NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.0%

Time: 10.2s

Alternatives: 12

Speedup: 1.0×

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: 78.3% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{a} + \frac{-1}{b}\\ t_1 := t_0 \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-273} \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}{\frac{1}{a} + \frac{1}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 a) (/ -1.0 b)))
        (t_1 (* t_0 (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))))))
   (if (or (<= t_1 -5e-273) (not (<= t_1 0.0)))
     (* t_0 (* 0.5 (/ (/ PI (+ a b)) (- b a))))
     (/ (* 0.5 (/ PI (* (* a b) (* a b)))) (+ (/ 1.0 a) (/ 1.0 b))))))

C

double code(double a, double b) {
	double t_0 = (1.0 / a) + (-1.0 / b);
	double t_1 = t_0 * ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a))));
	double tmp;
	if ((t_1 <= -5e-273) || !(t_1 <= 0.0)) {
		tmp = t_0 * (0.5 * ((((double) M_PI) / (a + b)) / (b - a)));
	} else {
		tmp = (0.5 * (((double) M_PI) / ((a * b) * (a * b)))) / ((1.0 / a) + (1.0 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (1.0 / a) + (-1.0 / b);
	double t_1 = t_0 * ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a))));
	double tmp;
	if ((t_1 <= -5e-273) || !(t_1 <= 0.0)) {
		tmp = t_0 * (0.5 * ((Math.PI / (a + b)) / (b - a)));
	} else {
		tmp = (0.5 * (Math.PI / ((a * b) * (a * b)))) / ((1.0 / a) + (1.0 / b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (1.0 / a) + (-1.0 / b)
	t_1 = t_0 * ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a))))
	tmp = 0
	if (t_1 <= -5e-273) or not (t_1 <= 0.0):
		tmp = t_0 * (0.5 * ((math.pi / (a + b)) / (b - a)))
	else:
		tmp = (0.5 * (math.pi / ((a * b) * (a * b)))) / ((1.0 / a) + (1.0 / b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(1.0 / a) + Float64(-1.0 / b))
	t_1 = Float64(t_0 * Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))))
	tmp = 0.0
	if ((t_1 <= -5e-273) || !(t_1 <= 0.0))
		tmp = Float64(t_0 * Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(Float64(a * b) * Float64(a * b)))) / Float64(Float64(1.0 / a) + Float64(1.0 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (1.0 / a) + (-1.0 / b);
	t_1 = t_0 * ((pi / 2.0) * (1.0 / ((b * b) - (a * a))));
	tmp = 0.0;
	if ((t_1 <= -5e-273) || ~((t_1 <= 0.0)))
		tmp = t_0 * (0.5 * ((pi / (a + b)) / (b - a)));
	else
		tmp = (0.5 * (pi / ((a * b) * (a * b)))) / ((1.0 / a) + (1.0 / b));
	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]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-273], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (/.f64 (PI.f64) 2) (/.f64 1 (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 1 a) (/.f64 1 b))) < -4.99999999999999965e-273 or 0.0 < (*.f64 (*.f64 (/.f64 (PI.f64) 2) (/.f64 1 (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 1 a) (/.f64 1 b)))

   1. Initial program 85.3%

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

      1. times-frac 84.7%

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

      2. *-commutative 84.7%

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

      3. times-frac 85.4%

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

      4. difference-of-squares 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%

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

   if -4.99999999999999965e-273 < (*.f64 (*.f64 (/.f64 (PI.f64) 2) (/.f64 1 (-.f64 (*.f64 b b) (*.f64 a a)))) (-.f64 (/.f64 1 a) (/.f64 1 b))) < 0.0

   1. Initial program 73.4%

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

   2. Taylor expanded in b around 0 40.3%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{\pi}{{a}^{2}}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   3. Step-by-step derivation

