NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.6%

Time: 9.6s

Alternatives: 13

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: 78.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. times-frac 77.6%

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

   2. *-commutative 77.6%

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

   3. times-frac 77.6%

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

   4. difference-of-squares 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* 87.1%

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

   6. metadata-eval 87.1%

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

   7. sub-neg 87.1%

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

   8. distribute-neg-frac 87.1%

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

   9. metadata-eval 87.1%

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

3. Simplified 87.1%

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

   1. clear-num 87.1%

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

   2. inv-pow 87.1%

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

5. Applied egg-rr 87.1%

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

   1. unpow-1 87.1%

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

   2. +-commutative 87.1%

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

7. Simplified 87.1%

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

   1. distribute-lft-in 81.2%

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

   2. associate-/l/ 80.8%

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

   3. associate-/l/ 80.6%

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

9. Applied egg-rr 80.6%

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

    1. distribute-lft-out 86.5%

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

    2. associate-*r* 86.5%

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

    3. associate-*l/ 86.6%

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

    4. *-lft-identity 86.6%

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

11. Simplified 86.6%

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

    1. *-un-lft-identity 86.6%

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

    2. times-frac 99.5%

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

13. Applied egg-rr 99.5%

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

    1. *-lft-identity 99.5%

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

    2. associate-/r/ 99.5%

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

15. Simplified 99.5%

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

    1. associate-*l/ 99.6%

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

17. Applied egg-rr 99.6%

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

18. Final simplification 99.6%

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

Alternative 2: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-175}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b - a} \cdot \left(\pi \cdot \frac{\frac{1}{a}}{a + b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.8e+123)
   (* (/ PI (* a b)) (/ 0.5 a))
   (if (<= a -1.65e-175)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (* 0.5 (/ (/ PI (+ a b)) (- b a))))
     (* (/ 0.5 (- b a)) (* PI (/ (/ 1.0 a) (+ a b)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.8e+123) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else if (a <= -1.65e-175) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((((double) M_PI) / (a + b)) / (b - a)));
	} else {
		tmp = (0.5 / (b - a)) * (((double) M_PI) * ((1.0 / a) / (a + b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.8e+123) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else if (a <= -1.65e-175) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((Math.PI / (a + b)) / (b - a)));
	} else {
		tmp = (0.5 / (b - a)) * (Math.PI * ((1.0 / a) / (a + b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.8e+123:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	elif a <= -1.65e-175:
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((math.pi / (a + b)) / (b - a)))
	else:
		tmp = (0.5 / (b - a)) * (math.pi * ((1.0 / a) / (a + b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.8e+123)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	elseif (a <= -1.65e-175)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))));
	else
		tmp = Float64(Float64(0.5 / Float64(b - a)) * Float64(pi * Float64(Float64(1.0 / a) / Float64(a + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.8e+123)
		tmp = (pi / (a * b)) * (0.5 / a);
	elseif (a <= -1.65e-175)
		tmp = ((1.0 / a) + (-1.0 / b)) * (0.5 * ((pi / (a + b)) / (b - a)));
	else
		tmp = (0.5 / (b - a)) * (pi * ((1.0 / a) / (a + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.8e+123], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.65e-175], 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], N[(N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(N[(1.0 / a), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.80000000000000011e123

   1. Initial program 55.8%

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

      1. times-frac 55.8%

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

      2. *-commutative 55.8%

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

      3. times-frac 55.8%

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

      4. difference-of-squares 72.1%

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

      5. associate-/r* 72.0%

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

      6. metadata-eval 72.0%

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

      7. sub-neg 72.0%

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

      8. distribute-neg-frac 72.0%

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

      9. metadata-eval 72.0%

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

   3. Simplified 72.0%

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

      1. clear-num 72.0%

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

      2. inv-pow 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. unpow-1 72.0%

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

      2. +-commutative 72.0%

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

   7. Simplified 72.0%

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

      1. distribute-lft-in 72.0%

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

      2. associate-/l/ 72.0%

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

      3. associate-/l/ 72.0%

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

   9. Applied egg-rr 72.0%

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

       1. distribute-lft-out 72.0%

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

       2. associate-*r* 72.0%

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

       3. associate-*l/ 72.0%

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

       4. *-lft-identity 72.0%

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

   11. Simplified 72.0%

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

       1. *-un-lft-identity 72.0%

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

       2. times-frac 99.7%

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

   13. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. associate-/r/ 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in b around 0 72.1%

