NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.6%

Time: 15.4s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.0% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. un-div-inv 82.9%

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

   2. difference-of-squares 92.7%

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

   3. associate-/r* 93.1%

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

   4. div-inv 93.1%

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

   5. metadata-eval 93.1%

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

4. Applied egg-rr 93.1%

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

   1. associate-/l/ 92.7%

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

   2. *-commutative 92.7%

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

   3. times-frac 93.1%

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

   4. +-commutative 93.1%

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

6. Simplified 93.1%

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

   1. clear-num 93.0%

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

   2. inv-pow 93.0%

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

8. Applied egg-rr 93.0%

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

   1. unpow-1 93.0%

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

10. Simplified 93.0%

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

    1. pow1 93.0%

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

    2. associate-*l* 99.6%

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

    3. clear-num 99.6%

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

    4. +-commutative 99.6%

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

12. Applied egg-rr 99.6%

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

    1. unpow1 99.6%

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

    2. +-commutative 99.6%

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

14. Simplified 99.6%

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

    1. associate-*l/ 99.7%

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

    2. *-commutative 99.7%

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

    3. +-commutative 99.7%

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

16. Applied egg-rr 99.7%

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

17. Final simplification 99.7%

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

Alternative 2: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+143}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{a \cdot b}\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{-235}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot \frac{\frac{0.5}{b - a}}{a + b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} + \frac{\pi}{b}}{b - a} \cdot \frac{0.5}{a + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.5e+143)
   (* (* 0.5 (/ PI a)) (/ 1.0 (* a b)))
   (if (<= a -8.6e-235)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (* PI (/ (/ 0.5 (- b a)) (+ a b))))
     (* (/ (+ (/ PI a) (/ PI b)) (- b a)) (/ 0.5 (+ a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.5e+143) {
		tmp = (0.5 * (((double) M_PI) / a)) * (1.0 / (a * b));
	} else if (a <= -8.6e-235) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (((double) M_PI) * ((0.5 / (b - a)) / (a + b)));
	} else {
		tmp = (((((double) M_PI) / a) + (((double) M_PI) / b)) / (b - a)) * (0.5 / (a + b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.5e+143) {
		tmp = (0.5 * (Math.PI / a)) * (1.0 / (a * b));
	} else if (a <= -8.6e-235) {
		tmp = ((1.0 / a) + (-1.0 / b)) * (Math.PI * ((0.5 / (b - a)) / (a + b)));
	} else {
		tmp = (((Math.PI / a) + (Math.PI / b)) / (b - a)) * (0.5 / (a + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.5e+143:
		tmp = (0.5 * (math.pi / a)) * (1.0 / (a * b))
	elif a <= -8.6e-235:
		tmp = ((1.0 / a) + (-1.0 / b)) * (math.pi * ((0.5 / (b - a)) / (a + b)))
	else:
		tmp = (((math.pi / a) + (math.pi / b)) / (b - a)) * (0.5 / (a + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.5e+143)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) * Float64(1.0 / Float64(a * b)));
	elseif (a <= -8.6e-235)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(pi * Float64(Float64(0.5 / Float64(b - a)) / Float64(a + b))));
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) + Float64(pi / b)) / Float64(b - a)) * Float64(0.5 / Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.5e+143)
		tmp = (0.5 * (pi / a)) * (1.0 / (a * b));
	elseif (a <= -8.6e-235)
		tmp = ((1.0 / a) + (-1.0 / b)) * (pi * ((0.5 / (b - a)) / (a + b)));
	else
		tmp = (((pi / a) + (pi / b)) / (b - a)) * (0.5 / (a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.5e+143], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -8.6e-235], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] + N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.50000000000000008e143

   1. Initial program 62.8%

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

      1. un-div-inv 62.8%

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

      2. difference-of-squares 84.4%

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

      3. associate-/r* 87.4%

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

      4. div-inv 87.4%

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

      5. metadata-eval 87.4%

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

   4. Applied egg-rr 87.4%

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

      1. associate-/l/ 84.4%

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

      2. *-commutative 84.4%

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

      3. times-frac 87.4%

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

      4. +-commutative 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in a around inf 87.4%

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

      1. *-commutative 87.4%

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

      2. frac-sub 87.4%

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

      3. associate-*l/ 87.4%

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

      4. frac-times 99.6%

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

      5. *-un-lft-identity 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. *-commutative 99.6%