      1. unpow2 40.3%

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

   4. Simplified 40.3%

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

      1. flip-- 33.4%

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

      2. div-inv 33.4%

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

      3. associate-/l/ 33.4%

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

   6. Applied egg-rr 33.4%

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

      1. associate-*r/ 33.4%

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

      2. *-rgt-identity 33.4%

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

      3. associate-/r* 33.4%

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

   8. Simplified 33.4%

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

      1. associate-*r/ 35.1%

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

      2. *-commutative 35.1%

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

      3. associate-/l/ 35.1%

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

      4. associate-/l/ 35.1%

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

   10. Applied egg-rr 35.1%

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

   11. Taylor expanded in a around inf 65.4%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot {b}^{2}}}}{\frac{1}{a} + \frac{1}{b}} \]
   12. Step-by-step derivation

       1. unpow2 65.4%

          \[\leadsto \frac{0.5 \cdot \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}}}{\frac{1}{a} + \frac{1}{b}} \]

       2. unpow2 65.4%

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

       3. unswap-sqr 99.8%

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

   13. Simplified 99.8%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}}{\frac{1}{a} + \frac{1}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \leq -5 \cdot 10^{-273} \lor \neg \left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \leq 0\right):\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}{\frac{1}{a} + \frac{1}{b}}\\ \end{array} \]

Alternative 2: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{0.5}{a} + \frac{-0.5}{b}\right) \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.25e+154)
   (* 0.5 (* PI (/ 1.0 (* a (* a b)))))
   (if (<= a -4.8e-155)
     (* (+ (/ 0.5 a) (/ -0.5 b)) (/ PI (- (* b b) (* a a))))
     (* (/ 1.0 a) (* 0.5 (/ (/ PI (+ a b)) (- b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.25e+154) {
		tmp = 0.5 * (((double) M_PI) * (1.0 / (a * (a * b))));
	} else if (a <= -4.8e-155) {
		tmp = ((0.5 / a) + (-0.5 / b)) * (((double) M_PI) / ((b * b) - (a * a)));
	} else {
		tmp = (1.0 / a) * (0.5 * ((((double) M_PI) / (a + b)) / (b - a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.25e+154) {
		tmp = 0.5 * (Math.PI * (1.0 / (a * (a * b))));
	} else if (a <= -4.8e-155) {
		tmp = ((0.5 / a) + (-0.5 / b)) * (Math.PI / ((b * b) - (a * a)));
	} else {
		tmp = (1.0 / a) * (0.5 * ((Math.PI / (a + b)) / (b - a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.25e+154:
		tmp = 0.5 * (math.pi * (1.0 / (a * (a * b))))
	elif a <= -4.8e-155:
		tmp = ((0.5 / a) + (-0.5 / b)) * (math.pi / ((b * b) - (a * a)))
	else:
		tmp = (1.0 / a) * (0.5 * ((math.pi / (a + b)) / (b - a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.25e+154)
		tmp = Float64(0.5 * Float64(pi * Float64(1.0 / Float64(a * Float64(a * b)))));
	elseif (a <= -4.8e-155)
		tmp = Float64(Float64(Float64(0.5 / a) + Float64(-0.5 / b)) * Float64(pi / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(1.0 / a) * Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.25e+154)
		tmp = 0.5 * (pi * (1.0 / (a * (a * b))));
	elseif (a <= -4.8e-155)
		tmp = ((0.5 / a) + (-0.5 / b)) * (pi / ((b * b) - (a * a)));
	else
		tmp = (1.0 / a) * (0.5 * ((pi / (a + b)) / (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.25e+154], N[(0.5 * N[(Pi * N[(1.0 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -4.8e-155], N[(N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] * N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.25000000000000001e154

   1. Initial program 58.2%

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

      1. *-commutative 58.2%

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

      2. associate-/r/ 58.2%

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

      3. associate-*l/ 58.2%

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

      4. *-commutative 58.2%

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

      5. associate-/r/ 58.2%

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

      6. times-frac 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in b around 0 58.2%