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

       1. unpow2 72.1%

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

       2. associate-*r* 99.7%

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

       3. associate-*r/ 99.7%

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

       4. *-commutative 99.7%

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

       5. *-commutative 99.7%

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

       6. times-frac 99.9%

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

   18. Simplified 99.9%

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

   if -2.80000000000000011e123 < a < -1.64999999999999999e-175

   1. Initial program 94.9%

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

      1. times-frac 94.9%

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

      2. *-commutative 94.9%

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

      3. times-frac 94.9%

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

      4. difference-of-squares 95.0%

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

      5. associate-/r* 95.9%

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

      6. metadata-eval 95.9%

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

      7. sub-neg 95.9%

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

      8. distribute-neg-frac 95.9%

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

      9. metadata-eval 95.9%

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

   3. Simplified 95.9%

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

   if -1.64999999999999999e-175 < a

   1. Initial program 73.9%

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

      1. times-frac 74.0%

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

      2. *-commutative 74.0%

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

      3. times-frac 74.0%

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

      4. difference-of-squares 85.9%

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

      5. associate-/r* 86.4%

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

      6. metadata-eval 86.4%

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

      7. sub-neg 86.4%

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

      8. distribute-neg-frac 86.4%

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

      9. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

      1. clear-num 86.4%

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

      2. inv-pow 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. unpow-1 86.4%

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

      2. +-commutative 86.4%

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

   7. Simplified 86.4%

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

      1. distribute-lft-in 78.7%

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

      2. associate-/l/ 78.5%

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

      3. associate-/l/ 78.1%

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

   9. Applied egg-rr 78.1%

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

       1. distribute-lft-out 85.8%

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

       2. associate-*r* 85.8%

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

       3. associate-*l/ 85.9%

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

       4. *-lft-identity 85.9%

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

   11. Simplified 85.9%

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

       1. *-un-lft-identity 85.9%

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

       2. times-frac 99.6%

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

   13. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

       2. associate-/r/ 99.6%

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

   15. Simplified 99.6%

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

   16. Taylor expanded in a around 0 78.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+123}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-175}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b - a} \cdot \left(\pi \cdot \frac{\frac{1}{a}}{a + b}\right)\\ \end{array} \]

Alternative 3: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{\pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b - a} \cdot \left(\pi \cdot \frac{\frac{1}{a}}{a + b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+123)
   (* (/ PI (* a b)) (/ 0.5 a))
   (if (<= a -4.2e-192)
     (* (/ PI (* (+ a b) (- b a))) (+ (/ 0.5 a) (/ -0.5 b)))
     (* (/ 0.5 (- b a)) (* PI (/ (/ 1.0 a) (+ a b)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+123) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else if (a <= -4.2e-192) {
		tmp = (((double) M_PI) / ((a + b) * (b - a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = (0.5 / (b - a)) * (((double) M_PI) * ((1.0 / a) / (a + b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e+123) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else if (a <= -4.2e-192) {
		tmp = (Math.PI / ((a + b) * (b - a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = (0.5 / (b - a)) * (Math.PI * ((1.0 / a) / (a + b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e+123:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	elif a <= -4.2e-192:
		tmp = (math.pi / ((a + b) * (b - a))) * ((0.5 / a) + (-0.5 / b))
	else:
		tmp = (0.5 / (b - a)) * (math.pi * ((1.0 / a) / (a + b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+123)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	elseif (a <= -4.2e-192)
		tmp = Float64(Float64(pi / Float64(Float64(a + b) * Float64(b - a))) * Float64(Float64(0.5 / a) + Float64(-0.5 / b)));
	else
		tmp = Float64(Float64(0.5 / Float64(b - a)) * Float64(pi * Float64(Float64(1.0 / a) / Float64(a + b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e+123)
		tmp = (pi / (a * b)) * (0.5 / a);
	elseif (a <= -4.2e-192)
		tmp = (pi / ((a + b) * (b - a))) * ((0.5 / a) + (-0.5 / b));
	else
		tmp = (0.5 / (b - a)) * (pi * ((1.0 / a) / (a + b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+123], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -4.2e-192], N[(N[(Pi / N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(N[(1.0 / a), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.99999999999999996e123