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

       2. *-commutative 99.6%

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

       3. associate-/l* 99.6%

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

       4. *-rgt-identity 99.6%

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

       5. *-commutative 99.6%

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

       6. associate-*l* 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in b around 0 99.9%

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

   if -3.50000000000000008e143 < a < -8.60000000000000048e-235

   1. Initial program 91.1%

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

      1. un-div-inv 91.2%

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

      2. difference-of-squares 97.4%

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

      3. associate-/r* 97.4%

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

      4. div-inv 97.4%

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

      5. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. associate-/l/ 97.4%

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

      2. *-commutative 97.4%

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

      3. times-frac 97.3%

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

      4. +-commutative 97.3%

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

   6. Simplified 97.3%

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

      1. clear-num 97.3%

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

      2. inv-pow 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. unpow-1 97.3%

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

   10. Simplified 97.3%

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

       1. clear-num 97.3%

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

       2. frac-times 97.3%

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

       3. metadata-eval 97.3%

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

       4. clear-num 97.3%

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

       5. clear-num 97.3%

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

       6. clear-num 97.3%

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

       7. +-commutative 97.3%

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

   12. Applied egg-rr 97.3%

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

       1. associate-/r* 97.4%

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

       2. associate-/r/ 97.4%

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

       3. +-commutative 97.4%

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

   14. Simplified 97.4%

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

   if -8.60000000000000048e-235 < a

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-*r* 83.4%

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

      3. associate-*r/ 83.4%

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

      4. associate-*r* 83.4%

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

      5. *-rgt-identity 83.4%

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

      6. sub-neg 83.4%

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

      7. distribute-neg-frac 83.4%

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

      8. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

      1. *-un-lft-identity 83.4%

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

      2. difference-of-squares 92.1%

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

      3. times-frac 99.7%

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

      4. add-sqr-sqrt 49.8%

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

      5. sqrt-unprod 79.8%

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

      6. frac-times 79.8%

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

      7. metadata-eval 79.8%

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

      8. metadata-eval 79.8%

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

      9. frac-times 79.8%

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

      10. sqrt-unprod 37.7%

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

      11. add-sqr-sqrt 70.5%

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

      12. div-inv 70.5%

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

      13. metadata-eval 70.5%

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

   6. Applied egg-rr 70.5%

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

      1. *-commutative 70.5%

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

      2. times-frac 65.7%

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

      3. *-rgt-identity 65.7%

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

      4. associate-*r* 65.7%

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

      5. *-commutative 65.7%

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

      6. distribute-rgt-in 65.8%

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

      7. associate-*l/ 65.8%

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

      8. *-lft-identity 65.8%

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

      9. associate-*l/ 65.8%

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

      10. *-lft-identity 65.8%

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

      11. +-commutative 65.8%

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

   8. Simplified 65.8%

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

      1. associate-/l* 65.8%

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

   10. Applied egg-rr 65.8%

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

       1. associate-*r/ 65.8%

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

       2. times-frac 70.5%

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

       3. +-commutative 70.5%

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

   12. Simplified 70.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+143}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{a \cdot b}\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{-235}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\pi \cdot \frac{\frac{0.5}{b - a}}{a + b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} + \frac{\pi}{b}}{b - a} \cdot \frac{0.5}{a + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{a \cdot b}\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-235}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\frac{0.5}{b - a} \cdot \frac{\pi}{a + b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} + \frac{\pi}{b}}{b - a} \cdot \frac{0.5}{a + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+143)
   (* (* 0.5 (/ PI a)) (/ 1.0 (* a b)))
   (if (<= a -8.5e-235)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (* (/ 0.5 (- b a)) (/ PI (+ a b))))
     (* (/ (+ (/ PI a) (/ PI b)) (- b a)) (/ 0.5 (+ a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+143) {
		tmp = (0.5 * (((double) M_PI) / a)) * (1.0 / (a * b));
	} else if (a <= -8.5e-235) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((0.5 / (b - a)) * (((double) M_PI) / (a + b)));
	} else {
		tmp = (((((double) M_PI) / a) + (((double) M_PI) / b)) / (b - a)) * (0.5 / (a + b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+143) {
		tmp = (0.5 * (Math.PI / a)) * (1.0 / (a * b));
	} else if (a <= -8.5e-235) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((0.5 / (b - a)) * (Math.PI / (a + b)));
	} else {
		tmp = (((Math.PI / a) + (Math.PI / b)) / (b - a)) * (0.5 / (a + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+143:
		tmp = (0.5 * (math.pi / a)) * (1.0 / (a * b))
	elif a <= -8.5e-235:
		tmp = ((1.0 / a) + (-1.0 / b)) * ((0.5 / (b - a)) * (math.pi / (a + b)))
	else:
		tmp = (((math.pi / a) + (math.pi / b)) / (b - a)) * (0.5 / (a + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+143)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) * Float64(1.0 / Float64(a * b)));
	elseif (a <= -8.5e-235)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(0.5 / Float64(b - a)) * Float64(pi / Float64(a + b))));
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) + Float64(pi / b)) / Float64(b - a)) * Float64(0.5 / Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+143)
		tmp = (0.5 * (pi / a)) * (1.0 / (a * b));
	elseif (a <= -8.5e-235)
		tmp = ((1.0 / a) + (-1.0 / b)) * ((0.5 / (b - a)) * (pi / (a + b)));
	else
		tmp = (((pi / a) + (pi / b)) / (b - a)) * (0.5 / (a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+143], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -8.5e-235], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] + N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.00000000000000012e143

