NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.8% → 99.6%

Time: 16.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. inv-pow 77.8%

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

   2. difference-of-squares 86.0%

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

   3. unpow-prod-down 86.2%

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

   4. inv-pow 86.2%

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

   5. inv-pow 86.2%

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

3. Applied egg-rr 86.2%

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

   1. associate-*r/ 86.3%

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

   2. *-rgt-identity 86.3%

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

   3. +-commutative 86.3%

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

5. Simplified 86.3%

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

   1. pow1 86.3%

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

   2. frac-times 86.3%

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

   3. +-commutative 86.3%

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

   4. div-inv 86.3%

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

   5. +-commutative 86.3%

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

   6. inv-pow 86.3%

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

   7. inv-pow 86.3%

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

7. Applied egg-rr 86.3%

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

   1. unpow1 86.3%

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

   2. associate-*l/ 99.7%

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

   3. unpow-1 99.7%

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

   4. unpow-1 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.35 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -2.45 \cdot 10^{-157}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.35e+78)
   (/ (/ PI a) (* 2.0 (* a b)))
   (if (<= a -2.45e-157)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (/ (/ PI 2.0) (- (* b b) (* a a))))
     (* 0.5 (/ PI (* b (* a b)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.35e+78) {
		tmp = (((double) M_PI) / a) / (2.0 * (a * b));
	} else if (a <= -2.45e-157) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((((double) M_PI) / 2.0) / ((b * b) - (a * a)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.35e+78) {
		tmp = (Math.PI / a) / (2.0 * (a * b));
	} else if (a <= -2.45e-157) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((Math.PI / 2.0) / ((b * b) - (a * a)));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.35e+78:
		tmp = (math.pi / a) / (2.0 * (a * b))
	elif a <= -2.45e-157:
		tmp = ((1.0 / a) + (-1.0 / b)) * ((math.pi / 2.0) / ((b * b) - (a * a)))
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.35e+78)
		tmp = Float64(Float64(pi / a) / Float64(2.0 * Float64(a * b)));
	elseif (a <= -2.45e-157)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(pi / 2.0) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.35e+78)
		tmp = (pi / a) / (2.0 * (a * b));
	elseif (a <= -2.45e-157)
		tmp = ((1.0 / a) + (-1.0 / b)) * ((pi / 2.0) / ((b * b) - (a * a)));
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.35e+78], N[(N[(Pi / a), $MachinePrecision] / N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.45e-157], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / 2.0), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.34999999999999983e78