   1. Initial program 55.8%

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

      1. times-frac 55.8%

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

      2. *-commutative 55.8%

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

      3. times-frac 55.8%

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

      4. difference-of-squares 72.1%

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

      5. associate-/r* 72.0%

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

      6. metadata-eval 72.0%

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

      7. sub-neg 72.0%

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

      8. distribute-neg-frac 72.0%

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

      9. metadata-eval 72.0%

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

   3. Simplified 72.0%

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

      1. clear-num 72.0%

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

      2. inv-pow 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. unpow-1 72.0%

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

      2. +-commutative 72.0%

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

   7. Simplified 72.0%

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

      1. distribute-lft-in 72.0%

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

      2. associate-/l/ 72.0%

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

      3. associate-/l/ 72.0%

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

   9. Applied egg-rr 72.0%

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

       1. distribute-lft-out 72.0%

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

       2. associate-*r* 72.0%

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

       3. associate-*l/ 72.0%

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

       4. *-lft-identity 72.0%

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

   11. Simplified 72.0%

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

       1. *-un-lft-identity 72.0%

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

       2. times-frac 99.7%

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

   13. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. associate-/r/ 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in b around 0 72.1%

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

       1. unpow2 72.1%

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

       2. associate-*r* 99.7%

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

       3. associate-*r/ 99.7%

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

       4. *-commutative 99.7%

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

       5. *-commutative 99.7%

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

       6. times-frac 99.9%

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

   18. Simplified 99.9%

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

   if -1.99999999999999996e123 < a < -4.19999999999999986e-192

   1. Initial program 95.1%

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

      1. times-frac 95.1%

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

      2. *-commutative 95.1%

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

      3. times-frac 95.1%

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

      4. difference-of-squares 95.1%

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

      5. associate-/r* 96.0%

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

      6. metadata-eval 96.0%

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

      7. sub-neg 96.0%

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

      8. distribute-neg-frac 96.0%

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

      9. metadata-eval 96.0%

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

   3. Simplified 96.0%

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

      1. distribute-lft-in 90.9%

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

      2. associate-/l/ 90.1%

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

      3. associate-/l/ 90.1%

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

   5. Applied egg-rr 90.1%

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

      1. distribute-lft-out 95.1%

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

      2. associate-*r* 95.1%

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

      3. associate-*l/ 95.2%

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

      4. *-commutative 95.2%

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

      5. difference-of-squares 95.1%

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

      6. associate-*l/ 95.1%

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

      7. distribute-lft-in 95.1%

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

      8. associate-*r/ 95.1%

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

      9. metadata-eval 95.1%

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

      10. associate-*r/ 95.1%

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

      11. metadata-eval 95.1%

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

   7. Simplified 95.1%

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

      1. difference-of-squares 95.1%

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

   9. Applied egg-rr 95.1%

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

   if -4.19999999999999986e-192 < a

   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. Step-by-step derivation

      1. times-frac 73.4%

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

      2. *-commutative 73.4%

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

      3. times-frac 73.4%

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

      4. difference-of-squares 85.6%

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

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

      1. clear-num 86.2%

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

      2. inv-pow 86.2%

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

   5. Applied egg-rr 86.2%

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

      1. unpow-1 86.2%

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

      2. +-commutative 86.2%

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

   7. Simplified 86.2%

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

      1. distribute-lft-in 78.3%

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

      2. associate-/l/ 78.0%

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

      3. associate-/l/ 77.6%

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

   9. Applied egg-rr 77.6%

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

       1. distribute-lft-out 85.5%

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

       2. associate-*r* 85.5%

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

       3. associate-*l/ 85.6%

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

       4. *-lft-identity 85.6%

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

   11. Simplified 85.6%

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

       1. *-un-lft-identity 85.6%

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

       2. times-frac 99.6%

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

   13. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

       2. associate-/r/ 99.6%

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

   15. Simplified 99.6%

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

   16. Taylor expanded in a around 0 77.7%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{\pi}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b - a} \cdot \left(\pi \cdot \frac{\frac{1}{a}}{a + b}\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. times-frac 77.6%

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

   2. *-commutative 77.6%

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

   3. times-frac 77.6%

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

   4. difference-of-squares 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* 87.1%