   1. Initial program 62.8%

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

      1. un-div-inv 62.8%

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

      2. difference-of-squares 84.4%

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

      3. associate-/r* 87.4%

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

      4. div-inv 87.4%

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

      5. metadata-eval 87.4%

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

   4. Applied egg-rr 87.4%

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

      1. associate-/l/ 84.4%

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

      2. *-commutative 84.4%

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

      3. times-frac 87.4%

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

      4. +-commutative 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in a around inf 87.4%

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

      1. *-commutative 87.4%

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

      2. frac-sub 87.4%

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

      3. associate-*l/ 87.4%

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

      4. frac-times 99.6%

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

      5. *-un-lft-identity 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. *-commutative 99.6%

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

       2. *-commutative 99.6%

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

       3. associate-/l* 99.6%

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

       4. *-rgt-identity 99.6%

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

       5. *-commutative 99.6%

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

       6. associate-*l* 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in b around 0 99.9%

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

   if -5.00000000000000012e143 < a < -8.49999999999999964e-235

   1. Initial program 91.1%

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

      1. un-div-inv 91.2%

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

      2. difference-of-squares 97.4%

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

      3. associate-/r* 97.4%

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

      4. div-inv 97.4%

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

      5. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. associate-/l/ 97.4%

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

      2. *-commutative 97.4%

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

      3. times-frac 97.3%

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

      4. +-commutative 97.3%

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

   6. Simplified 97.3%

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

   if -8.49999999999999964e-235 < a

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-*r* 83.4%

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

      3. associate-*r/ 83.4%

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

      4. associate-*r* 83.4%

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

      5. *-rgt-identity 83.4%

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

      6. sub-neg 83.4%

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

      7. distribute-neg-frac 83.4%

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

      8. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

      1. *-un-lft-identity 83.4%

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

      2. difference-of-squares 92.1%

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

      3. times-frac 99.7%

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

      4. add-sqr-sqrt 49.8%

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

      5. sqrt-unprod 79.8%

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

      6. frac-times 79.8%

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

      7. metadata-eval 79.8%

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

      8. metadata-eval 79.8%

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

      9. frac-times 79.8%

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

      10. sqrt-unprod 37.7%

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

      11. add-sqr-sqrt 70.5%

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

      12. div-inv 70.5%

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

      13. metadata-eval 70.5%

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

   6. Applied egg-rr 70.5%

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

      1. *-commutative 70.5%

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

      2. times-frac 65.7%

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

      3. *-rgt-identity 65.7%

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

      4. associate-*r* 65.7%

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

      5. *-commutative 65.7%

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

      6. distribute-rgt-in 65.8%

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

      7. associate-*l/ 65.8%

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

      8. *-lft-identity 65.8%

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

      9. associate-*l/ 65.8%

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

      10. *-lft-identity 65.8%

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

      11. +-commutative 65.8%

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

   8. Simplified 65.8%

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

