

      1. expm1-log1p-u 74.8%

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

      2. expm1-udef 59.2%

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

      3. associate-*l* 59.2%

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

      4. associate-/l/ 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. expm1-def 73.1%

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

      2. expm1-log1p 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-/r* 76.8%

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

      5. associate-*r/ 99.7%

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

      6. associate-*r/ 99.7%

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

      7. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

       1. associate-*l/ 99.8%

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

   11. Applied egg-rr 99.8%

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

   if -1.99999999999999999e89 < a < -1.7000000000000001e-151 or 2.6000000000000001e-107 < a < 7.99999999999999947e146

   1. Initial program 99.6%

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

      1. times-frac 99.6%

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

      2. *-commutative 99.6%

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

      3. times-frac 99.6%

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

      4. difference-of-squares 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. distribute-lft-in 99.6%

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

      2. associate-/l/ 99.5%

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

      3. associate-/l/ 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*r* 99.6%

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

      3. associate-*l/ 99.6%

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

      4. *-commutative 99.6%

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

      5. difference-of-squares 99.6%

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

      6. associate-*l/ 99.6%

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

      7. distribute-lft-in 99.6%

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

      8. associate-*r/ 99.6%

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

      9. metadata-eval 99.6%

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

      10. associate-*r/ 99.6%

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

      11. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   if -1.7000000000000001e-151 < a < 2.6000000000000001e-107

   1. Initial program 76.4%

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

      1. times-frac 76.5%

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

      2. *-commutative 76.5%

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

      3. times-frac 76.5%

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

      4. difference-of-squares 84.9%

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

      5. associate-/r* 84.9%

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

      6. metadata-eval 84.9%

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

      7. sub-neg 84.9%

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

      8. distribute-neg-frac 84.9%

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

      9. metadata-eval 84.9%

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

   3. Simplified 84.9%

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

      1. frac-add 84.8%

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

      2. *-un-lft-identity 84.8%

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

   5. Applied egg-rr 84.8%

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

      1. *-commutative 84.8%

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

      2. neg-mul-1 84.8%

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

      3. sub-neg 84.8%

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

   7. Simplified 84.8%

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

      1. expm1-log1p-u 53.3%

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

      2. expm1-udef 47.7%

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

   9. Applied egg-rr 47.7%

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

       1. expm1-def 53.3%

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

       2. expm1-log1p 84.8%

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

       3. *-commutative 84.8%

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

       4. associate-*l* 85.1%

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

   11. Simplified 85.1%

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

       1. associate-*l/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in a around 0 91.3%

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

   if 7.99999999999999947e146 < a

   1. Initial program 61.7%

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

      1. times-frac 61.7%

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

      2. *-commutative 61.7%

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

      3. times-frac 61.7%

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

      4. difference-of-squares 78.4%

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

      5. associate-/r* 81.0%

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

      6. metadata-eval 81.0%

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

      7. sub-neg 81.0%

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

      8. distribute-neg-frac 81.0%

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

      9. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in a around inf 81.0%

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

   5. Taylor expanded in b around 0 81.0%

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

      1. expm1-log1p-u 81.0%

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

      2. expm1-udef 78.4%

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

      3. associate-*l* 78.4%

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

      4. associate-/l/ 78.4%

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

   7. Applied egg-rr 78.4%

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

      1. expm1-def 78.4%

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

      2. expm1-log1p 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-/r* 81.0%

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

      5. associate-*r/ 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

       1. frac-times 99.9%

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

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot \pi}{b \cdot \left(b - a\right)}}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b - a}}{b}}{a}\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+146}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot \left(b - a\right)}}{a}\\ \end{array} \]

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b - a}}{b}}{a}\\ \mathbf{elif}\;a \leq -1.08 \cdot 10^{-166}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot \left(b - a\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e+87)
   (/ (/ (* -0.5 (/ PI (- b a))) b) a)
   (if (<= a -1.08e-166)
     (* (* 0.5 (/ (/ PI (+ a b)) (- b a))) (+ (/ 1.0 a) (/ -1.0 b)))
     (if (<= a 2.6e-107)
       (/ (* 0.5 (/ PI (* a b))) (- b a))
       (if (<= a 1.8e+147)
         (* (/ PI (- (* b b) (* a a))) (+ (/ 0.5 a) (/ -0.5 b)))
         (/ (/ (* PI -0.5) (* b (- b a))) a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e+87) {
		tmp = ((-0.5 * (((double) M_PI) / (b - a))) / b) / a;
	} else if (a <= -1.08e-166) {
		tmp = (0.5 * ((((double) M_PI) / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	} else if (a <= 2.6e-107) {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	} else if (a <= 1.8e+147) {
		tmp = (((double) M_PI) / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = ((((double) M_PI) * -0.5) / (b * (b - a))) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.2e+87) {
		tmp = ((-0.5 * (Math.PI / (b - a))) / b) / a;
	} else if (a <= -1.08e-166) {
		tmp = (0.5 * ((Math.PI / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	} else if (a <= 2.6e-107) {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	} else if (a <= 1.8e+147) {
		tmp = (Math.PI / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = ((Math.PI * -0.5) / (b * (b - a))) / a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.2e+87:
		tmp = ((-0.5 * (math.pi / (b - a))) / b) / a
	elif a <= -1.08e-166:
		tmp = (0.5 * ((math.pi / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b))
	elif a <= 2.6e-107:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	elif a <= 1.8e+147:
		tmp = (math.pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b))
	else:
		tmp = ((math.pi * -0.5) / (b * (b - a))) / a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e+87)
		tmp = Float64(Float64(Float64(-0.5 * Float64(pi / Float64(b - a))) / b) / a);
	elseif (a <= -1.08e-166)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))) * Float64(Float64(1.0 / a) + Float64(-1.0 / b)));
	elseif (a <= 2.6e-107)
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	elseif (a <= 1.8e+147)
		tmp = Float64(Float64(pi / Float64(Float64(b * b) - Float64(a * a))) * Float64(Float64(0.5 / a) + Float64(-0.5 / b)));
	else
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * Float64(b - a))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.2e+87)
		tmp = ((-0.5 * (pi / (b - a))) / b) / a;
	elseif (a <= -1.08e-166)
		tmp = (0.5 * ((pi / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	elseif (a <= 2.6e-107)
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	elseif (a <= 1.8e+147)
		tmp = (pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	else
		tmp = ((pi * -0.5) / (b * (b - a))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e+87], N[(N[(N[(-0.5 * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, -1.08e-166], N[(N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.6e-107], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e+147], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -1.19999999999999991e87