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

      1. associate-*r/ 58.2%

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

      2. mul-1-neg 58.2%

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

   6. Simplified 58.2%

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

   7. Taylor expanded in b around 0 83.2%

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

      1. unpow2 83.2%

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

      2. associate-*l* 99.9%

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

   9. Simplified 99.9%

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

       1. div-inv 99.9%

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

       2. *-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

   if -1.25000000000000001e154 < a < -4.8e-155

   1. Initial program 98.3%

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

      1. times-frac 97.0%

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

      2. *-commutative 97.0%

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

      3. times-frac 98.4%

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

      4. difference-of-squares 98.4%

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

      5. associate-/r* 98.3%

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

      6. metadata-eval 98.3%

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

      7. sub-neg 98.3%

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

      8. distribute-neg-frac 98.3%

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

      9. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. distribute-lft-in 93.9%

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

      2. associate-/l/ 93.9%

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

      3. associate-/l/ 93.9%

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

   5. Applied egg-rr 93.9%

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

      1. distribute-lft-out 98.4%

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

      2. associate-*r* 98.4%

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

      3. associate-*l/ 98.3%

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

      4. *-commutative 98.3%

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

      5. difference-of-squares 98.3%

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

      6. associate-*l/ 98.4%

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

      7. distribute-lft-in 98.4%

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

      8. associate-*r/ 98.4%

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

      9. metadata-eval 98.4%

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

      10. associate-*r/ 98.4%

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

      11. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

   if -4.8e-155 < a

   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 76.4%

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

      2. *-commutative 76.4%

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

      3. times-frac 76.4%

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

      4. difference-of-squares 85.2%

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

      5. associate-/r* 85.8%

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

      6. metadata-eval 85.8%

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

      7. sub-neg 85.8%

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

      8. distribute-neg-frac 85.8%

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

      9. metadata-eval 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in a around 0 65.5%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-155}:\\ \;\;\;\;\left(\frac{0.5}{a} + \frac{-0.5}{b}\right) \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \end{array} \]

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e+114)
   (* 0.5 (* PI (/ 1.0 (* a (* a b)))))
   (* (+ (/ 1.0 a) (/ -1.0 b)) (* 0.5 (/ (/ PI (+ a b)) (- b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e+114) {
		tmp = 0.5 * (((double) M_PI) * (1.0 / (a * (a * b))));
	} else {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((((double) M_PI) / (a + b)) / (b - a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.2e+114) {
		tmp = 0.5 * (Math.PI * (1.0 / (a * (a * b))));
	} else {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((Math.PI / (a + b)) / (b - a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.2e+114:
		tmp = 0.5 * (math.pi * (1.0 / (a * (a * b))))
	else:
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((math.pi / (a + b)) / (b - a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e+114)
		tmp = Float64(0.5 * Float64(pi * Float64(1.0 / Float64(a * Float64(a * b)))));
	else
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.2e+114)
		tmp = 0.5 * (pi * (1.0 / (a * (a * b))));
	else
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((pi / (a + b)) / (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e+114], N[(0.5 * N[(Pi * N[(1.0 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.2e114

   1. Initial program 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-/r/ 65.6%

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

      3. associate-*l/ 65.6%

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

      4. *-commutative 65.6%

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

      5. associate-/r/ 65.6%

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

      6. times-frac 65.6%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in b around 0 65.5%

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

      1. associate-*r/ 65.5%

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

      2. mul-1-neg 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in b around 0 86.1%