      1. associate-/l* 65.8%

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

   10. Applied egg-rr 65.8%

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

       1. associate-*r/ 65.8%

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

       2. times-frac 70.5%

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

       3. +-commutative 70.5%

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

   12. Simplified 70.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{a \cdot b}\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-235}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\frac{0.5}{b - a} \cdot \frac{\pi}{a + b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} + \frac{\pi}{b}}{b - a} \cdot \frac{0.5}{a + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{a \cdot b}\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{-235}:\\ \;\;\;\;\pi \cdot \left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{\left(b - a\right) \cdot \left(a + b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} + \frac{\pi}{b}}{b - a} \cdot \frac{0.5}{a + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+143)
   (* (* 0.5 (/ PI a)) (/ 1.0 (* a b)))
   (if (<= a -8.6e-235)
     (* PI (* (+ (/ 1.0 a) (/ -1.0 b)) (/ 0.5 (* (- b a) (+ a b)))))
     (* (/ (+ (/ PI a) (/ PI b)) (- b a)) (/ 0.5 (+ a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+143) {
		tmp = (0.5 * (((double) M_PI) / a)) * (1.0 / (a * b));
	} else if (a <= -8.6e-235) {
		tmp = ((double) M_PI) * (((1.0 / a) + (-1.0 / b)) * (0.5 / ((b - a) * (a + b))));
	} else {
		tmp = (((((double) M_PI) / a) + (((double) M_PI) / b)) / (b - a)) * (0.5 / (a + b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+143) {
		tmp = (0.5 * (Math.PI / a)) * (1.0 / (a * b));
	} else if (a <= -8.6e-235) {
		tmp = Math.PI * (((1.0 / a) + (-1.0 / b)) * (0.5 / ((b - a) * (a + b))));
	} else {
		tmp = (((Math.PI / a) + (Math.PI / b)) / (b - a)) * (0.5 / (a + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+143:
		tmp = (0.5 * (math.pi / a)) * (1.0 / (a * b))
	elif a <= -8.6e-235:
		tmp = math.pi * (((1.0 / a) + (-1.0 / b)) * (0.5 / ((b - a) * (a + b))))
	else:
		tmp = (((math.pi / a) + (math.pi / b)) / (b - a)) * (0.5 / (a + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+143)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) * Float64(1.0 / Float64(a * b)));
	elseif (a <= -8.6e-235)
		tmp = Float64(pi * Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(0.5 / Float64(Float64(b - a) * Float64(a + b)))));
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) + Float64(pi / b)) / Float64(b - a)) * Float64(0.5 / Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+143)
		tmp = (0.5 * (pi / a)) * (1.0 / (a * b));
	elseif (a <= -8.6e-235)
		tmp = pi * (((1.0 / a) + (-1.0 / b)) * (0.5 / ((b - a) * (a + b))));
	else
		tmp = (((pi / a) + (pi / b)) / (b - a)) * (0.5 / (a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+143], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -8.6e-235], N[(Pi * N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] + N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.00000000000000012e143

   1. Initial program 62.8%

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

      1. un-div-inv 62.8%

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

      2. difference-of-squares 84.4%

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

      3. associate-/r* 87.4%

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

      4. div-inv 87.4%

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

      5. metadata-eval 87.4%

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

   4. Applied egg-rr 87.4%

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

      1. associate-/l/ 84.4%

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

      2. *-commutative 84.4%

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

      3. times-frac 87.4%

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

      4. +-commutative 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in a around inf 87.4%

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

      1. *-commutative 87.4%

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

      2. frac-sub 87.4%

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

      3. associate-*l/ 87.4%

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

      4. frac-times 99.6%

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

      5. *-un-lft-identity 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. *-commutative 99.6%

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

       2. *-commutative 99.6%

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

       3. associate-/l* 99.6%

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

       4. *-rgt-identity 99.6%

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

       5. *-commutative 99.6%

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

       6. associate-*l* 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in b around 0 99.9%

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

   if -5.00000000000000012e143 < a < -8.60000000000000048e-235

   1. Initial program 91.1%

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

      1. un-div-inv 91.2%

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

      2. difference-of-squares 97.4%

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

      3. associate-/r* 97.4%

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

      4. div-inv 97.4%

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

      5. metadata-eval 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. associate-/l/ 97.4%