   1. Initial program 61.2%

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

      1. times-frac 61.4%

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

      2. *-commutative 61.4%

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

      3. times-frac 61.4%

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

      4. difference-of-squares 75.1%

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

      5. associate-/r* 76.8%

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

      6. metadata-eval 76.8%

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

      7. sub-neg 76.8%

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

      8. distribute-neg-frac 76.8%

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

      9. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in a around inf 76.8%

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

   5. Taylor expanded in b around 0 76.8%

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

      1. expm1-log1p-u 74.8%

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

      2. expm1-udef 59.2%

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

      3. associate-*l* 59.2%

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

      4. associate-/l/ 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. expm1-def 73.1%

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

      2. expm1-log1p 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-/r* 76.8%

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

      5. associate-*r/ 99.7%

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

      6. associate-*r/ 99.7%

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

      7. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

       1. associate-*l/ 99.8%

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

   11. Applied egg-rr 99.8%

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

   if -1.19999999999999991e87 < a < -1.08e-166

   1. Initial program 97.7%

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

      1. times-frac 97.7%

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

      2. *-commutative 97.7%

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

      3. times-frac 97.7%

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

      4. difference-of-squares 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   if -1.08e-166 < a < 2.6000000000000001e-107

   1. Initial program 75.4%

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

      1. times-frac 75.6%

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

      2. *-commutative 75.6%

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

      3. times-frac 75.6%

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

      4. difference-of-squares 83.3%

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

      5. associate-/r* 83.3%

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

      6. metadata-eval 83.3%

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

      7. sub-neg 83.3%

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

      8. distribute-neg-frac 83.3%

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

      9. metadata-eval 83.3%

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

   3. Simplified 83.3%

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

      1. frac-add 83.2%

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

      2. *-un-lft-identity 83.2%

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

   5. Applied egg-rr 83.2%

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

      1. *-commutative 83.2%

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

      2. neg-mul-1 83.2%

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

      3. sub-neg 83.2%

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

   7. Simplified 83.2%

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

      1. expm1-log1p-u 55.9%

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

      2. expm1-udef 49.8%

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

   9. Applied egg-rr 49.8%

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

       1. expm1-def 55.9%

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

       2. expm1-log1p 83.2%

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

       3. *-commutative 83.2%

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

       4. associate-*l* 83.5%

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

   11. Simplified 83.5%

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

       1. associate-*l/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in a around 0 91.9%

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

   if 2.6000000000000001e-107 < a < 1.8000000000000001e147

   1. Initial program 99.5%

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

      1. times-frac 99.6%

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

      2. *-commutative 99.6%

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

      3. times-frac 99.6%

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

      4. difference-of-squares 99.6%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. associate-/l/ 99.5%

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

      3. associate-/l/ 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*r* 99.6%

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

      3. associate-*l/ 99.6%

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

      4. *-commutative 99.6%

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

      5. difference-of-squares 99.6%

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

      6. associate-*l/ 99.6%

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

      7. distribute-lft-in 99.6%

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

      8. associate-*r/ 99.6%

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

      9. metadata-eval 99.6%

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

      10. associate-*r/ 99.6%

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

      11. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   if 1.8000000000000001e147 < a

   1. Initial program 61.7%

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

      1. times-frac 61.7%

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

      2. *-commutative 61.7%

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

      3. times-frac 61.7%

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

      4. difference-of-squares 78.4%

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

      5. associate-/r* 81.0%

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

      6. metadata-eval 81.0%

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

      7. sub-neg 81.0%

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

      8. distribute-neg-frac 81.0%

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

      9. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in a around inf 81.0%

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

   5. Taylor expanded in b around 0 81.0%

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

      1. expm1-log1p-u 81.0%

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

      2. expm1-udef 78.4%

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

      3. associate-*l* 78.4%

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

      4. associate-/l/ 78.4%

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

   7. Applied egg-rr 78.4%

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

      1. expm1-def 78.4%

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

      2. expm1-log1p 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-/r* 81.0%

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

      5. associate-*r/ 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

       1. frac-times 99.9%

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

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot \pi}{b \cdot \left(b - a\right)}}}{a} \]
3. Recombined 5 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b - a}}{b}}{a}\\ \mathbf{elif}\;a \leq -1.08 \cdot 10^{-166}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot \left(b - a\right)}}{a}\\ \end{array} \]