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

   6. metadata-eval 87.1%

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

   7. sub-neg 87.1%

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

   8. distribute-neg-frac 87.1%

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

   9. metadata-eval 87.1%

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

3. Simplified 87.1%

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

   1. clear-num 87.1%

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

   2. inv-pow 87.1%

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

5. Applied egg-rr 87.1%

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

   1. unpow-1 87.1%

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

   2. +-commutative 87.1%

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

7. Simplified 87.1%

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

   1. distribute-lft-in 81.2%

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

   2. associate-/l/ 80.8%

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

   3. associate-/l/ 80.6%

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

9. Applied egg-rr 80.6%

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

    1. distribute-lft-out 86.5%

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

    2. associate-*r* 86.5%

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

    3. associate-*l/ 86.6%

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

    4. *-lft-identity 86.6%

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

11. Simplified 86.6%

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

    1. *-un-lft-identity 86.6%

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

    2. times-frac 99.5%

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

13. Applied egg-rr 99.5%

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

    1. *-lft-identity 99.5%

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

    2. associate-/r/ 99.5%

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

15. Simplified 99.5%

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

16. Final simplification 99.5%

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

Alternative 5: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \frac{\frac{-\pi}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e+124)
   (* (/ PI (* a b)) (/ 0.5 a))
   (if (<= a -1.1e-119)
     (* 0.5 (/ (/ (- PI) b) (- (* b b) (* a a))))
     (/ (* 0.5 PI) (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.5e+124) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else if (a <= -1.1e-119) {
		tmp = 0.5 * ((-((double) M_PI) / b) / ((b * b) - (a * a)));
	} else {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e+124) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else if (a <= -1.1e-119) {
		tmp = 0.5 * ((-Math.PI / b) / ((b * b) - (a * a)));
	} else {
		tmp = (0.5 * Math.PI) / (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.5e+124:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	elif a <= -1.1e-119:
		tmp = 0.5 * ((-math.pi / b) / ((b * b) - (a * a)))
	else:
		tmp = (0.5 * math.pi) / (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.5e+124)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	elseif (a <= -1.1e-119)
		tmp = Float64(0.5 * Float64(Float64(Float64(-pi) / b) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e+124)
		tmp = (pi / (a * b)) * (0.5 / a);
	elseif (a <= -1.1e-119)
		tmp = 0.5 * ((-pi / b) / ((b * b) - (a * a)));
	else
		tmp = (0.5 * pi) / (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.5e+124], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.1e-119], N[(0.5 * N[(N[((-Pi) / b), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.5000000000000004e124

   1. Initial program 55.8%

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

      1. times-frac 55.8%

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

      2. *-commutative 55.8%

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

      3. times-frac 55.8%

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

      4. difference-of-squares 72.1%

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

      5. associate-/r* 72.0%

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

      6. metadata-eval 72.0%

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

      7. sub-neg 72.0%

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

      8. distribute-neg-frac 72.0%

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

      9. metadata-eval 72.0%

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

   3. Simplified 72.0%

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

      1. clear-num 72.0%

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

      2. inv-pow 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. unpow-1 72.0%

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

      2. +-commutative 72.0%

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

   7. Simplified 72.0%

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

      1. distribute-lft-in 72.0%

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

      2. associate-/l/ 72.0%

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

      3. associate-/l/ 72.0%

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

   9. Applied egg-rr 72.0%

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

       1. distribute-lft-out 72.0%

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

       2. associate-*r* 72.0%

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

       3. associate-*l/ 72.0%

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

       4. *-lft-identity 72.0%

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

   11. Simplified 72.0%

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

       1. *-un-lft-identity 72.0%

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

       2. times-frac 99.7%

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

   13. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. associate-/r/ 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in b around 0 72.1%

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

       1. unpow2 72.1%

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

       2. associate-*r* 99.7%

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

       3. associate-*r/ 99.7%

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

       4. *-commutative 99.7%

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

       5. *-commutative 99.7%

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

       6. times-frac 99.9%

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

   18. Simplified 99.9%

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

   if -4.5000000000000004e124 < a < -1.1e-119

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-/r/ 98.1%

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

      3. associate-*l/ 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-/r/ 98.1%

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

      6. times-frac 98.1%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in b around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. mul-1-neg 75.5%

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

   6. Simplified 75.5%

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

   if -1.1e-119 < a

   1. Initial program 74.4%

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

      1. times-frac 74.5%

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

      2. *-commutative 74.5%

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

      3. times-frac 74.5%

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

      4. difference-of-squares 85.4%

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

      5. associate-/r* 85.9%

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

      6. metadata-eval 85.9%

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

      7. sub-neg 85.9%

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

      8. distribute-neg-frac 85.9%

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

      9. metadata-eval 85.9%

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

   3. Simplified 85.9%

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

      1. clear-num 85.9%

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

      2. inv-pow 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. unpow-1 85.9%

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

      2. +-commutative 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in a around 0 60.6%