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

      2. *-commutative 97.4%

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

      3. times-frac 97.3%

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

      4. +-commutative 97.3%

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

   6. Simplified 97.3%

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

      1. pow1 97.3%

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

      2. associate-*r/ 97.4%

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

      3. *-commutative 97.4%

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

   8. Applied egg-rr 97.4%

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

      1. unpow1 97.4%

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

      2. associate-/l* 97.4%

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

      3. associate-/r* 97.3%

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

      4. associate-*l* 97.3%

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

      5. sub-neg 97.3%

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

      6. distribute-neg-frac 97.3%

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

      7. metadata-eval 97.3%

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

   10. Simplified 97.3%

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

   if -8.60000000000000048e-235 < a

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-*r* 83.4%

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

      3. associate-*r/ 83.4%

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

      4. associate-*r* 83.4%

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

      5. *-rgt-identity 83.4%

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

      6. sub-neg 83.4%

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

      7. distribute-neg-frac 83.4%

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

      8. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

      1. *-un-lft-identity 83.4%

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

      2. difference-of-squares 92.1%

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

      3. times-frac 99.7%

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

      4. add-sqr-sqrt 49.8%

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

      5. sqrt-unprod 79.8%

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

      6. frac-times 79.8%

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

      7. metadata-eval 79.8%

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

      8. metadata-eval 79.8%

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

      9. frac-times 79.8%

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

      10. sqrt-unprod 37.7%

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

      11. add-sqr-sqrt 70.5%

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

      12. div-inv 70.5%

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

      13. metadata-eval 70.5%

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

   6. Applied egg-rr 70.5%

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

      1. *-commutative 70.5%

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

      2. times-frac 65.7%

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

      3. *-rgt-identity 65.7%

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

      4. associate-*r* 65.7%

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

      5. *-commutative 65.7%

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

      6. distribute-rgt-in 65.8%

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

      7. associate-*l/ 65.8%

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

      8. *-lft-identity 65.8%

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

      9. associate-*l/ 65.8%

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

      10. *-lft-identity 65.8%

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

      11. +-commutative 65.8%

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

   8. Simplified 65.8%

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

      1. associate-/l* 65.8%

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

   10. Applied egg-rr 65.8%

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

       1. associate-*r/ 65.8%

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

       2. times-frac 70.5%

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

       3. +-commutative 70.5%

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

   12. Simplified 70.5%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{a \cdot b}\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{-235}:\\ \;\;\;\;\pi \cdot \left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{\left(b - a\right) \cdot \left(a + b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} + \frac{\pi}{b}}{b - a} \cdot \frac{0.5}{a + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-141}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} + \frac{\pi}{b}}{b - a} \cdot \frac{0.5}{a + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.1e-141)
   (* (* 0.5 (/ PI a)) (/ 1.0 (* a b)))
   (* (/ (+ (/ PI a) (/ PI b)) (- b a)) (/ 0.5 (+ a b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.1e-141) {
		tmp = (0.5 * (((double) M_PI) / a)) * (1.0 / (a * b));
	} else {
		tmp = (((((double) M_PI) / a) + (((double) M_PI) / b)) / (b - a)) * (0.5 / (a + b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.1e-141) {
		tmp = (0.5 * (Math.PI / a)) * (1.0 / (a * b));
	} else {
		tmp = (((Math.PI / a) + (Math.PI / b)) / (b - a)) * (0.5 / (a + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.1e-141:
		tmp = (0.5 * (math.pi / a)) * (1.0 / (a * b))
	else:
		tmp = (((math.pi / a) + (math.pi / b)) / (b - a)) * (0.5 / (a + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.1e-141)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) * Float64(1.0 / Float64(a * b)));
	else
		tmp = Float64(Float64(Float64(Float64(pi / a) + Float64(pi / b)) / Float64(b - a)) * Float64(0.5 / Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.1e-141)
		tmp = (0.5 * (pi / a)) * (1.0 / (a * b));
	else
		tmp = (((pi / a) + (pi / b)) / (b - a)) * (0.5 / (a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.1e-141], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi / a), $MachinePrecision] + N[(Pi / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.10000000000000027e-141

   1. Initial program 80.3%

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

      1. un-div-inv 80.3%

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

      2. difference-of-squares 92.7%

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

      3. associate-/r* 93.4%

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

      4. div-inv 93.4%

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

      5. metadata-eval 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. associate-/l/ 92.7%