Alternative 5: 83.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-102} \lor \neg \left(a \leq 3.3 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.05e-102) (not (<= a 3.3e-96)))
   (* (/ PI (- b a)) (/ (/ -0.5 b) a))
   (* (/ 0.5 a) (/ PI (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.05e-102) || !(a <= 3.3e-96)) {
		tmp = (((double) M_PI) / (b - a)) * ((-0.5 / b) / a);
	} 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 <= -1.05e-102) || !(a <= 3.3e-96)) {
		tmp = (Math.PI / (b - a)) * ((-0.5 / b) / a);
	} else {
		tmp = (0.5 / a) * (Math.PI / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.05e-102) or not (a <= 3.3e-96):
		tmp = (math.pi / (b - a)) * ((-0.5 / b) / a)
	else:
		tmp = (0.5 / a) * (math.pi / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.05e-102) || !(a <= 3.3e-96))
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(Float64(-0.5 / b) / a));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.05e-102) || ~((a <= 3.3e-96)))
		tmp = (pi / (b - a)) * ((-0.5 / b) / a);
	else
		tmp = (0.5 / a) * (pi / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.05e-102], N[Not[LessEqual[a, 3.3e-96]], $MachinePrecision]], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05e-102 or 3.2999999999999999e-96 < a

   1. Initial program 80.6%

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

      1. times-frac 80.7%

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

      2. *-commutative 80.7%

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

      3. times-frac 80.7%

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

      4. difference-of-squares 88.1%

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

      5. associate-/r* 89.1%

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

      6. metadata-eval 89.1%

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

      7. sub-neg 89.1%

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

      8. distribute-neg-frac 89.1%

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

      9. metadata-eval 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in a around inf 81.0%

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

   5. Taylor expanded in b around 0 80.9%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 58.2%

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

      3. associate-*l* 58.2%

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

      4. associate-/l/ 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. expm1-def 67.3%

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

      2. expm1-log1p 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/r* 80.9%

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

      5. associate-*r/ 91.4%

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

      6. associate-*r/ 91.4%

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

      7. metadata-eval 91.4%

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

   9. Simplified 91.4%

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

       1. expm1-log1p-u 78.8%

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

       2. expm1-udef 58.2%

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

       3. associate-/l* 58.2%

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

   11. Applied egg-rr 58.2%

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

       1. expm1-def 67.8%

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

       2. expm1-log1p 80.4%

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

       3. associate-/r/ 91.4%

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

   13. Simplified 91.4%

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

   if -1.05e-102 < a < 3.2999999999999999e-96

   1. Initial program 79.0%

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

      1. times-frac 79.1%

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

      2. *-commutative 79.1%

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

      3. times-frac 79.1%

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

      4. difference-of-squares 86.5%

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

      5. associate-/r* 86.6%

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

      6. metadata-eval 86.6%

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

      7. sub-neg 86.6%

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

      8. distribute-neg-frac 86.6%

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

      9. metadata-eval 86.6%

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

   3. Simplified 86.6%

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

      1. frac-add 86.5%

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

      2. *-un-lft-identity 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. neg-mul-1 86.5%

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

      3. sub-neg 86.5%

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

   7. Simplified 86.5%

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

      1. expm1-log1p-u 53.5%

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

      2. expm1-udef 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. expm1-def 53.5%

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

       2. expm1-log1p 86.5%

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

       3. *-commutative 86.5%

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

       4. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

   12. Taylor expanded in a around 0 81.5%

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

       1. associate-*r/ 81.5%

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

       2. *-commutative 81.5%

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

       3. *-commutative 81.5%

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

       4. times-frac 81.5%

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

       5. unpow2 81.5%

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

   14. Simplified 81.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-102} \lor \neg \left(a \leq 3.3 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \]

Alternative 6: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-103} \lor \neg \left(a \leq 9.2 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.6e-103) (not (<= a 9.2e-99)))
   (* (/ PI (- b a)) (/ (/ -0.5 b) a))
   (/ (* 0.5 (/ PI (* a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.6e-103) || !(a <= 9.2e-99)) {
		tmp = (((double) M_PI) / (b - a)) * ((-0.5 / b) / a);
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.6e-103) || !(a <= 9.2e-99)) {
		tmp = (Math.PI / (b - a)) * ((-0.5 / b) / a);
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.6e-103) or not (a <= 9.2e-99):
		tmp = (math.pi / (b - a)) * ((-0.5 / b) / a)
	else:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.6e-103) || !(a <= 9.2e-99))
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(Float64(-0.5 / b) / a));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.6e-103) || ~((a <= 9.2e-99)))
		tmp = (pi / (b - a)) * ((-0.5 / b) / a);
	else
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.6e-103], N[Not[LessEqual[a, 9.2e-99]], $MachinePrecision]], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.59999999999999996e-103 or 9.1999999999999994e-99 < a

   1. Initial program 80.6%

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

      1. times-frac 80.7%

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

      2. *-commutative 80.7%

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

      3. times-frac 80.7%

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

      4. difference-of-squares 88.1%

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

      5. associate-/r* 89.1%

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

      6. metadata-eval 89.1%

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

      7. sub-neg 89.1%

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

      8. distribute-neg-frac 89.1%

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

      9. metadata-eval 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in a around inf 81.0%

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

   5. Taylor expanded in b around 0 80.9%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 58.2%

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

      3. associate-*l* 58.2%

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

      4. associate-/l/ 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. expm1-def 67.3%