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

      1. unpow2 86.1%

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

      2. associate-*l* 99.9%

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

   9. Simplified 99.9%

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

       1. div-inv 99.9%

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

       2. *-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

   if -1.2e114 < a

   1. Initial program 82.5%

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

      1. times-frac 82.5%

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

      2. *-commutative 82.5%

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

      3. times-frac 82.5%

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

      4. difference-of-squares 88.8%

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

      5. associate-/r* 89.3%

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

      6. metadata-eval 89.3%

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

      7. sub-neg 89.3%

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

      8. distribute-neg-frac 89.3%

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

      9. metadata-eval 89.3%

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

   3. Simplified 89.3%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \end{array} \]

Alternative 4: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} + \frac{-0.5}{b}}{\left(a + b\right) \cdot \frac{b - a}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e+93)
   (* 0.5 (* PI (/ 1.0 (* a (* a b)))))
   (/ (+ (/ 0.5 a) (/ -0.5 b)) (* (+ a b) (/ (- b a) PI)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e+93) {
		tmp = 0.5 * (((double) M_PI) * (1.0 / (a * (a * b))));
	} else {
		tmp = ((0.5 / a) + (-0.5 / b)) / ((a + b) * ((b - a) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9e+93) {
		tmp = 0.5 * (Math.PI * (1.0 / (a * (a * b))));
	} else {
		tmp = ((0.5 / a) + (-0.5 / b)) / ((a + b) * ((b - a) / Math.PI));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9e+93:
		tmp = 0.5 * (math.pi * (1.0 / (a * (a * b))))
	else:
		tmp = ((0.5 / a) + (-0.5 / b)) / ((a + b) * ((b - a) / math.pi))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e+93)
		tmp = Float64(0.5 * Float64(pi * Float64(1.0 / Float64(a * Float64(a * b)))));
	else
		tmp = Float64(Float64(Float64(0.5 / a) + Float64(-0.5 / b)) / Float64(Float64(a + b) * Float64(Float64(b - a) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e+93)
		tmp = 0.5 * (pi * (1.0 / (a * (a * b))));
	else
		tmp = ((0.5 / a) + (-0.5 / b)) / ((a + b) * ((b - a) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e+93], N[(0.5 * N[(Pi * N[(1.0 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.99999999999999981e93

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

      2. associate-/r/ 69.1%

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

      3. associate-*l/ 69.1%

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

      4. *-commutative 69.1%

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

      5. associate-/r/ 69.1%

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

      6. times-frac 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in b around 0 69.1%

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

      1. associate-*r/ 69.1%

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

      2. mul-1-neg 69.1%

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

   6. Simplified 69.1%

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

   7. Taylor expanded in b around 0 87.5%

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

      1. unpow2 87.5%

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

      2. associate-*l* 99.8%

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

   9. Simplified 99.8%

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

       1. div-inv 99.8%

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

       2. *-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   if -8.99999999999999981e93 < a

   1. Initial program 82.2%

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

      1. times-frac 82.2%

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

      2. *-commutative 82.2%

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

      3. times-frac 82.2%

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

      4. difference-of-squares 88.6%

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

      5. associate-/r* 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. Step-by-step derivation

      1. clear-num 88.6%

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

      2. inv-pow 88.6%

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

   5. Applied egg-rr 88.6%

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

      1. distribute-lft-in 81.7%

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

      2. *-commutative 81.7%

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

      3. unpow-1 81.7%

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

      4. associate-/r/ 81.7%

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

      5. *-commutative 81.7%

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

      6. unpow-1 81.7%

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

      7. associate-/r/ 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. metadata-eval 81.7%

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

      2. distribute-neg-frac 81.7%

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

      3. distribute-lft-out 88.6%

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

      4. distribute-neg-frac 88.6%

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

      5. metadata-eval 88.6%

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

      6. *-commutative 88.6%

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

      7. metadata-eval 88.6%

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

      8. distribute-neg-frac 88.6%

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

      9. sub-neg 88.6%

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

      10. associate-*r/ 88.6%

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

      11. metadata-eval 88.6%

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

   9. Simplified 88.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} + \frac{-0.5}{b}}{\left(a + b\right) \cdot \frac{b - a}{\pi}}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi \cdot \left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right)}{a + b}}{b - a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (/ (* PI (* 0.5 (+ (/ 1.0 a) (/ -1.0 b)))) (+ a b)) (- b a)))

C

double code(double a, double b) {
	return ((((double) M_PI) * (0.5 * ((1.0 / a) + (-1.0 / b)))) / (a + b)) / (b - a);
}

Java

public static double code(double a, double b) {
	return ((Math.PI * (0.5 * ((1.0 / a) + (-1.0 / b)))) / (a + b)) / (b - a);
}

Python

def code(a, b):
	return ((math.pi * (0.5 * ((1.0 / a) + (-1.0 / b)))) / (a + b)) / (b - a)

Julia

function code(a, b)
	return Float64(Float64(Float64(pi * Float64(0.5 * Float64(Float64(1.0 / a) + Float64(-1.0 / b)))) / Float64(a + b)) / Float64(b - a))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi * (0.5 * ((1.0 / a) + (-1.0 / b)))) / (a + b)) / (b - a);
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi * N[(0.5 * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. times-frac 79.9%