   1. Initial program 62.9%

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

      1. div-inv 63.0%

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

      2. *-commutative 63.0%

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

      3. frac-sub 63.0%

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

      4. div-inv 63.0%

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

      5. associate-*l/ 63.0%

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

      6. frac-times 63.0%

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

      7. *-un-lft-identity 63.0%

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

   3. Applied egg-rr 63.0%

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

   4. Taylor expanded in b around 0 99.0%

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

   if -3.34999999999999983e78 < a < -2.4499999999999999e-157

   1. Initial program 91.4%

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

      1. associate-*r/ 91.5%

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

      2. *-rgt-identity 91.5%

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

      3. sub-neg 91.5%

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

      4. distribute-neg-frac 91.5%

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

      5. metadata-eval 91.5%

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

   3. Simplified 91.5%

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

   if -2.4499999999999999e-157 < a

   1. Initial program 80.7%

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

      1. inv-pow 80.7%

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

      2. difference-of-squares 87.2%

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

      3. unpow-prod-down 87.7%

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

      4. inv-pow 87.7%

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

      5. inv-pow 87.7%

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

   3. Applied egg-rr 87.7%

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

      1. associate-*r/ 87.7%

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

      2. *-rgt-identity 87.7%

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

      3. +-commutative 87.7%

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

   5. Simplified 87.7%

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

      1. pow1 87.7%

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

      2. frac-times 87.8%

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

      3. +-commutative 87.8%

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

      4. div-inv 87.8%

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

      5. +-commutative 87.8%

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

      6. inv-pow 87.8%

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

      7. inv-pow 87.8%

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

   7. Applied egg-rr 87.8%

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

      1. unpow1 87.8%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 67.4%

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

       1. *-commutative 67.4%

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

       2. unpow2 67.4%

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

       3. associate-*l* 76.0%

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

   12. Simplified 76.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.35 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq -2.45 \cdot 10^{-157}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 3: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{-\pi}{a}}{b}}{2 \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{a}}{b - a}}{\frac{\left(a + b\right) \cdot 2}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.85e-77)
   (/ (/ (/ (- PI) a) b) (* 2.0 (- b a)))
   (/ (/ (/ 1.0 a) (- b a)) (/ (* (+ a b) 2.0) PI))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.85e-77) {
		tmp = ((-((double) M_PI) / a) / b) / (2.0 * (b - a));
	} else {
		tmp = ((1.0 / a) / (b - a)) / (((a + b) * 2.0) / ((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.85e-77) {
		tmp = ((-Math.PI / a) / b) / (2.0 * (b - a));
	} else {
		tmp = ((1.0 / a) / (b - a)) / (((a + b) * 2.0) / Math.PI);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.85e-77:
		tmp = ((-math.pi / a) / b) / (2.0 * (b - a))
	else:
		tmp = ((1.0 / a) / (b - a)) / (((a + b) * 2.0) / math.pi)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.85e-77)
		tmp = Float64(Float64(Float64(Float64(-pi) / a) / b) / Float64(2.0 * Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(1.0 / a) / Float64(b - a)) / Float64(Float64(Float64(a + b) * 2.0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.85e-77)
		tmp = ((-pi / a) / b) / (2.0 * (b - a));
	else
		tmp = ((1.0 / a) / (b - a)) / (((a + b) * 2.0) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.85e-77], N[(N[(N[((-Pi) / a), $MachinePrecision] / b), $MachinePrecision] / N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.84999999999999998e-77

   1. Initial program 72.6%

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

      1. inv-pow 72.6%

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

      2. difference-of-squares 84.5%

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

      3. unpow-prod-down 84.5%

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

      4. inv-pow 84.5%

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

      5. inv-pow 84.5%

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

   3. Applied egg-rr 84.5%

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

      1. associate-*r/ 84.5%

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

      2. *-rgt-identity 84.5%

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

      3. +-commutative 84.5%

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

   5. Simplified 84.5%

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

      1. pow1 84.5%

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

      2. frac-times 84.4%

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

      3. +-commutative 84.4%

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

      4. div-inv 84.6%

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

      5. +-commutative 84.6%

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

      6. inv-pow 84.6%

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

      7. inv-pow 84.6%

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

   7. Applied egg-rr 84.6%

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

      1. unpow1 84.6%

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

      2. associate-*l/ 99.7%

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

      3. unpow-1 99.7%

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

      4. unpow-1 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in a around inf 94.2%