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

      2. expm1-log1p 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/r* 80.9%

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

      5. associate-*r/ 91.4%

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

      6. associate-*r/ 91.4%

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

      7. metadata-eval 91.4%

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

   9. Simplified 91.4%

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

       1. expm1-log1p-u 78.8%

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

       2. expm1-udef 58.2%

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

       3. associate-/l* 58.2%

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

   11. Applied egg-rr 58.2%

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

       1. expm1-def 67.8%

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

       2. expm1-log1p 80.4%

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

       3. associate-/r/ 91.4%

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

   13. Simplified 91.4%

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

   if -2.59999999999999996e-103 < a < 9.1999999999999994e-99

   1. Initial program 79.0%

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

      1. times-frac 79.1%

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

      2. *-commutative 79.1%

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

      3. times-frac 79.1%

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

      4. difference-of-squares 86.5%

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

      5. associate-/r* 86.6%

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

      6. metadata-eval 86.6%

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

      7. sub-neg 86.6%

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

      8. distribute-neg-frac 86.6%

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

      9. metadata-eval 86.6%

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

   3. Simplified 86.6%

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

      1. frac-add 86.5%

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

      2. *-un-lft-identity 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. neg-mul-1 86.5%

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

      3. sub-neg 86.5%

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

   7. Simplified 86.5%

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

      1. expm1-log1p-u 53.5%

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

      2. expm1-udef 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. expm1-def 53.5%

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

       2. expm1-log1p 86.5%

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

       3. *-commutative 86.5%

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

       4. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

       1. associate-*l/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in a around 0 88.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{-103} \lor \neg \left(a \leq 9.2 \cdot 10^{-99}\right):\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]

Alternative 7: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-104} \lor \neg \left(a \leq 2.35 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot \left(b - a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.9e-104) (not (<= a 2.35e-101)))
   (/ (/ (* PI -0.5) (* b (- b a))) a)
   (/ (* 0.5 (/ PI (* a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.9e-104) || !(a <= 2.35e-101)) {
		tmp = ((((double) M_PI) * -0.5) / (b * (b - a))) / a;
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.9e-104) || !(a <= 2.35e-101)) {
		tmp = ((Math.PI * -0.5) / (b * (b - a))) / a;
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.9e-104) or not (a <= 2.35e-101):
		tmp = ((math.pi * -0.5) / (b * (b - a))) / a
	else:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.9e-104) || !(a <= 2.35e-101))
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * Float64(b - a))) / a);
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.9e-104) || ~((a <= 2.35e-101)))
		tmp = ((pi * -0.5) / (b * (b - a))) / a;
	else
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.9e-104], N[Not[LessEqual[a, 2.35e-101]], $MachinePrecision]], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9e-104 or 2.35e-101 < a

   1. Initial program 80.6%

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

      1. times-frac 80.7%

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

      2. *-commutative 80.7%

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

      3. times-frac 80.7%

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

      4. difference-of-squares 88.1%

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

      5. associate-/r* 89.1%

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

      6. metadata-eval 89.1%

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

      7. sub-neg 89.1%

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

      8. distribute-neg-frac 89.1%

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

      9. metadata-eval 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in a around inf 81.0%

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

   5. Taylor expanded in b around 0 80.9%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 58.2%

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

      3. associate-*l* 58.2%

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

      4. associate-/l/ 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. expm1-def 67.3%

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

      2. expm1-log1p 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/r* 80.9%

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

      5. associate-*r/ 91.4%

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

      6. associate-*r/ 91.4%

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

      7. metadata-eval 91.4%

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

   9. Simplified 91.4%

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

       1. frac-times 91.5%

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

   11. Applied egg-rr 91.5%

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

   if -1.9e-104 < a < 2.35e-101

   1. Initial program 79.0%

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

      1. times-frac 79.1%

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

      2. *-commutative 79.1%

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

      3. times-frac 79.1%

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

      4. difference-of-squares 86.5%

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

      5. associate-/r* 86.6%

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

      6. metadata-eval 86.6%

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

      7. sub-neg 86.6%

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

      8. distribute-neg-frac 86.6%

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

      9. metadata-eval 86.6%

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

   3. Simplified 86.6%

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

      1. frac-add 86.5%

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

      2. *-un-lft-identity 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. neg-mul-1 86.5%

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

      3. sub-neg 86.5%

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

   7. Simplified 86.5%

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

      1. expm1-log1p-u 53.5%

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

      2. expm1-udef 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. expm1-def 53.5%

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

       2. expm1-log1p 86.5%

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

       3. *-commutative 86.5%

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

       4. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

       1. associate-*l/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in a around 0 88.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-104} \lor \neg \left(a \leq 2.35 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot \left(b - a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]

Alternative 8: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-102} \lor \neg \left(a \leq 1.7 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b - a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.1e-102) (not (<= a 1.7e-101)))
   (/ (/ (* -0.5 (/ PI (- b a))) b) a)
   (/ (* 0.5 (/ PI (* a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.1e-102) || !(a <= 1.7e-101)) {
		tmp = ((-0.5 * (((double) M_PI) / (b - a))) / b) / a;
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.1e-102) || !(a <= 1.7e-101)) {
		tmp = ((-0.5 * (Math.PI / (b - a))) / b) / a;
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.1e-102) or not (a <= 1.7e-101):
		tmp = ((-0.5 * (math.pi / (b - a))) / b) / a
	else:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.1e-102) || !(a <= 1.7e-101))
		tmp = Float64(Float64(Float64(-0.5 * Float64(pi / Float64(b - a))) / b) / a);
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.1e-102) || ~((a <= 1.7e-101)))
		tmp = ((-0.5 * (pi / (b - a))) / b) / a;
	else
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.1e-102], N[Not[LessEqual[a, 1.7e-101]], $MachinePrecision]], N[(N[(N[(-0.5 * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.1e-102 or 1.69999999999999995e-101 < a