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

   2. *-commutative 79.9%

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

   3. times-frac 80.3%

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

   4. difference-of-squares 88.5%

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

   5. associate-/r* 88.8%

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

   6. metadata-eval 88.8%

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

   7. sub-neg 88.8%

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

   8. distribute-neg-frac 88.8%

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

   9. metadata-eval 88.8%

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

3. Simplified 88.8%

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

   1. div-inv 88.8%

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

5. Applied egg-rr 88.8%

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

   1. pow1 88.8%

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

   2. associate-*l* 88.8%

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

   3. un-div-inv 88.9%

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

7. Applied egg-rr 88.9%

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

   1. associate-*l/ 99.7%

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

9. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(0.5 \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\right) \cdot \frac{\pi}{a + b}}{b - a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (* (* 0.5 (+ (/ 1.0 a) (/ -1.0 b))) (/ PI (+ a b))) (- b a)))

C

double code(double a, double b) {
	return ((0.5 * ((1.0 / a) + (-1.0 / b))) * (((double) M_PI) / (a + b))) / (b - a);
}

Java

public static double code(double a, double b) {
	return ((0.5 * ((1.0 / a) + (-1.0 / b))) * (Math.PI / (a + b))) / (b - a);
}

Python

def code(a, b):
	return ((0.5 * ((1.0 / a) + (-1.0 / b))) * (math.pi / (a + b))) / (b - a)

Julia

function code(a, b)
	return Float64(Float64(Float64(0.5 * Float64(Float64(1.0 / a) + Float64(-1.0 / b))) * Float64(pi / Float64(a + b))) / Float64(b - a))
end

MATLAB

function tmp = code(a, b)
	tmp = ((0.5 * ((1.0 / a) + (-1.0 / b))) * (pi / (a + b))) / (b - a);
end

Wolfram

code[a_, b_] := N[(N[(N[(0.5 * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. times-frac 79.9%

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

   2. *-commutative 79.9%

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

   3. times-frac 80.3%

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

   4. difference-of-squares 88.5%

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

   5. associate-/r* 88.8%

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

   6. metadata-eval 88.8%

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

   7. sub-neg 88.8%

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

   8. distribute-neg-frac 88.8%

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

   9. metadata-eval 88.8%

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

3. Simplified 88.8%

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

   1. div-inv 88.8%

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

5. Applied egg-rr 88.8%

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

   1. pow1 88.8%

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

   2. associate-*l* 88.8%

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

   3. un-div-inv 88.9%

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

7. Applied egg-rr 88.9%

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

   1. associate-*l/ 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 7: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.14 \cdot 10^{-88}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.14e-88)
   (* 0.5 (* PI (/ 1.0 (* a (* a b)))))
   (* (/ 1.0 a) (* 0.5 (/ (/ PI (+ a b)) (- b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.14e-88) {
		tmp = 0.5 * (((double) M_PI) * (1.0 / (a * (a * b))));
	} else {
		tmp = (1.0 / a) * (0.5 * ((((double) M_PI) / (a + b)) / (b - a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.14e-88) {
		tmp = 0.5 * (Math.PI * (1.0 / (a * (a * b))));
	} else {
		tmp = (1.0 / a) * (0.5 * ((Math.PI / (a + b)) / (b - a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.14e-88:
		tmp = 0.5 * (math.pi * (1.0 / (a * (a * b))))
	else:
		tmp = (1.0 / a) * (0.5 * ((math.pi / (a + b)) / (b - a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.14e-88)
		tmp = Float64(0.5 * Float64(pi * Float64(1.0 / Float64(a * Float64(a * b)))));
	else
		tmp = Float64(Float64(1.0 / a) * Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.14e-88)
		tmp = 0.5 * (pi * (1.0 / (a * (a * b))));
	else
		tmp = (1.0 / a) * (0.5 * ((pi / (a + b)) / (b - a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.14e-88], N[(0.5 * N[(Pi * N[(1.0 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] * N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.14e-88