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

       1. mul-1-neg 94.2%

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

       2. associate-/r* 94.1%

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

       3. distribute-neg-frac 94.1%

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

   12. Simplified 94.1%

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

   if -1.84999999999999998e-77 < a

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l/ 80.7%

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

      3. associate-*r/ 80.7%

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

      4. associate-/l* 80.7%

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

      5. sub-neg 80.7%

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

      6. distribute-neg-frac 80.7%

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

      7. metadata-eval 80.7%

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

      8. associate-*r/ 80.7%

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

      9. *-rgt-identity 80.7%

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

      10. difference-of-squares 86.8%

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

      11. associate-/r* 86.8%

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

   3. Simplified 86.8%

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

      1. associate-/r/ 86.8%

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

   5. Applied egg-rr 86.8%

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

   6. Taylor expanded in a around 0 73.4%

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

      1. expm1-log1p-u 54.9%

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

      2. expm1-udef 49.1%

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

      3. *-commutative 49.1%

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

      4. associate-/r/ 49.1%

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

      5. +-commutative 49.1%

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

   8. Applied egg-rr 49.1%

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

      1. expm1-def 54.9%

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

      2. expm1-log1p 73.3%

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

      3. associate-/r* 82.7%

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

      4. *-commutative 82.7%

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

      5. associate-*r/ 82.7%

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

      6. +-commutative 82.7%

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

   10. Simplified 82.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{-\pi}{a}}{b}}{2 \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{a}}{b - a}}{\frac{\left(a + b\right) \cdot 2}{\pi}}\\ \end{array} \]

Alternative 4: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.08 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{\frac{-\pi}{a}}{b}}{2 \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.08e-116)
   (/ (/ (/ (- PI) a) b) (* 2.0 (- b a)))
   (* 0.5 (/ PI (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.08e-116) {
		tmp = ((-((double) M_PI) / a) / b) / (2.0 * (b - a));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.08e-116) {
		tmp = ((-Math.PI / a) / b) / (2.0 * (b - a));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.08e-116:
		tmp = ((-math.pi / a) / b) / (2.0 * (b - a))
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.08e-116)
		tmp = Float64(Float64(Float64(Float64(-pi) / a) / b) / Float64(2.0 * Float64(b - a)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.08e-116)
		tmp = ((-pi / a) / b) / (2.0 * (b - a));
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.08e-116], N[(N[(N[((-Pi) / a), $MachinePrecision] / b), $MachinePrecision] / N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.08000000000000001e-116

   1. Initial program 73.1%

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

      1. inv-pow 73.1%

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

      2. difference-of-squares 84.4%

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

      3. unpow-prod-down 84.4%

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

      4. inv-pow 84.4%

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

      5. inv-pow 84.4%

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

   3. Applied egg-rr 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

      3. +-commutative 84.4%

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

   5. Simplified 84.4%

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

      1. pow1 84.4%

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

      2. frac-times 84.3%

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

      3. +-commutative 84.3%

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

      4. div-inv 84.5%

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

      5. +-commutative 84.5%

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

      6. inv-pow 84.5%

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

      7. inv-pow 84.5%

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

   7. Applied egg-rr 84.5%

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

      1. unpow1 84.5%

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

      2. associate-*l/ 99.7%

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

      3. unpow-1 99.7%

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

      4. unpow-1 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in a around inf 90.6%

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

       1. mul-1-neg 90.6%

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

       2. associate-/r* 90.5%

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

       3. distribute-neg-frac 90.5%

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

   12. Simplified 90.5%

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

   if -1.08000000000000001e-116 < a

   1. Initial program 80.7%

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

      1. inv-pow 80.7%

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

      2. difference-of-squares 87.0%

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

      3. unpow-prod-down 87.4%

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

      4. inv-pow 87.4%

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

      5. inv-pow 87.4%

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

   3. Applied egg-rr 87.4%

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

      1. associate-*r/ 87.4%

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

      2. *-rgt-identity 87.4%

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

      3. +-commutative 87.4%

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

   5. Simplified 87.4%

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

      1. pow1 87.4%

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

      2. frac-times 87.5%

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

      3. +-commutative 87.5%

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

      4. div-inv 87.5%

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

      5. +-commutative 87.5%

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

      6. inv-pow 87.5%

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

      7. inv-pow 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. unpow1 87.5%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 68.0%