   1. Initial program 80.6%

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

      1. times-frac 80.7%

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

      2. *-commutative 80.7%

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

      3. times-frac 80.7%

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

      4. difference-of-squares 88.1%

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

      5. associate-/r* 89.1%

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

      6. metadata-eval 89.1%

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

      7. sub-neg 89.1%

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

      8. distribute-neg-frac 89.1%

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

      9. metadata-eval 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in a around inf 81.0%

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

   5. Taylor expanded in b around 0 80.9%

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

      1. expm1-log1p-u 68.3%

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

      2. expm1-udef 58.2%

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

      3. associate-*l* 58.2%

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

      4. associate-/l/ 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. expm1-def 67.3%

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

      2. expm1-log1p 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/r* 80.9%

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

      5. associate-*r/ 91.4%

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

      6. associate-*r/ 91.4%

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

      7. metadata-eval 91.4%

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

   9. Simplified 91.4%

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

       1. associate-*l/ 91.5%

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

   11. Applied egg-rr 91.5%

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

   if -2.1e-102 < a < 1.69999999999999995e-101

   1. Initial program 79.0%

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

      1. times-frac 79.1%

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

      2. *-commutative 79.1%

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

      3. times-frac 79.1%

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

      4. difference-of-squares 86.5%

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

      5. associate-/r* 86.6%

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

      6. metadata-eval 86.6%

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

      7. sub-neg 86.6%

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

      8. distribute-neg-frac 86.6%

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

      9. metadata-eval 86.6%

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

   3. Simplified 86.6%

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

      1. frac-add 86.5%

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

      2. *-un-lft-identity 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. neg-mul-1 86.5%

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

      3. sub-neg 86.5%

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

   7. Simplified 86.5%

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

      1. expm1-log1p-u 53.5%

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

      2. expm1-udef 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. expm1-def 53.5%

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

       2. expm1-log1p 86.5%

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

       3. *-commutative 86.5%

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

       4. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

       1. associate-*l/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in a around 0 88.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-102} \lor \neg \left(a \leq 1.7 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{\frac{-0.5 \cdot \frac{\pi}{b - a}}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]

Alternative 9: 85.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{b - a}\\ \mathbf{if}\;a \leq -1.86 \cdot 10^{-104}:\\ \;\;\;\;\frac{t_0 \cdot \frac{-0.5}{b}}{a}\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{-0.5}{b}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (- b a))))
   (if (<= a -1.86e-104)
     (/ (* t_0 (/ -0.5 b)) a)
     (if (<= a 6.6e-105)
       (/ (* 0.5 (/ PI (* a b))) (- b a))
       (* t_0 (/ (/ -0.5 b) a))))))

C

double code(double a, double b) {
	double t_0 = ((double) M_PI) / (b - a);
	double tmp;
	if (a <= -1.86e-104) {
		tmp = (t_0 * (-0.5 / b)) / a;
	} else if (a <= 6.6e-105) {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	} else {
		tmp = t_0 * ((-0.5 / b) / a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.PI / (b - a);
	double tmp;
	if (a <= -1.86e-104) {
		tmp = (t_0 * (-0.5 / b)) / a;
	} else if (a <= 6.6e-105) {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	} else {
		tmp = t_0 * ((-0.5 / b) / a);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pi / (b - a)
	tmp = 0
	if a <= -1.86e-104:
		tmp = (t_0 * (-0.5 / b)) / a
	elif a <= 6.6e-105:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	else:
		tmp = t_0 * ((-0.5 / b) / a)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(pi / Float64(b - a))
	tmp = 0.0
	if (a <= -1.86e-104)
		tmp = Float64(Float64(t_0 * Float64(-0.5 / b)) / a);
	elseif (a <= 6.6e-105)
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(t_0 * Float64(Float64(-0.5 / b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = pi / (b - a);
	tmp = 0.0;
	if (a <= -1.86e-104)
		tmp = (t_0 * (-0.5 / b)) / a;
	elseif (a <= 6.6e-105)
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	else
		tmp = t_0 * ((-0.5 / b) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.86e-104], N[(N[(t$95$0 * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, 6.6e-105], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(-0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8599999999999999e-104

   1. Initial program 76.8%

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

      1. times-frac 76.9%

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

      2. *-commutative 76.9%

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

      3. times-frac 76.9%

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

      4. difference-of-squares 85.0%

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

      5. associate-/r* 86.1%

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

      6. metadata-eval 86.1%

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

      7. sub-neg 86.1%

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

      8. distribute-neg-frac 86.1%

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

      9. metadata-eval 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in a around inf 78.5%

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

   5. Taylor expanded in b around 0 78.3%

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

      1. expm1-log1p-u 64.9%

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

      2. expm1-udef 52.4%

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

      3. associate-*l* 52.4%

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

      4. associate-/l/ 52.4%

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

   7. Applied egg-rr 52.4%

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

      1. expm1-def 63.8%

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

      2. expm1-log1p 77.3%

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

      3. *-commutative 77.3%

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

      4. associate-/r* 78.3%

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

      5. associate-*r/ 91.9%

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

      6. associate-*r/ 91.9%

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

      7. metadata-eval 91.9%

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

   9. Simplified 91.9%

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

   if -1.8599999999999999e-104 < a < 6.5999999999999997e-105

   1. Initial program 79.0%

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

      1. times-frac 79.1%

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

      2. *-commutative 79.1%

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

      3. times-frac 79.1%

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

      4. difference-of-squares 86.5%

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

      5. associate-/r* 86.6%

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

      6. metadata-eval 86.6%

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

      7. sub-neg 86.6%

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

      8. distribute-neg-frac 86.6%

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

      9. metadata-eval 86.6%

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

   3. Simplified 86.6%

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

      1. frac-add 86.5%

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

      2. *-un-lft-identity 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. neg-mul-1 86.5%