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

      1. associate-*r/ 60.6%

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

      2. unpow2 60.6%

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

      3. associate-*r* 69.5%

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

   10. Simplified 69.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \frac{\frac{-\pi}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 6: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-121}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e+122)
   (* (/ PI (* a b)) (/ 0.5 a))
   (if (<= a -1.55e-121)
     (* (/ -0.5 b) (/ PI (- (* b b) (* a a))))
     (/ (* 0.5 PI) (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e+122) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else if (a <= -1.55e-121) {
		tmp = (-0.5 / b) * (((double) M_PI) / ((b * b) - (a * a)));
	} else {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.8e+122) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else if (a <= -1.55e-121) {
		tmp = (-0.5 / b) * (Math.PI / ((b * b) - (a * a)));
	} else {
		tmp = (0.5 * Math.PI) / (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.8e+122:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	elif a <= -1.55e-121:
		tmp = (-0.5 / b) * (math.pi / ((b * b) - (a * a)))
	else:
		tmp = (0.5 * math.pi) / (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.8e+122)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	elseif (a <= -1.55e-121)
		tmp = Float64(Float64(-0.5 / b) * Float64(pi / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.8e+122)
		tmp = (pi / (a * b)) * (0.5 / a);
	elseif (a <= -1.55e-121)
		tmp = (-0.5 / b) * (pi / ((b * b) - (a * a)));
	else
		tmp = (0.5 * pi) / (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.8e+122], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.55e-121], N[(N[(-0.5 / b), $MachinePrecision] * N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.8e122

   1. Initial program 55.8%

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

      1. times-frac 55.8%

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

      2. *-commutative 55.8%

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

      3. times-frac 55.8%

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

      4. difference-of-squares 72.1%

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

      5. associate-/r* 72.0%

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

      6. metadata-eval 72.0%

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

      7. sub-neg 72.0%

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

      8. distribute-neg-frac 72.0%

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

      9. metadata-eval 72.0%

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

   3. Simplified 72.0%

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

      1. clear-num 72.0%

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

      2. inv-pow 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. unpow-1 72.0%

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

      2. +-commutative 72.0%

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

   7. Simplified 72.0%

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

      1. distribute-lft-in 72.0%

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

      2. associate-/l/ 72.0%

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

      3. associate-/l/ 72.0%

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

   9. Applied egg-rr 72.0%

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

       1. distribute-lft-out 72.0%

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

       2. associate-*r* 72.0%

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

       3. associate-*l/ 72.0%

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

       4. *-lft-identity 72.0%

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

   11. Simplified 72.0%

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

       1. *-un-lft-identity 72.0%

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

       2. times-frac 99.7%

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

   13. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. associate-/r/ 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in b around 0 72.1%

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

       1. unpow2 72.1%

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

       2. associate-*r* 99.7%

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

       3. associate-*r/ 99.7%

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

       4. *-commutative 99.7%

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

       5. *-commutative 99.7%

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

       6. times-frac 99.9%

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

   18. Simplified 99.9%

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

   if -6.8e122 < a < -1.5499999999999999e-121

   1. Initial program 98.0%

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

      1. times-frac 98.0%

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

      2. *-commutative 98.0%

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

      3. times-frac 98.0%

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

      4. difference-of-squares 98.0%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. sub-neg 99.2%

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

      8. distribute-neg-frac 99.2%

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

      9. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. distribute-lft-in 97.6%

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

      2. associate-/l/ 96.6%

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

      3. associate-/l/ 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. distribute-lft-out 98.0%

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

      2. associate-*r* 98.0%

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

      3. associate-*l/ 98.1%

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

      4. *-commutative 98.1%

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

      5. difference-of-squares 98.1%

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

      6. associate-*l/ 98.0%

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

      7. distribute-lft-in 98.0%

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

      8. associate-*r/ 98.0%

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

      9. metadata-eval 98.0%

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

      10. associate-*r/ 98.0%

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

      11. metadata-eval 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in a around inf 74.2%