   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. *-commutative 79.0%

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

      2. associate-/r/ 79.0%

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

      3. associate-*l/ 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-/r/ 79.0%

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

      6. times-frac 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in b around 0 62.0%

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

      1. associate-*r/ 62.0%

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

      2. mul-1-neg 62.0%

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

   6. Simplified 62.0%

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

   7. Taylor expanded in b around 0 65.1%

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

      1. unpow2 65.1%

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

      2. associate-*l* 71.9%

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

   9. Simplified 71.9%

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

       1. div-inv 71.9%

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

       2. *-commutative 71.9%

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

   11. Applied egg-rr 71.9%

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

   if 1.14e-88 < b

   1. Initial program 82.7%

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

      1. times-frac 82.7%

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

      2. *-commutative 82.7%

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

      3. times-frac 82.7%

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

      4. difference-of-squares 87.2%

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

      5. associate-/r* 88.0%

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

      6. metadata-eval 88.0%

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

      7. sub-neg 88.0%

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

      8. distribute-neg-frac 88.0%

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

      9. metadata-eval 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in a around 0 73.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.14 \cdot 10^{-88}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \end{array} \]

Alternative 8: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00048:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.00048)
   (* 0.5 (* PI (/ 1.0 (* a (* a b)))))
   (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.00048) {
		tmp = 0.5 * (((double) M_PI) * (1.0 / (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 <= -0.00048) {
		tmp = 0.5 * (Math.PI * (1.0 / (a * (a * b))));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.00048:
		tmp = 0.5 * (math.pi * (1.0 / (a * (a * b))))
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.00048)
		tmp = Float64(0.5 * Float64(pi * Float64(1.0 / 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 <= -0.00048)
		tmp = 0.5 * (pi * (1.0 / (a * (a * b))));
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.00048], N[(0.5 * N[(Pi * N[(1.0 / N[(a * N[(a * b), $MachinePrecision]), $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 < -4.80000000000000012e-4

   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. *-commutative 80.6%

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

      2. associate-/r/ 80.5%

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

      3. associate-*l/ 80.5%

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

      4. *-commutative 80.5%

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

      5. associate-/r/ 80.5%

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

      6. times-frac 80.5%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 75.1%

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

      1. associate-*r/ 75.1%

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

      2. mul-1-neg 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in b around 0 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*l* 89.7%

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

   9. Simplified 89.7%

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

       1. div-inv 89.7%

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

       2. *-commutative 89.7%

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

   11. Applied egg-rr 89.7%

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

   if -4.80000000000000012e-4 < a

   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. *-commutative 80.1%

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

      2. associate-/r/ 80.1%

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

      3. associate-*l/ 80.1%

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

      4. *-commutative 80.1%

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

      5. associate-/r/ 80.1%

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

      6. times-frac 80.1%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in b around inf 61.7%

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

      1. unpow2 61.7%

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

   6. Simplified 61.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00048:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 9: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00039:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{\frac{\pi}{b}}{b \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.00039)
   (* 0.5 (* PI (/ 1.0 (* a (* a b)))))
   (* (/ 1.0 a) (/ (/ PI b) (* b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.00039) {
		tmp = 0.5 * (((double) M_PI) * (1.0 / (a * (a * b))));
	} else {
		tmp = (1.0 / a) * ((((double) M_PI) / b) / (b * 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.00039) {
		tmp = 0.5 * (Math.PI * (1.0 / (a * (a * b))));
	} else {
		tmp = (1.0 / a) * ((Math.PI / b) / (b * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.00039:
		tmp = 0.5 * (math.pi * (1.0 / (a * (a * b))))
	else:
		tmp = (1.0 / a) * ((math.pi / b) / (b * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.00039)
		tmp = Float64(0.5 * Float64(pi * Float64(1.0 / Float64(a * Float64(a * b)))));
	else
		tmp = Float64(Float64(1.0 / a) * Float64(Float64(pi / b) / Float64(b * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.00039)
		tmp = 0.5 * (pi * (1.0 / (a * (a * b))));
	else
		tmp = (1.0 / a) * ((pi / b) / (b * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.00039], N[(0.5 * N[(Pi * N[(1.0 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.89999999999999993e-4