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

      2. *-commutative 92.7%

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

      3. times-frac 93.3%

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

      4. +-commutative 93.3%

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

   6. Simplified 93.3%

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

   7. Taylor expanded in a around inf 67.6%

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

      1. *-commutative 67.6%

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

      2. frac-sub 67.5%

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

      3. associate-*l/ 67.6%

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

      4. frac-times 77.1%

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

      5. *-un-lft-identity 77.1%

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

   9. Applied egg-rr 77.1%

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

       1. *-commutative 77.1%

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

       2. *-commutative 77.1%

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

       3. associate-/l* 79.6%

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

       4. *-rgt-identity 79.6%

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

       5. *-commutative 79.6%

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

       6. associate-*l* 79.7%

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

   11. Simplified 79.7%

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

   12. Taylor expanded in b around 0 72.2%

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

   if 3.10000000000000027e-141 < b

   1. Initial program 87.4%

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

      1. *-commutative 87.4%

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

      2. associate-*r* 87.3%

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

      3. associate-*r/ 87.3%

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

      4. associate-*r* 87.3%

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

      5. *-rgt-identity 87.3%

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

      6. sub-neg 87.3%

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

      7. distribute-neg-frac 87.3%

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

      8. metadata-eval 87.3%

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

   3. Simplified 87.3%

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

      1. *-un-lft-identity 87.3%

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

      2. difference-of-squares 92.6%

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

      3. times-frac 99.7%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 86.5%

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

      6. frac-times 86.5%

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

      7. metadata-eval 86.5%

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

      8. metadata-eval 86.5%

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

      9. frac-times 86.5%

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

      10. sqrt-unprod 86.5%

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

      11. add-sqr-sqrt 86.5%

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

      12. div-inv 86.5%

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

      13. metadata-eval 86.5%

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

   6. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. times-frac 80.6%

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

      3. *-rgt-identity 80.6%

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

      4. associate-*r* 80.6%

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

      5. *-commutative 80.6%

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

      6. distribute-rgt-in 80.6%

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

      7. associate-*l/ 80.6%

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

      8. *-lft-identity 80.6%

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

      9. associate-*l/ 80.6%

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

      10. *-lft-identity 80.6%

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

      11. +-commutative 80.6%

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

   8. Simplified 80.6%

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

      1. associate-/l* 80.6%

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

   10. Applied egg-rr 80.6%

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

       1. associate-*r/ 80.6%

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

       2. times-frac 86.5%

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

       3. +-commutative 86.5%

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

   12. Simplified 86.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-141}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{1}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} + \frac{\pi}{b}}{b - a} \cdot \frac{0.5}{a + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-82}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{b - a}{a \cdot \left(b \cdot \left(b - a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\frac{0.5}{b - a} \cdot \frac{\pi}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e-82)
   (* (* 0.5 (/ PI a)) (/ (- b a) (* a (* b (- b a)))))
   (* (+ (/ 1.0 a) (/ -1.0 b)) (* (/ 0.5 (- b a)) (/ PI b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-82) {
		tmp = (0.5 * (((double) M_PI) / a)) * ((b - a) / (a * (b * (b - a))));
	} else {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((0.5 / (b - a)) * (((double) M_PI) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-82) {
		tmp = (0.5 * (Math.PI / a)) * ((b - a) / (a * (b * (b - a))));
	} else {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((0.5 / (b - a)) * (Math.PI / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.8e-82:
		tmp = (0.5 * (math.pi / a)) * ((b - a) / (a * (b * (b - a))))
	else:
		tmp = ((1.0 / a) + (-1.0 / b)) * ((0.5 / (b - a)) * (math.pi / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.8e-82)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) * Float64(Float64(b - a) / Float64(a * Float64(b * Float64(b - a)))));
	else
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(0.5 / Float64(b - a)) * Float64(pi / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.8e-82)
		tmp = (0.5 * (pi / a)) * ((b - a) / (a * (b * (b - a))));
	else
		tmp = ((1.0 / a) + (-1.0 / b)) * ((0.5 / (b - a)) * (pi / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.8e-82], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] / N[(a * N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.79999999999999999e-82

   1. Initial program 83.0%

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

      1. un-div-inv 83.0%

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

      2. difference-of-squares 92.8%

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

      3. associate-/r* 94.1%

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

      4. div-inv 94.1%

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

      5. metadata-eval 94.1%

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

   4. Applied egg-rr 94.1%

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

      1. associate-/l/ 92.8%

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

      2. *-commutative 92.8%

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

      3. times-frac 94.1%

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

      4. +-commutative 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in a around inf 79.9%