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

       1. *-commutative 68.0%

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

       2. unpow2 68.0%

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

       3. associate-*l* 76.8%

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

   12. Simplified 76.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.08 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{\frac{-\pi}{a}}{b}}{2 \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 5: 75.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{2 \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.6e-117)
   (/ (/ PI a) (* 2.0 (* a b)))
   (/ (/ PI (* a b)) (* 2.0 (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.6e-117) {
		tmp = (((double) M_PI) / a) / (2.0 * (a * b));
	} else {
		tmp = (((double) M_PI) / (a * b)) / (2.0 * (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.6e-117) {
		tmp = (Math.PI / a) / (2.0 * (a * b));
	} else {
		tmp = (Math.PI / (a * b)) / (2.0 * (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.6e-117:
		tmp = (math.pi / a) / (2.0 * (a * b))
	else:
		tmp = (math.pi / (a * b)) / (2.0 * (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.6e-117)
		tmp = Float64(Float64(pi / a) / Float64(2.0 * Float64(a * b)));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) / Float64(2.0 * Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.6e-117)
		tmp = (pi / a) / (2.0 * (a * b));
	else
		tmp = (pi / (a * b)) / (2.0 * (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.6e-117], N[(N[(Pi / a), $MachinePrecision] / N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.59999999999999989e-117

   1. Initial program 80.0%

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

      1. div-inv 80.0%

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

      2. *-commutative 80.0%

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

      3. frac-sub 80.0%

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

      4. div-inv 80.0%

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

      5. associate-*l/ 80.0%

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

      6. frac-times 80.0%

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

      7. *-un-lft-identity 80.0%

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

   3. Applied egg-rr 80.0%

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

   4. Taylor expanded in b around 0 66.5%

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

   if 4.59999999999999989e-117 < b

   1. Initial program 72.9%

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

      1. inv-pow 72.9%

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

      2. difference-of-squares 83.0%

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

      3. unpow-prod-down 83.9%

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

      4. inv-pow 83.9%

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

      5. inv-pow 83.9%

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

   3. Applied egg-rr 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-rgt-identity 83.9%

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

      3. +-commutative 83.9%

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

   5. Simplified 83.9%

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

      1. pow1 83.9%

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

      2. frac-times 84.0%

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

      3. +-commutative 84.0%

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

      4. div-inv 84.0%

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

      5. +-commutative 84.0%

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

      6. inv-pow 84.0%

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

      7. inv-pow 84.0%

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

   7. Applied egg-rr 84.0%

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

      1. unpow1 84.0%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 89.7%

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

       1. *-commutative 89.7%

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

   12. Simplified 89.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{2 \cdot \left(b - a\right)}\\ \end{array} \]

Alternative 6: 70.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.2e-76)
   (* 0.5 (/ (/ PI (* a a)) b))
   (* 0.5 (/ PI (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.2e-76) {
		tmp = 0.5 * ((((double) M_PI) / (a * a)) / b);
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9.2e-76) {
		tmp = 0.5 * ((Math.PI / (a * a)) / b);
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9.2e-76:
		tmp = 0.5 * ((math.pi / (a * a)) / b)
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.2e-76)
		tmp = Float64(0.5 * Float64(Float64(pi / Float64(a * a)) / b));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9.2e-76)
		tmp = 0.5 * ((pi / (a * a)) / b);
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.2e-76], N[(0.5 * N[(N[(Pi / N[(a * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.20000000000000025e-76