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

   if -1.5499999999999999e-121 < a

   1. Initial program 74.3%

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

      1. times-frac 74.3%

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

      2. *-commutative 74.3%

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

      3. times-frac 74.3%

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

      4. difference-of-squares 85.3%

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

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

      1. clear-num 85.8%

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

      2. inv-pow 85.8%

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

   5. Applied egg-rr 85.8%

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

      1. unpow-1 85.8%

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

      2. +-commutative 85.8%

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

   7. Simplified 85.8%

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

   8. Taylor expanded in a around 0 60.3%

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

      1. associate-*r/ 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*r* 69.3%

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

   10. Simplified 69.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-121}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 7: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.92 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b - a} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.92e-53)
   (* 0.5 (/ (/ PI a) (* a b)))
   (* (/ 0.5 (- b a)) (/ PI (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.92e-53) {
		tmp = 0.5 * ((((double) M_PI) / a) / (a * b));
	} else {
		tmp = (0.5 / (b - a)) * (((double) M_PI) / (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.92e-53) {
		tmp = 0.5 * ((Math.PI / a) / (a * b));
	} else {
		tmp = (0.5 / (b - a)) * (Math.PI / (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.92e-53:
		tmp = 0.5 * ((math.pi / a) / (a * b))
	else:
		tmp = (0.5 / (b - a)) * (math.pi / (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.92e-53)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 / Float64(b - a)) * Float64(pi / Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.92e-53)
		tmp = 0.5 * ((pi / a) / (a * b));
	else
		tmp = (0.5 / (b - a)) * (pi / (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.92e-53], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.92000000000000011e-53

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/r/ 77.7%

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

      3. associate-*l/ 77.7%

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

      4. *-commutative 77.7%

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

      5. associate-/r/ 77.7%

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

      6. times-frac 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in b around 0 55.9%

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

      1. associate-*r/ 55.9%

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

      2. mul-1-neg 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in b around 0 59.8%

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

      1. unpow2 59.8%

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

      2. associate-*l* 69.5%

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

      3. *-commutative 69.5%

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

   9. Simplified 69.5%

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

   10. Taylor expanded in a around 0 59.8%

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

       1. unpow2 59.8%

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

       2. associate-*r* 69.5%

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

       3. associate-/r* 70.0%

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

   12. Simplified 70.0%

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

   if 1.92000000000000011e-53 < b

   1. Initial program 77.3%

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

      1. times-frac 77.4%

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

      2. *-commutative 77.4%

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

      3. times-frac 77.4%

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

      4. difference-of-squares 87.0%

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

      5. associate-/r* 88.4%

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

      6. metadata-eval 88.4%

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

      7. sub-neg 88.4%

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

      8. distribute-neg-frac 88.4%

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

      9. metadata-eval 88.4%

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

   3. Simplified 88.4%

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

      1. clear-num 88.3%

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

      2. inv-pow 88.3%

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

   5. Applied egg-rr 88.3%

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

      1. unpow-1 88.3%

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

      2. +-commutative 88.3%

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

   7. Simplified 88.3%

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

      1. distribute-lft-in 88.4%

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

      2. associate-/l/ 86.9%

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

      3. associate-/l/ 86.9%

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

   9. Applied egg-rr 86.9%

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

       1. distribute-lft-out 86.9%

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

       2. associate-*r* 86.9%

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

       3. associate-*l/ 87.1%

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

       4. *-lft-identity 87.1%

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

   11. Simplified 87.1%

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

       1. *-un-lft-identity 87.1%

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

       2. times-frac 99.4%

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

   13. Applied egg-rr 99.4%

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

       1. *-lft-identity 99.4%

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

       2. associate-/r/ 99.5%

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

   15. Simplified 99.5%

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

   16. Taylor expanded in a around 0 90.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.92 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b - a} \cdot \frac{\pi}{a \cdot b}\\ \end{array} \]

Alternative 8: 66.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-121}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.55e-121)
   (* (/ PI (* a b)) (/ 0.5 a))
   (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-121) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} 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 <= -1.55e-121) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.55e-121:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.55e-121)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	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 <= -1.55e-121)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.55e-121], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $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 < -1.5499999999999999e-121

   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 82.6%

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

      2. *-commutative 82.6%

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

      3. times-frac 82.6%

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

      4. difference-of-squares 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* 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. clear-num 89.1%

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

      2. inv-pow 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. unpow-1 89.1%

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

      2. +-commutative 89.1%

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

   7. Simplified 89.1%

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

      1. distribute-lft-in 88.1%

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

      2. associate-/l/ 87.5%

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

      3. associate-/l/ 87.5%

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

   9. Applied egg-rr 87.5%

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

       1. distribute-lft-out 88.5%

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

       2. associate-*r* 88.5%

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

       3. associate-*l/ 88.5%

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

       4. *-lft-identity 88.5%

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

   11. Simplified 88.5%

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

       1. *-un-lft-identity 88.5%

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

       2. times-frac 99.5%

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

   13. Applied egg-rr 99.5%

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

       1. *-lft-identity 99.5%

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

       2. associate-/r/ 99.5%

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

   15. Simplified 99.5%

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

   16. Taylor expanded in b around 0 67.9%

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

       1. unpow2 67.9%

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

       2. associate-*r* 78.0%

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

       3. associate-*r/ 78.0%

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

       4. *-commutative 78.0%

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

       5. *-commutative 78.0%

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

       6. times-frac 78.1%

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

   18. Simplified 78.1%

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

   if -1.5499999999999999e-121 < a

   1. Initial program 74.3%

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

      1. *-commutative 74.3%

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

      2. associate-/r/ 74.3%

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

      3. associate-*l/ 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-/r/ 74.3%

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

      6. times-frac 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in b around inf 60.3%