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

      1. *-commutative 79.9%

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

      2. frac-sub 79.8%

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

      3. associate-*l/ 79.8%

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

      4. frac-times 91.7%

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

      5. *-un-lft-identity 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. *-commutative 91.7%

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

       2. *-commutative 91.7%

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

       3. associate-/l* 91.7%

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

       4. *-rgt-identity 91.7%

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

       5. *-commutative 91.7%

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

       6. associate-*l* 91.7%

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

   11. Simplified 91.7%

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

   if -1.79999999999999999e-82 < a

   1. Initial program 82.8%

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

      1. un-div-inv 82.9%

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

      2. difference-of-squares 92.6%

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

      3. associate-/r* 92.7%

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

      4. div-inv 92.7%

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

      5. metadata-eval 92.7%

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

   4. Applied egg-rr 92.7%

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

      1. associate-/l/ 92.6%

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

      2. *-commutative 92.6%

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

      3. times-frac 92.5%

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

      4. +-commutative 92.5%

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

   6. Simplified 92.5%

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

   7. Taylor expanded in a around 0 69.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-82}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{a}\right) \cdot \frac{b - a}{a \cdot \left(b \cdot \left(b - a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\frac{0.5}{b - a} \cdot \frac{\pi}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. un-div-inv 82.9%

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

   2. difference-of-squares 92.7%

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

   3. associate-/r* 93.1%

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

   4. div-inv 93.1%

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

   5. metadata-eval 93.1%

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

4. Applied egg-rr 93.1%

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

   1. associate-/l/ 92.7%

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

   2. *-commutative 92.7%

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

   3. times-frac 93.1%

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

   4. +-commutative 93.1%

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

6. Simplified 93.1%

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

   1. clear-num 93.0%

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

   2. inv-pow 93.0%

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

8. Applied egg-rr 93.0%

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

   1. unpow-1 93.0%

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

10. Simplified 93.0%

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

    1. pow1 93.0%

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

    2. associate-*l* 99.6%

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

    3. clear-num 99.6%

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

    4. +-commutative 99.6%

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

12. Applied egg-rr 99.6%

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

    1. unpow1 99.6%

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

    2. +-commutative 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 8: 70.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. un-div-inv 82.9%

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

   2. difference-of-squares 92.7%

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

   3. associate-/r* 93.1%

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

   4. div-inv 93.1%

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

   5. metadata-eval 93.1%

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

4. Applied egg-rr 93.1%

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

   1. associate-/l/ 92.7%

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

   2. *-commutative 92.7%

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

   3. times-frac 93.1%

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

   4. +-commutative 93.1%

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

6. Simplified 93.1%

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

7. Taylor expanded in a around inf 60.8%

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

   1. *-commutative 60.8%

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

   2. frac-sub 60.7%

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

   3. associate-*l/ 60.7%

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

   4. frac-times 67.3%

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

   5. *-un-lft-identity 67.3%

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

9. Applied egg-rr 67.3%

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

    1. *-commutative 67.3%

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

    2. *-commutative 67.3%

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

    3. associate-/l* 71.2%

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

    4. *-rgt-identity 71.2%

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

    5. *-commutative 71.2%

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

    6. associate-*l* 71.4%

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

11. Simplified 71.4%

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

Alternative 9: 63.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

   1. un-div-inv 82.9%

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

   2. difference-of-squares 92.7%

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

   3. associate-/r* 93.1%

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

   4. div-inv 93.1%

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

   5. metadata-eval 93.1%

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

4. Applied egg-rr 93.1%

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

   1. associate-/l/ 92.7%

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

   2. *-commutative 92.7%

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

   3. times-frac 93.1%

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

   4. +-commutative 93.1%

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

6. Simplified 93.1%

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

7. Taylor expanded in a around inf 60.8%

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

   1. *-commutative 60.8%

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

   2. frac-sub 60.7%

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

   3. associate-*l/ 60.7%

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

   4. frac-times 67.3%

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

   5. *-un-lft-identity 67.3%

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

9. Applied egg-rr 67.3%

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

    1. *-commutative 67.3%

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

    2. *-commutative 67.3%

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

    3. associate-/l* 71.2%

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

    4. *-rgt-identity 71.2%

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

    5. *-commutative 71.2%

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

    6. associate-*l* 71.4%

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

11. Simplified 71.4%

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

12. Taylor expanded in b around 0 64.0%

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

Reproduce

herbie shell --seed 2024111 
(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))))