   1. Initial program 72.6%

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

      1. inv-pow 72.6%

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

      2. difference-of-squares 84.5%

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

      3. unpow-prod-down 84.5%

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

      4. inv-pow 84.5%

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

      5. inv-pow 84.5%

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

   3. Applied egg-rr 84.5%

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

      1. associate-*r/ 84.5%

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

      2. *-rgt-identity 84.5%

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

      3. +-commutative 84.5%

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

   5. Simplified 84.5%

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

   6. Taylor expanded in a around inf 69.1%

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

      1. associate-/r* 69.6%

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

      2. unpow2 69.6%

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

   8. Simplified 69.6%

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

   if -9.20000000000000025e-76 < a

   1. Initial program 80.7%

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

      1. inv-pow 80.7%

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

      2. difference-of-squares 86.8%

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

      3. unpow-prod-down 87.2%

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

      4. inv-pow 87.2%

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

      5. inv-pow 87.2%

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

   3. Applied egg-rr 87.2%

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

      1. associate-*r/ 87.3%

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

      2. *-rgt-identity 87.3%

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

      3. +-commutative 87.3%

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

   5. Simplified 87.3%

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

      1. pow1 87.3%

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

      2. frac-times 87.3%

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

      3. +-commutative 87.3%

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

      4. div-inv 87.3%

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

      5. +-commutative 87.3%

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

      6. inv-pow 87.3%

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

      7. inv-pow 87.3%

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

   7. Applied egg-rr 87.3%

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

      1. unpow1 87.3%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 68.5%

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

       1. *-commutative 68.5%

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

       2. unpow2 68.5%

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

       3. associate-*l* 77.5%

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

   12. Simplified 77.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 7: 73.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-77}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.8e-77)
   (* PI (/ 0.5 (* a (* a b))))
   (* 0.5 (/ PI (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.8e-77) {
		tmp = ((double) M_PI) * (0.5 / (a * (a * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.8e-77) {
		tmp = Math.PI * (0.5 / (a * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.8e-77:
		tmp = math.pi * (0.5 / (a * (a * b)))
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.8e-77)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.8e-77)
		tmp = pi * (0.5 / (a * (a * b)));
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.8e-77], N[(Pi * N[(0.5 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.7999999999999998e-77

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*l/ 72.6%

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

      3. associate-*r/ 72.5%

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

      4. associate-/l* 72.6%

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

      5. sub-neg 72.6%

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

      6. distribute-neg-frac 72.6%

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

      7. metadata-eval 72.6%

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

      8. associate-*r/ 72.6%

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

      9. *-rgt-identity 72.6%

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

      10. difference-of-squares 84.6%

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

      11. associate-/r* 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in b around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. unpow2 66.1%

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

   6. Simplified 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. associate-/r/ 66.1%

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

      3. inv-pow 66.1%

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

   8. Applied egg-rr 66.1%

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

      1. *-lft-identity 66.1%

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

      2. *-commutative 66.1%

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

      3. +-commutative 66.1%

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

      4. unpow-1 66.1%

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

      5. unpow2 66.1%

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

      6. *-commutative 66.1%

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

      7. unpow2 66.1%

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

   10. Simplified 66.1%

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

   11. Taylor expanded in b around 0 69.0%

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

       1. unpow2 69.0%

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

       2. associate-*l* 83.0%

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

       3. *-commutative 83.0%

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

   13. Simplified 83.0%

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

   if -4.7999999999999998e-77 < a

   1. Initial program 80.7%

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

      1. inv-pow 80.7%

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

      2. difference-of-squares 86.8%

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

      3. unpow-prod-down 87.2%

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

      4. inv-pow 87.2%

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

      5. inv-pow 87.2%

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

   3. Applied egg-rr 87.2%

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

      1. associate-*r/ 87.3%

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

      2. *-rgt-identity 87.3%

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

      3. +-commutative 87.3%

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

   5. Simplified 87.3%

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

      1. pow1 87.3%

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

      2. frac-times 87.3%

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

      3. +-commutative 87.3%

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

      4. div-inv 87.3%

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

      5. +-commutative 87.3%

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

      6. inv-pow 87.3%

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

      7. inv-pow 87.3%

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

   7. Applied egg-rr 87.3%

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

      1. unpow1 87.3%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 68.5%

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

       1. *-commutative 68.5%

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

       2. unpow2 68.5%

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

       3. associate-*l* 77.5%

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

   12. Simplified 77.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-77}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 8: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-75}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e-75) (/ PI (* a (* 2.0 (* a b)))) (* 0.5 (/ PI (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e-75) {
		tmp = ((double) M_PI) / (a * (2.0 * (a * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e-75) {
		tmp = Math.PI / (a * (2.0 * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e-75:
		tmp = math.pi / (a * (2.0 * (a * b)))
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e-75)
		tmp = Float64(pi / Float64(a * Float64(2.0 * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e-75)
		tmp = pi / (a * (2.0 * (a * b)));
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e-75], N[(Pi / N[(a * N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999996e-76