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

      1. unpow2 60.3%

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

   6. Simplified 60.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-121}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 9: 66.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.3e-120)
   (* 0.5 (/ (/ PI a) (* a b)))
   (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.3e-120) {
		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 <= -1.3e-120) {
		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 <= -1.3e-120:
		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 <= -1.3e-120)
		tmp = Float64(0.5 * Float64(Float64(pi / 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 <= -1.3e-120)
		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, -1.3e-120], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(a * b), $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 < -1.3000000000000001e-120

   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. *-commutative 82.5%

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

      2. associate-/r/ 82.6%

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

      3. associate-*l/ 82.6%

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

      4. *-commutative 82.6%

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

      5. associate-/r/ 82.6%

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

      6. times-frac 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in b around 0 67.5%

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

      1. associate-*r/ 67.5%

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

      2. mul-1-neg 67.5%

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

   6. Simplified 67.5%

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

   7. Taylor expanded in b around 0 67.9%

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

      1. unpow2 67.9%

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

      2. associate-*l* 78.0%

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

      3. *-commutative 78.0%

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

   9. Simplified 78.0%

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

   10. Taylor expanded in a around 0 67.9%

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

       1. unpow2 67.9%

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

       2. associate-*r* 78.0%

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

       3. associate-/r* 78.1%

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

   12. Simplified 78.1%

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

   if -1.3000000000000001e-120 < a

   1. Initial program 74.3%

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

      1. *-commutative 74.3%

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

      2. associate-/r/ 74.3%

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

      3. associate-*l/ 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-/r/ 74.3%

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

      6. times-frac 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in b around inf 60.3%

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

      1. unpow2 60.3%

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

   6. Simplified 60.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 10: 66.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.2e-120)
   (* 0.5 (/ (/ PI a) (* a b)))
   (* 0.5 (/ (/ PI a) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.2e-120) {
		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 <= -8.2e-120) {
		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 <= -8.2e-120:
		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 <= -8.2e-120)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(a * b)));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.2e-120)
		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, -8.2e-120], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.20000000000000068e-120

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-/r/ 82.4%

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

      3. associate-*l/ 82.4%

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

      4. *-commutative 82.4%

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

      5. associate-/r/ 82.4%

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

      6. times-frac 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in b around 0 68.2%

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

      1. associate-*r/ 68.2%

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

      2. mul-1-neg 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in b around 0 67.6%

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

      1. unpow2 67.6%

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

      2. associate-*l* 77.8%

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

      3. *-commutative 77.8%

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

   9. Simplified 77.8%

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

   10. Taylor expanded in a around 0 67.6%

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

       1. unpow2 67.6%

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

       2. associate-*r* 77.8%

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

       3. associate-/r* 77.9%

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

   12. Simplified 77.9%

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

   if -8.20000000000000068e-120 < a

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

      2. associate-/r/ 74.5%

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

      3. associate-*l/ 74.5%

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

      4. *-commutative 74.5%

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

      5. associate-/r/ 74.5%

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

      6. times-frac 74.5%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in b around inf 60.6%

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

      1. unpow2 60.6%

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

   6. Simplified 60.6%

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

   7. Taylor expanded in a around 0 60.6%

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

      1. associate-/r* 60.6%

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

      2. unpow2 60.6%

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

   9. Simplified 60.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \]

Alternative 11: 72.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.2e-89)
   (* 0.5 (/ (/ PI a) (* a b)))
   (/ (* 0.5 PI) (* b (* a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.2e-89) {
		tmp = 0.5 * ((((double) M_PI) / a) / (a * b));
	} else {
		tmp = (0.5 * ((double) M_PI)) / (b * (a * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.2e-89) {
		tmp = 0.5 * ((Math.PI / a) / (a * b));
	} else {
		tmp = (0.5 * Math.PI) / (b * (a * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.2e-89:
		tmp = 0.5 * ((math.pi / a) / (a * b))
	else:
		tmp = (0.5 * math.pi) / (b * (a * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.2e-89)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 * pi) / Float64(b * Float64(a * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.2e-89)
		tmp = 0.5 * ((pi / a) / (a * b));
	else
		tmp = (0.5 * pi) / (b * (a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.2e-89], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * Pi), $MachinePrecision] / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.2000000000000002e-89

