NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.7%

Time: 10.9s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. associate-*l* 79.2%

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

   2. associate-*l/ 79.2%

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

   3. *-lft-identity 79.2%

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

   4. difference-of-squares 87.0%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a around 0 99.6%

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

   1. associate-*r/ 99.6%

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

   2. associate-/l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. expm1-log1p-u 78.1%

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

   5. expm1-udef 53.0%

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

7. Applied egg-rr 53.0%

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

   1. expm1-def 77.5%

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

   2. expm1-log1p 99.0%

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

   3. *-rgt-identity 99.0%

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

   4. *-commutative 99.0%

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

   5. times-frac 99.0%

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

   6. *-commutative 99.0%

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

   7. associate-/r* 99.6%

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

   8. associate-/l/ 99.6%

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

   9. times-frac 99.6%

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

   10. associate-*r/ 99.6%

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

   11. associate-*r/ 99.7%

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

   12. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in b around 0 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{\frac{\pi}{a \cdot b}}{2 \cdot \left(a + b\right)} \]
12. Add Preprocessing

Alternative 2: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-306} \lor \neg \left(b \leq 4.3 \cdot 10^{-203}\right):\\ \;\;\;\;\frac{\pi}{2 \cdot \left(b \cdot \left(a \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b 1.7e-306) (not (<= b 4.3e-203)))
   (/ PI (* 2.0 (* b (* a b))))
   (* (/ PI (* a b)) (/ -0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= 1.7e-306) || !(b <= 4.3e-203)) {
		tmp = ((double) M_PI) / (2.0 * (b * (a * b)));
	} else {
		tmp = (((double) M_PI) / (a * b)) * (-0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= 1.7e-306) || !(b <= 4.3e-203)) {
		tmp = Math.PI / (2.0 * (b * (a * b)));
	} else {
		tmp = (Math.PI / (a * b)) * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= 1.7e-306) or not (b <= 4.3e-203):
		tmp = math.pi / (2.0 * (b * (a * b)))
	else:
		tmp = (math.pi / (a * b)) * (-0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= 1.7e-306) || !(b <= 4.3e-203))
		tmp = Float64(pi / Float64(2.0 * Float64(b * Float64(a * b))));
	else
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= 1.7e-306) || ~((b <= 4.3e-203)))
		tmp = pi / (2.0 * (b * (a * b)));
	else
		tmp = (pi / (a * b)) * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, 1.7e-306], N[Not[LessEqual[b, 4.3e-203]], $MachinePrecision]], N[(Pi / N[(2.0 * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6999999999999999e-306 or 4.30000000000000027e-203 < b

   1. Initial program 80.5%

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

      1. associate-*l* 80.5%

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

      2. associate-*l/ 80.5%

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

      3. *-lft-identity 80.5%

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

      4. difference-of-squares 88.4%

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

      5. associate-/l/ 99.6%

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

      6. sub-neg 99.6%

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

      7. distribute-neg-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around 0 99.7%

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

      1. div-inv 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/l/ 99.1%

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

      6. *-commutative 99.1%

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

      7. metadata-eval 99.1%

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

      8. div-inv 99.1%

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

      9. frac-times 99.1%

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

      10. *-un-lft-identity 99.1%

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

      11. +-commutative 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. expm1-log1p-u 66.1%

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

      2. expm1-udef 35.6%

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

      3. +-commutative 35.6%

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

      4. *-commutative 35.6%

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

   9. Applied egg-rr 35.6%

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

       1. expm1-def 66.1%

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

       2. expm1-log1p 99.1%

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

       3. *-commutative 99.1%

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

       4. associate-*l* 95.4%

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

       5. +-commutative 95.4%

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

   11. Simplified 95.4%

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

   12. Taylor expanded in a around 0 64.5%

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

   if 1.6999999999999999e-306 < b < 4.30000000000000027e-203

   1. Initial program 69.0%

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

      1. associate-*l* 69.0%

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

      2. associate-*l/ 69.0%

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

      3. *-lft-identity 69.0%

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

      4. difference-of-squares 75.9%

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

      5. associate-/l/ 99.2%

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

      6. sub-neg 99.2%

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

      7. distribute-neg-frac 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in a around 0 99.5%

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

      1. div-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l/ 99.2%

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

      6. associate-/l/ 75.9%

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

      7. *-commutative 75.9%

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

      8. metadata-eval 75.9%

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

      9. div-inv 75.9%

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

      10. frac-2neg 75.9%

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

      11. frac-times 76.0%

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

      12. +-commutative 76.0%

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

      13. metadata-eval 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-/r* 76.0%