   1. Initial program 72.6%

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

      1. div-inv 72.6%

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

      2. *-commutative 72.6%

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

      3. frac-sub 72.6%

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

      4. div-inv 72.5%

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

      5. associate-*l/ 72.5%

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

      6. frac-times 72.6%

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

      7. *-un-lft-identity 72.6%

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

   3. Applied egg-rr 72.6%

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

   4. Taylor expanded in b around 0 83.7%

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

      1. expm1-log1p-u 73.7%

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

      2. expm1-udef 58.9%

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

      3. associate-/l/ 58.9%

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

      4. associate-*l* 58.9%

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

   6. Applied egg-rr 58.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{\left(a \cdot \left(b \cdot 2\right)\right) \cdot a}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 73.0%

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

      2. expm1-log1p 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-*r* 83.1%

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

      5. *-commutative 83.1%

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

      6. *-commutative 83.1%

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

   8. Simplified 83.1%

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

   if -9.9999999999999996e-76 < a

   1. Initial program 80.7%

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

      1. inv-pow 80.7%

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

      2. difference-of-squares 86.8%

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

      3. unpow-prod-down 87.2%

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

      4. inv-pow 87.2%

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

      5. inv-pow 87.2%

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

   3. Applied egg-rr 87.2%

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

      1. associate-*r/ 87.3%

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

      2. *-rgt-identity 87.3%

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

      3. +-commutative 87.3%

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

   5. Simplified 87.3%

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

      1. pow1 87.3%

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

      2. frac-times 87.3%

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

      3. +-commutative 87.3%

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

      4. div-inv 87.3%

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

      5. +-commutative 87.3%

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

      6. inv-pow 87.3%

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

      7. inv-pow 87.3%

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

   7. Applied egg-rr 87.3%

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

      1. unpow1 87.3%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 68.5%

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

       1. *-commutative 68.5%

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

       2. unpow2 68.5%

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

       3. associate-*l* 77.5%

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

   12. Simplified 77.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-75}:\\ \;\;\;\;\frac{\pi}{a \cdot \left(2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 9: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e-75) (/ (/ PI a) (* 2.0 (* a b))) (* 0.5 (/ PI (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e-75) {
		tmp = (((double) M_PI) / a) / (2.0 * (a * b));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e-75) {
		tmp = (Math.PI / a) / (2.0 * (a * b));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e-75:
		tmp = (math.pi / a) / (2.0 * (a * b))
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e-75)
		tmp = Float64(Float64(pi / a) / Float64(2.0 * Float64(a * b)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e-75)
		tmp = (pi / a) / (2.0 * (a * b));
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e-75], N[(N[(Pi / a), $MachinePrecision] / N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999996e-76

   1. Initial program 72.6%

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

      1. div-inv 72.6%

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

      2. *-commutative 72.6%

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

      3. frac-sub 72.6%

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

      4. div-inv 72.5%

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

      5. associate-*l/ 72.5%

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

      6. frac-times 72.6%

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

      7. *-un-lft-identity 72.6%

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

   3. Applied egg-rr 72.6%

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

   4. Taylor expanded in b around 0 83.7%

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

   if -9.9999999999999996e-76 < a

   1. Initial program 80.7%

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

      1. inv-pow 80.7%

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

      2. difference-of-squares 86.8%

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

      3. unpow-prod-down 87.2%

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

      4. inv-pow 87.2%

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

      5. inv-pow 87.2%

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

   3. Applied egg-rr 87.2%

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

      1. associate-*r/ 87.3%

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

      2. *-rgt-identity 87.3%

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

      3. +-commutative 87.3%

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

   5. Simplified 87.3%

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

      1. pow1 87.3%

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

      2. frac-times 87.3%

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

      3. +-commutative 87.3%

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

      4. div-inv 87.3%

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

      5. +-commutative 87.3%

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

      6. inv-pow 87.3%

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

      7. inv-pow 87.3%

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

   7. Applied egg-rr 87.3%

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

      1. unpow1 87.3%

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

      2. associate-*l/ 99.6%

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

      3. unpow-1 99.6%

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

      4. unpow-1 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in a around 0 68.5%