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/r/ 82.0%

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

      3. associate-*l/ 82.0%

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

      4. *-commutative 82.0%

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

      5. associate-/r/ 82.0%

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

      6. times-frac 82.0%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in b around 0 69.6%

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

      1. associate-*r/ 69.6%

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

      2. mul-1-neg 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in b around 0 69.9%

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

      1. unpow2 69.9%

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

      2. associate-*l* 81.1%

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

      3. *-commutative 81.1%

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

   9. Simplified 81.1%

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

   10. Taylor expanded in a around 0 69.9%

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

       1. unpow2 69.9%

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

       2. associate-*r* 81.1%

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

       3. associate-/r* 81.2%

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

   12. Simplified 81.2%

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

   if -4.2000000000000002e-89 < a

   1. Initial program 75.1%

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

      1. times-frac 75.2%

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

      2. *-commutative 75.2%

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

      3. times-frac 75.2%

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

      4. difference-of-squares 85.5%

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

      5. associate-/r* 86.4%

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

      6. metadata-eval 86.4%

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

      7. sub-neg 86.4%

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

      8. distribute-neg-frac 86.4%

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

      9. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

      1. clear-num 86.4%

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

      2. inv-pow 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. unpow-1 86.4%

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

      2. +-commutative 86.4%

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

   7. Simplified 86.4%

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

   8. Taylor expanded in a around 0 60.1%

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

      1. associate-*r/ 60.1%

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

      2. unpow2 60.1%

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

      3. associate-*r* 69.1%

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

   10. Simplified 69.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-89}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 12: 57.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. times-frac 77.6%

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

   2. *-commutative 77.6%

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

   3. times-frac 77.6%

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

   4. difference-of-squares 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* 87.1%

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

   6. metadata-eval 87.1%

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

   7. sub-neg 87.1%

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

   8. distribute-neg-frac 87.1%

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

   9. metadata-eval 87.1%

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

3. Simplified 87.1%

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

   1. clear-num 87.1%

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

   2. inv-pow 87.1%

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

5. Applied egg-rr 87.1%

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

   1. unpow-1 87.1%

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

   2. +-commutative 87.1%

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

7. Simplified 87.1%

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

8. Taylor expanded in a around inf 55.1%

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

   1. associate-*r/ 55.1%

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

   2. *-commutative 55.1%

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

   3. *-commutative 55.1%

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

   4. times-frac 55.4%

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

   5. unpow2 55.4%

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

10. Simplified 55.4%

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

11. Final simplification 55.4%

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

Alternative 13: 63.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. times-frac 77.6%

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

   2. *-commutative 77.6%

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

   3. times-frac 77.6%

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

   4. difference-of-squares 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* 87.1%

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

   6. metadata-eval 87.1%

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

   7. sub-neg 87.1%

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

   8. distribute-neg-frac 87.1%

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

   9. metadata-eval 87.1%

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

3. Simplified 87.1%

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

   1. clear-num 87.1%

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

   2. inv-pow 87.1%

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

5. Applied egg-rr 87.1%

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

   1. unpow-1 87.1%

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

   2. +-commutative 87.1%

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

7. Simplified 87.1%

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

   1. distribute-lft-in 81.2%

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

   2. associate-/l/ 80.8%

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

   3. associate-/l/ 80.6%

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

9. Applied egg-rr 80.6%

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

    1. distribute-lft-out 86.5%

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

    2. associate-*r* 86.5%

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

    3. associate-*l/ 86.6%

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

    4. *-lft-identity 86.6%

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

11. Simplified 86.6%

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

    1. *-un-lft-identity 86.6%

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

    2. times-frac 99.5%

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

13. Applied egg-rr 99.5%

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

    1. *-lft-identity 99.5%

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

    2. associate-/r/ 99.5%

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

15. Simplified 99.5%

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

16. Taylor expanded in b around 0 55.1%

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

    1. unpow2 55.1%

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

    2. associate-*r* 62.1%

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

    3. associate-*r/ 62.1%

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

    4. *-commutative 62.1%

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

    5. *-commutative 62.1%

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

    6. times-frac 62.4%

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

18. Simplified 62.4%

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

19. Final simplification 62.4%

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

Reproduce

herbie shell --seed 2023257 
(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))))