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

      3. associate-*l/ 76.0%

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

      4. *-lft-identity 76.0%

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

      5. *-commutative 76.0%

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

   9. Simplified 76.0%

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

   10. Taylor expanded in a around 0 15.9%

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

       1. expm1-log1p-u 12.3%

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

       2. expm1-udef 12.3%

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

       3. div-inv 12.3%

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

       4. times-frac 12.3%

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

       5. add-sqr-sqrt 0.0%

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

       6. sqrt-unprod 17.8%

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

       7. sqr-neg 17.8%

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

       8. sqrt-unprod 17.8%

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

       9. add-sqr-sqrt 17.8%

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

       10. metadata-eval 17.8%

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

   12. Applied egg-rr 17.8%

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

       1. expm1-def 17.8%

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

       2. expm1-log1p 18.4%

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

       3. associate-/l/ 18.4%

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

   14. Simplified 18.4%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-306} \lor \neg \left(b \leq 4.3 \cdot 10^{-203}\right):\\ \;\;\;\;\frac{\pi}{2 \cdot \left(b \cdot \left(a \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{\pi}{2 \cdot \left(b \cdot \left(a \cdot b\right)\right)}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-203}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a \cdot \left(b \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6e-309)
   (/ PI (* 2.0 (* b (* a b))))
   (if (<= b 5.2e-203)
     (* (/ PI (* a b)) (/ -0.5 b))
     (/ (/ PI b) (* a (* b 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6e-309) {
		tmp = ((double) M_PI) / (2.0 * (b * (a * b)));
	} else if (b <= 5.2e-203) {
		tmp = (((double) M_PI) / (a * b)) * (-0.5 / b);
	} else {
		tmp = (((double) M_PI) / b) / (a * (b * 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6e-309) {
		tmp = Math.PI / (2.0 * (b * (a * b)));
	} else if (b <= 5.2e-203) {
		tmp = (Math.PI / (a * b)) * (-0.5 / b);
	} else {
		tmp = (Math.PI / b) / (a * (b * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6e-309:
		tmp = math.pi / (2.0 * (b * (a * b)))
	elif b <= 5.2e-203:
		tmp = (math.pi / (a * b)) * (-0.5 / b)
	else:
		tmp = (math.pi / b) / (a * (b * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6e-309)
		tmp = Float64(pi / Float64(2.0 * Float64(b * Float64(a * b))));
	elseif (b <= 5.2e-203)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(-0.5 / b));
	else
		tmp = Float64(Float64(pi / b) / Float64(a * Float64(b * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6e-309)
		tmp = pi / (2.0 * (b * (a * b)));
	elseif (b <= 5.2e-203)
		tmp = (pi / (a * b)) * (-0.5 / b);
	else
		tmp = (pi / b) / (a * (b * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6e-309], N[(Pi / N[(2.0 * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-203], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(a * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.000000000000001e-309

   1. Initial program 79.5%

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

      1. associate-*l* 79.5%

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

      2. associate-*l/ 79.5%

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

      3. *-lft-identity 79.5%

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

      4. difference-of-squares 88.5%

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

      5. associate-/l/ 99.6%

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

      6. sub-neg 99.6%

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

      7. distribute-neg-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around 0 99.7%

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

      1. div-inv 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

      8. div-inv 99.7%

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

      9. frac-times 99.7%

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

      10. *-un-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. expm1-log1p-u 52.6%

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

      2. expm1-udef 17.8%

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

      3. +-commutative 17.8%

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

      4. *-commutative 17.8%

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

   9. Applied egg-rr 17.8%

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

       1. expm1-def 52.6%

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

       2. expm1-log1p 99.7%

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

       3. *-commutative 99.7%

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

       4. associate-*l* 95.9%

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

       5. +-commutative 95.9%

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

   11. Simplified 95.9%

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

   12. Taylor expanded in a around 0 60.6%

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

   if -6.000000000000001e-309 < b < 5.19999999999999951e-203

   1. Initial program 69.0%

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

      1. associate-*l* 69.0%

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

      2. associate-*l/ 69.0%

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

      3. *-lft-identity 69.0%

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

      4. difference-of-squares 75.9%

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

      5. associate-/l/ 99.2%

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

      6. sub-neg 99.2%

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

      7. distribute-neg-frac 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in a around 0 99.5%

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

      1. div-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/l/ 99.2%

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

      6. associate-/l/ 75.9%

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

      7. *-commutative 75.9%

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

      8. metadata-eval 75.9%

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

      9. div-inv 75.9%

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

      10. frac-2neg 75.9%

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

      11. frac-times 76.0%

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

      12. +-commutative 76.0%

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

      13. metadata-eval 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-/r* 76.0%