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

       1. *-commutative 68.5%

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

       2. unpow2 68.5%

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

       3. associate-*l* 77.5%

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

   12. Simplified 77.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 10: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 2.0 (* a b))))
   (if (<= a -3.8e-65) (/ (/ PI a) t_0) (/ (/ PI b) t_0))))

C

double code(double a, double b) {
	double t_0 = 2.0 * (a * b);
	double tmp;
	if (a <= -3.8e-65) {
		tmp = (((double) M_PI) / a) / t_0;
	} else {
		tmp = (((double) M_PI) / b) / t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 2.0 * (a * b);
	double tmp;
	if (a <= -3.8e-65) {
		tmp = (Math.PI / a) / t_0;
	} else {
		tmp = (Math.PI / b) / t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 2.0 * (a * b)
	tmp = 0
	if a <= -3.8e-65:
		tmp = (math.pi / a) / t_0
	else:
		tmp = (math.pi / b) / t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(2.0 * Float64(a * b))
	tmp = 0.0
	if (a <= -3.8e-65)
		tmp = Float64(Float64(pi / a) / t_0);
	else
		tmp = Float64(Float64(pi / b) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 2.0 * (a * b);
	tmp = 0.0;
	if (a <= -3.8e-65)
		tmp = (pi / a) / t_0;
	else
		tmp = (pi / b) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(2.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.8e-65], N[(N[(Pi / a), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8000000000000002e-65

   1. Initial program 72.6%

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

      1. div-inv 72.7%

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

      2. *-commutative 72.7%

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

      3. frac-sub 72.6%

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

      4. div-inv 72.6%

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

      5. associate-*l/ 72.6%

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

      6. frac-times 72.6%

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

      7. *-un-lft-identity 72.6%

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

   3. Applied egg-rr 72.6%

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

   4. Taylor expanded in b around 0 85.4%

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

   if -3.8000000000000002e-65 < a

   1. Initial program 80.6%

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

      1. div-inv 80.6%

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

      2. *-commutative 80.6%

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

      3. frac-sub 80.6%

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

      4. div-inv 80.6%

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

      5. associate-*l/ 80.6%

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

      6. frac-times 80.6%

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

      7. *-un-lft-identity 80.6%

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

   3. Applied egg-rr 80.6%

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

   4. Taylor expanded in b around inf 78.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{2 \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 11: 62.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. inv-pow 77.8%

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

   2. difference-of-squares 86.0%

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

   3. unpow-prod-down 86.2%

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

   4. inv-pow 86.2%

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

   5. inv-pow 86.2%

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

3. Applied egg-rr 86.2%

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

   1. associate-*r/ 86.3%

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

   2. *-rgt-identity 86.3%

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

   3. +-commutative 86.3%

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

5. Simplified 86.3%

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

   1. pow1 86.3%

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

   2. frac-times 86.3%

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

   3. +-commutative 86.3%

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

   4. div-inv 86.3%

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

   5. +-commutative 86.3%

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

   6. inv-pow 86.3%

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

   7. inv-pow 86.3%

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

7. Applied egg-rr 86.3%

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

   1. unpow1 86.3%

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

   2. associate-*l/ 99.7%

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

   3. unpow-1 99.7%

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

   4. unpow-1 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in a around 0 60.5%

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

    1. *-commutative 60.5%

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

    2. unpow2 60.5%

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

    3. associate-*l* 66.4%

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

12. Simplified 66.4%

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

13. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2023274 
(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))))