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

      3. sub-neg 86.5%

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

   7. Simplified 86.5%

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

      1. expm1-log1p-u 53.5%

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

      2. expm1-udef 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. expm1-def 53.5%

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

       2. expm1-log1p 86.5%

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

       3. *-commutative 86.5%

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

       4. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

       1. associate-*l/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in a around 0 88.4%

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

   if 6.5999999999999997e-105 < a

   1. Initial program 84.2%

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

      1. times-frac 84.3%

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

      2. *-commutative 84.3%

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

      3. times-frac 84.3%

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

      4. difference-of-squares 91.0%

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

      5. associate-/r* 92.0%

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

      6. metadata-eval 92.0%

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

      7. sub-neg 92.0%

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

      8. distribute-neg-frac 92.0%

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

      9. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in a around inf 83.5%

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

   5. Taylor expanded in b around 0 83.4%

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

      1. expm1-log1p-u 71.6%

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

      2. expm1-udef 63.8%

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

      3. associate-*l* 63.8%

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

      4. associate-/l/ 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. expm1-def 70.6%

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

      2. expm1-log1p 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-/r* 83.4%

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

      5. associate-*r/ 91.0%

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

      6. associate-*r/ 91.0%

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

      7. metadata-eval 91.0%

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

   9. Simplified 91.0%

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

       1. expm1-log1p-u 79.2%

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

       2. expm1-udef 63.8%

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

       3. associate-/l* 63.8%

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

   11. Applied egg-rr 63.8%

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

       1. expm1-def 71.6%

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

       2. expm1-log1p 83.5%

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

       3. associate-/r/ 91.1%

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

   13. Simplified 91.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.86 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{\pi}{b - a} \cdot \frac{-0.5}{b}}{a}\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{\frac{-0.5}{b}}{a}\\ \end{array} \]

Alternative 10: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-65} \lor \neg \left(a \leq 1.12 \cdot 10^{-38}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.2e-65) (not (<= a 1.12e-38)))
   (* 0.5 (/ PI (* a (* a b))))
   (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.2e-65) || !(a <= 1.12e-38)) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.2e-65) || !(a <= 1.12e-38)) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.2e-65) or not (a <= 1.12e-38):
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.2e-65) || !(a <= 1.12e-38))
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.2e-65) || ~((a <= 1.12e-38)))
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.2e-65], N[Not[LessEqual[a, 1.12e-38]], $MachinePrecision]], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.20000000000000021e-65 or 1.1200000000000001e-38 < a

   1. Initial program 78.1%

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

      1. times-frac 78.2%

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

      2. *-commutative 78.2%

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

      3. times-frac 78.2%

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

      4. difference-of-squares 86.6%

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

      5. associate-/r* 87.7%

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

      6. metadata-eval 87.7%

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

      7. sub-neg 87.7%

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

      8. distribute-neg-frac 87.7%

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

      9. metadata-eval 87.7%

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

   3. Simplified 87.7%

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

      1. frac-add 87.7%

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

      2. *-un-lft-identity 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. *-commutative 87.7%

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

      2. neg-mul-1 87.7%

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

      3. sub-neg 87.7%

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

   7. Simplified 87.7%

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

      1. expm1-log1p-u 76.1%

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

      2. expm1-udef 58.6%

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

   9. Applied egg-rr 58.6%

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

       1. expm1-def 76.1%

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

       2. expm1-log1p 87.7%

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

       3. *-commutative 87.7%

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

       4. associate-*l* 87.7%

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

   11. Simplified 87.7%

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

   12. Taylor expanded in a around inf 77.6%

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

       1. unpow2 77.6%

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

       2. associate-*l* 89.6%

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

   14. Simplified 89.6%

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

   if -2.20000000000000021e-65 < a < 1.1200000000000001e-38

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. associate-/r/ 83.2%

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

      3. associate-*l/ 83.1%

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

      4. *-commutative 83.1%

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

      5. associate-/r/ 83.1%

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

      6. times-frac 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in b around inf 80.2%

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

      1. unpow2 80.2%

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

   6. Simplified 80.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-65} \lor \neg \left(a \leq 1.12 \cdot 10^{-38}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 11: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-65} \lor \neg \left(a \leq 1.55 \cdot 10^{-38}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.2e-65) (not (<= a 1.55e-38)))
   (* 0.5 (/ PI (* a (* a b))))
   (* (/ 0.5 a) (/ PI (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4.2e-65) || !(a <= 1.55e-38)) {
		tmp = 0.5 * (((double) M_PI) / (a * (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 <= -4.2e-65) || !(a <= 1.55e-38)) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = (0.5 / a) * (Math.PI / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -4.2e-65) or not (a <= 1.55e-38):
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = (0.5 / a) * (math.pi / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -4.2e-65) || !(a <= 1.55e-38))
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -4.2e-65) || ~((a <= 1.55e-38)))
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = (0.5 / a) * (pi / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -4.2e-65], N[Not[LessEqual[a, 1.55e-38]], $MachinePrecision]], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.20000000000000006e-65 or 1.54999999999999991e-38 < a

   1. Initial program 78.1%

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

      1. times-frac 78.2%

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

      2. *-commutative 78.2%

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

      3. times-frac 78.2%

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

      4. difference-of-squares 86.6%

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

      5. associate-/r* 87.7%

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

      6. metadata-eval 87.7%

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

      7. sub-neg 87.7%

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

      8. distribute-neg-frac 87.7%

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

      9. metadata-eval 87.7%

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

   3. Simplified 87.7%

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

      1. frac-add 87.7%

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

      2. *-un-lft-identity 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. *-commutative 87.7%