   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. *-commutative 80.6%

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

      2. associate-/r/ 80.5%

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

      3. associate-*l/ 80.5%

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

      4. *-commutative 80.5%

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

      5. associate-/r/ 80.5%

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

      6. times-frac 80.5%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 75.1%

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

      1. associate-*r/ 75.1%

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

      2. mul-1-neg 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in b around 0 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*l* 89.7%

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

   9. Simplified 89.7%

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

       1. div-inv 89.7%

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

       2. *-commutative 89.7%

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

   11. Applied egg-rr 89.7%

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

   if -3.89999999999999993e-4 < a

   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. Taylor expanded in b around inf 54.3%

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

      1. unpow2 54.3%

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

      2. associate-/r* 54.6%

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

   4. Simplified 54.6%

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

   5. Taylor expanded in a around 0 62.0%

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

      1. frac-times 62.0%

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

   7. Applied egg-rr 62.0%

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

      1. associate-*r/ 62.0%

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

      2. *-rgt-identity 62.0%

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

   9. Simplified 62.0%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{b}}{2 \cdot b}} \cdot \frac{1}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00039:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{1}{a \cdot \left(a \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{\frac{\pi}{b}}{b \cdot 2}\\ \end{array} \]

Alternative 10: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.8e-8) (* 0.5 (/ PI (* a (* a b)))) (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.8e-8) {
		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 <= -4.8e-8) {
		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 <= -4.8e-8:
		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 <= -4.8e-8)
		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 <= -4.8e-8)
		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[LessEqual[a, -4.8e-8], 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 < -4.79999999999999997e-8

   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. *-commutative 80.6%

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

      2. associate-/r/ 80.5%

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

      3. associate-*l/ 80.5%

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

      4. *-commutative 80.5%

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

      5. associate-/r/ 80.5%

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

      6. times-frac 80.5%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 75.1%

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

      1. associate-*r/ 75.1%

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

      2. mul-1-neg 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in b around 0 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*l* 89.7%

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

   9. Simplified 89.7%

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

   if -4.79999999999999997e-8 < a

   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. *-commutative 80.1%

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

      2. associate-/r/ 80.1%

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

      3. associate-*l/ 80.1%

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

      4. *-commutative 80.1%

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

      5. associate-/r/ 80.1%

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

      6. times-frac 80.1%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in b around inf 61.7%

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

      1. unpow2 61.7%

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

   6. Simplified 61.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 11: 56.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. times-frac 79.9%

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

   2. *-commutative 79.9%

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

   3. times-frac 80.3%

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

   4. difference-of-squares 88.5%

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

   5. associate-/r* 88.8%

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

   6. metadata-eval 88.8%

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

   7. sub-neg 88.8%

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

   8. distribute-neg-frac 88.8%

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

   9. metadata-eval 88.8%

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

3. Simplified 88.8%

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

   1. div-inv 88.8%

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

5. Applied egg-rr 88.8%

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

6. Taylor expanded in b around 0 56.8%

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

   1. associate-*r/ 56.8%

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

   2. *-commutative 56.8%

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

   3. times-frac 56.8%

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

   4. unpow2 56.8%

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

8. Simplified 56.8%

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

9. Final simplification 56.8%

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

Alternative 12: 62.6% 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 80.2%

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

   1. *-commutative 80.2%

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

   2. associate-/r/ 80.2%

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

   3. associate-*l/ 80.2%

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

   4. *-commutative 80.2%

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

   5. associate-/r/ 80.2%

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

   6. times-frac 80.2%

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

3. Simplified 80.3%

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

4. Taylor expanded in b around 0 59.6%

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

   1. associate-*r/ 59.6%

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

   2. mul-1-neg 59.6%

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

6. Simplified 59.6%

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

7. Taylor expanded in b around 0 56.8%

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

   1. unpow2 56.8%

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

   2. associate-*l* 61.2%

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

9. Simplified 61.2%

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

10. Final simplification 61.2%

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

Reproduce

herbie shell --seed 2023240 
(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))))