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

      2. neg-mul-1 87.7%

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

      3. sub-neg 87.7%

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

   7. Simplified 87.7%

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

      1. expm1-log1p-u 76.1%

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

      2. expm1-udef 58.6%

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

   9. Applied egg-rr 58.6%

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

       1. expm1-def 76.1%

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

       2. expm1-log1p 87.7%

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

       3. *-commutative 87.7%

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

       4. associate-*l* 87.7%

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

   11. Simplified 87.7%

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

   12. Taylor expanded in a around inf 77.6%

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

       1. unpow2 77.6%

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

       2. associate-*l* 89.6%

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

   14. Simplified 89.6%

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

   if -4.20000000000000006e-65 < a < 1.54999999999999991e-38

   1. Initial program 83.1%

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

      1. times-frac 83.2%

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

      2. *-commutative 83.2%

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

      3. times-frac 83.2%

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

      4. difference-of-squares 89.1%

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

      5. associate-/r* 89.2%

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

      6. metadata-eval 89.2%

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

      7. sub-neg 89.2%

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

      8. distribute-neg-frac 89.2%

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

      9. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

      1. frac-add 89.1%

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

      2. *-un-lft-identity 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

   5. Applied egg-rr 89.1%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
   6. Step-by-step derivation

      1. *-commutative 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

      2. neg-mul-1 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

      3. sub-neg 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

   7. Simplified 89.1%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 57.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b - a}{a \cdot b}\right)\right)} \]

      2. expm1-udef 50.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b - a}{a \cdot b}\right)} - 1} \]

   9. Applied egg-rr 50.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 57.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}\right)\right)} \]

       2. expm1-log1p 89.1%

          \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}} \]

       3. *-commutative 89.1%

          \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{a + b}}{b - a} \cdot 0.5\right)} \cdot \frac{b - a}{a \cdot b} \]

       4. associate-*l* 89.3%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{a + b}}{b - a} \cdot \left(0.5 \cdot \frac{b - a}{a \cdot b}\right)} \]

   11. Simplified 89.3%

       \[\leadsto \color{blue}{\frac{\frac{\pi}{a + b}}{b - a} \cdot \left(0.5 \cdot \frac{b - a}{a \cdot b}\right)} \]

   12. Taylor expanded in a around 0 80.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   13. Step-by-step derivation

       1. associate-*r/ 80.2%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

       2. *-commutative 80.2%

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

       3. *-commutative 80.2%

          \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{{b}^{2} \cdot a}} \]

       4. times-frac 80.2%

          \[\leadsto \color{blue}{\frac{\pi}{{b}^{2}} \cdot \frac{0.5}{a}} \]

       5. unpow2 80.2%

          \[\leadsto \frac{\pi}{\color{blue}{b \cdot b}} \cdot \frac{0.5}{a} \]

   14. Simplified 80.2%

       \[\leadsto \color{blue}{\frac{\pi}{b \cdot b} \cdot \frac{0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-65} \lor \neg \left(a \leq 1.55 \cdot 10^{-38}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \]

Alternative 12: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-65} \lor \neg \left(a \leq 9.4 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.2e-65) (not (<= a 9.4e-39)))
   (* (/ 0.5 a) (/ (/ PI b) a))
   (* (/ 0.5 a) (/ PI (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4.2e-65) || !(a <= 9.4e-39)) {
		tmp = (0.5 / a) * ((((double) M_PI) / b) / a);
	} 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 <= -4.2e-65) || !(a <= 9.4e-39)) {
		tmp = (0.5 / a) * ((Math.PI / b) / a);
	} else {
		tmp = (0.5 / a) * (Math.PI / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -4.2e-65) or not (a <= 9.4e-39):
		tmp = (0.5 / a) * ((math.pi / b) / a)
	else:
		tmp = (0.5 / a) * (math.pi / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -4.2e-65) || !(a <= 9.4e-39))
		tmp = Float64(Float64(0.5 / a) * Float64(Float64(pi / b) / a));
	else
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -4.2e-65) || ~((a <= 9.4e-39)))
		tmp = (0.5 / a) * ((pi / b) / a);
	else
		tmp = (0.5 / a) * (pi / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -4.2e-65], N[Not[LessEqual[a, 9.4e-39]], $MachinePrecision]], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.20000000000000006e-65 or 9.4000000000000005e-39 < a

   1. Initial program 78.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 78.2%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 78.2%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 78.2%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 86.6%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 87.7%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 87.7%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 87.7%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 87.7%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 87.7%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. clear-num 87.7%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. inv-pow 87.7%

         \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   5. Applied egg-rr 87.7%

      \[\leadsto \left(\frac{\color{blue}{{\left(\frac{b + a}{\pi}\right)}^{-1}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]
   6. Step-by-step derivation

      1. unpow-1 87.7%

         \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{b + a}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

      2. +-commutative 87.7%

         \[\leadsto \left(\frac{\frac{1}{\frac{\color{blue}{a + b}}{\pi}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   7. Simplified 87.7%

      \[\leadsto \left(\frac{\color{blue}{\frac{1}{\frac{a + b}{\pi}}}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right) \]

   8. Taylor expanded in a around inf 77.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
   9. Step-by-step derivation

      1. associate-*r/ 77.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

      3. *-commutative 77.6%

         \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{b \cdot {a}^{2}}} \]

      4. times-frac 77.7%

         \[\leadsto \color{blue}{\frac{\pi}{b} \cdot \frac{0.5}{{a}^{2}}} \]

      5. unpow2 77.7%

         \[\leadsto \frac{\pi}{b} \cdot \frac{0.5}{\color{blue}{a \cdot a}} \]

   10. Simplified 77.7%

       \[\leadsto \color{blue}{\frac{\pi}{b} \cdot \frac{0.5}{a \cdot a}} \]
   11. Step-by-step derivation

       1. associate-*r/ 77.7%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{b} \cdot 0.5}{a \cdot a}} \]

   12. Applied egg-rr 77.7%

       \[\leadsto \color{blue}{\frac{\frac{\pi}{b} \cdot 0.5}{a \cdot a}} \]
   13. Step-by-step derivation

       1. times-frac 90.1%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{a}} \]

   14. Applied egg-rr 90.1%

       \[\leadsto \color{blue}{\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{a}} \]

   if -4.20000000000000006e-65 < a < 9.4000000000000005e-39

   1. Initial program 83.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 83.2%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 83.2%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 83.2%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 89.1%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 89.2%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 89.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 89.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 89.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 89.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. frac-add 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

      2. *-un-lft-identity 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

   5. Applied egg-rr 89.1%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
   6. Step-by-step derivation

      1. *-commutative 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

      2. neg-mul-1 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

      3. sub-neg 89.1%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

   7. Simplified 89.1%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 57.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b - a}{a \cdot b}\right)\right)} \]

      2. expm1-udef 50.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b - a}{a \cdot b}\right)} - 1} \]

   9. Applied egg-rr 50.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 57.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}\right)\right)} \]

       2. expm1-log1p 89.1%

          \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}} \]

       3. *-commutative 89.1%

          \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{a + b}}{b - a} \cdot 0.5\right)} \cdot \frac{b - a}{a \cdot b} \]

       4. associate-*l* 89.3%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{a + b}}{b - a} \cdot \left(0.5 \cdot \frac{b - a}{a \cdot b}\right)} \]

   11. Simplified 89.3%

       \[\leadsto \color{blue}{\frac{\frac{\pi}{a + b}}{b - a} \cdot \left(0.5 \cdot \frac{b - a}{a \cdot b}\right)} \]

   12. Taylor expanded in a around 0 80.2%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   13. Step-by-step derivation

       1. associate-*r/ 80.2%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

       2. *-commutative 80.2%

          \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

       3. *-commutative 80.2%

          \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{{b}^{2} \cdot a}} \]

       4. times-frac 80.2%

          \[\leadsto \color{blue}{\frac{\pi}{{b}^{2}} \cdot \frac{0.5}{a}} \]

       5. unpow2 80.2%

          \[\leadsto \frac{\pi}{\color{blue}{b \cdot b}} \cdot \frac{0.5}{a} \]

   14. Simplified 80.2%

       \[\leadsto \color{blue}{\frac{\pi}{b \cdot b} \cdot \frac{0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-65} \lor \neg \left(a \leq 9.4 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\frac{\pi}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{b \cdot b}\\ \end{array} \]

Alternative 13: 63.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* 0.5 (/ PI (* a (* a b)))))

C

double code(double a, double b) {
	return 0.5 * (((double) M_PI) / (a * (a * b)));
}

Java

public static double code(double a, double b) {
	return 0.5 * (Math.PI / (a * (a * b)));
}

Python

def code(a, b):
	return 0.5 * (math.pi / (a * (a * b)))

Julia

function code(a, b)
	return Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 * (pi / (a * (a * b)));
end

Wolfram

code[a_, b_] := N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. times-frac 80.2%

      \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. *-commutative 80.2%

      \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. times-frac 80.2%

      \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. difference-of-squares 87.6%

      \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. associate-/r* 88.3%

      \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. metadata-eval 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. sub-neg 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

   8. distribute-neg-frac 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

   9. metadata-eval 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Simplified 88.3%

   \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
4. Step-by-step derivation

   1. frac-add 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

   2. *-un-lft-identity 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

5. Applied egg-rr 88.3%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
6. Step-by-step derivation

   1. *-commutative 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

   2. neg-mul-1 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

   3. sub-neg 88.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

7. Simplified 88.3%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 68.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b - a}{a \cdot b}\right)\right)} \]

   2. expm1-udef 55.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b - a}{a \cdot b}\right)} - 1} \]

9. Applied egg-rr 55.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 68.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}\right)\right)} \]

    2. expm1-log1p 88.3%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \frac{b - a}{a \cdot b}} \]

    3. *-commutative 88.3%

       \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{a + b}}{b - a} \cdot 0.5\right)} \cdot \frac{b - a}{a \cdot b} \]

    4. associate-*l* 88.3%

       \[\leadsto \color{blue}{\frac{\frac{\pi}{a + b}}{b - a} \cdot \left(0.5 \cdot \frac{b - a}{a \cdot b}\right)} \]

11. Simplified 88.3%

    \[\leadsto \color{blue}{\frac{\frac{\pi}{a + b}}{b - a} \cdot \left(0.5 \cdot \frac{b - a}{a \cdot b}\right)} \]

12. Taylor expanded in a around inf 57.8%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
13. Step-by-step derivation

    1. unpow2 57.8%

       \[\leadsto 0.5 \cdot \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]

    2. associate-*l* 65.1%

       \[\leadsto 0.5 \cdot \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \]

14. Simplified 65.1%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}} \]

15. Final simplification 65.1%

    \[\leadsto 0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)} \]

Reproduce

herbie shell --seed 2023171 
(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))))