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

      3. associate-*l/ 76.0%

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

      4. *-lft-identity 76.0%

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

      5. *-commutative 76.0%

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

   9. Simplified 76.0%

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

   10. Taylor expanded in a around 0 15.9%

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

       1. expm1-log1p-u 12.3%

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

       2. expm1-udef 12.3%

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

       3. div-inv 12.3%

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

       4. times-frac 12.3%

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

       5. add-sqr-sqrt 0.0%

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

       6. sqrt-unprod 17.8%

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

       7. sqr-neg 17.8%

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

       8. sqrt-unprod 17.8%

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

       9. add-sqr-sqrt 17.8%

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

       10. metadata-eval 17.8%

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

   12. Applied egg-rr 17.8%

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

       1. expm1-def 17.8%

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

       2. expm1-log1p 18.4%

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

       3. associate-/l/ 18.4%

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

   14. Simplified 18.4%

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

   if 5.19999999999999951e-203 < b

   1. Initial program 81.6%

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

      1. associate-*l* 81.7%

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

      2. associate-*l/ 81.7%

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

      3. *-lft-identity 81.7%

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

      4. difference-of-squares 88.4%

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

      5. associate-/l/ 99.6%

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

      6. sub-neg 99.6%

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

      7. distribute-neg-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around 0 99.6%

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

      1. div-inv 99.5%

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

      2. associate-/l/ 99.6%

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

      3. +-commutative 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. frac-times 96.0%

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

      4. *-un-lft-identity 96.0%

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

      5. times-frac 96.0%

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

      6. associate-*l/ 96.0%

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

      7. *-un-lft-identity 96.0%

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

      8. associate-*l* 96.0%

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

      9. +-commutative 96.0%

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

   9. Applied egg-rr 96.0%

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

   10. Taylor expanded in b around inf 70.2%

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

       1. associate-*r* 70.2%

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

       2. *-commutative 70.2%

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

       3. associate-*l* 70.2%

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

       4. *-commutative 70.2%

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

   12. Simplified 70.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{\pi}{2 \cdot \left(b \cdot \left(a \cdot b\right)\right)}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-203}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a \cdot \left(b \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{1}{b}}{a \cdot 2} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a \cdot \left(b \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e-44)
   (* (/ (/ 1.0 b) (* a 2.0)) (/ PI a))
   (/ (/ PI b) (* a (* b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-44) {
		tmp = ((1.0 / b) / (a * 2.0)) * (((double) M_PI) / a);
	} else {
		tmp = (((double) M_PI) / b) / (a * (b * 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-44) {
		tmp = ((1.0 / b) / (a * 2.0)) * (Math.PI / a);
	} else {
		tmp = (Math.PI / b) / (a * (b * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.7e-44:
		tmp = ((1.0 / b) / (a * 2.0)) * (math.pi / a)
	else:
		tmp = (math.pi / b) / (a * (b * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e-44)
		tmp = Float64(Float64(Float64(1.0 / b) / Float64(a * 2.0)) * Float64(pi / a));
	else
		tmp = Float64(Float64(pi / b) / Float64(a * Float64(b * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.7e-44)
		tmp = ((1.0 / b) / (a * 2.0)) * (pi / a);
	else
		tmp = (pi / b) / (a * (b * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e-44], N[(N[(N[(1.0 / b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(a * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6999999999999999e-44

   1. Initial program 80.2%

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

      1. associate-*l* 80.3%

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

      2. associate-*l/ 80.2%

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

      3. *-lft-identity 80.2%

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

      4. difference-of-squares 91.9%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around 0 99.6%

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

      1. div-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l/ 99.5%

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

      6. *-commutative 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.5%

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

      9. frac-times 99.7%

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

      10. *-un-lft-identity 99.7%

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

      11. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-/r* 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. *-commutative 99.7%

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

      4. times-frac 99.7%

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

      5. associate-/l/ 99.6%

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

      6. times-frac 91.9%

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

      7. *-commutative 91.9%

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

      8. associate-/r* 91.9%

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

      9. associate-*l* 91.9%

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

      10. *-commutative 91.9%

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

      11. times-frac 99.6%

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

      12. +-commutative 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in b around 0 90.2%

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

   if -2.6999999999999999e-44 < a

   1. Initial program 78.9%

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

      1. associate-*l* 78.8%

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

      2. associate-*l/ 78.9%

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

      3. *-lft-identity 78.9%

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

      4. difference-of-squares 85.5%

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

      5. associate-/l/ 99.6%

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

      6. sub-neg 99.6%

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

      7. distribute-neg-frac 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around 0 99.6%

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

      1. div-inv 99.6%

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

      2. associate-/l/ 99.5%

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

      3. +-commutative 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. frac-times 94.3%

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

      4. *-un-lft-identity 94.3%

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

      5. times-frac 94.2%

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

      6. associate-*l/ 94.3%

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

      7. *-un-lft-identity 94.3%

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

      8. associate-*l* 94.3%

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

      9. +-commutative 94.3%

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

   9. Applied egg-rr 94.3%

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

   10. Taylor expanded in b around inf 65.0%

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

       1. associate-*r* 65.5%

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

       2. *-commutative 65.5%

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

       3. associate-*l* 65.0%

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

       4. *-commutative 65.0%

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

   12. Simplified 65.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{1}{b}}{a \cdot 2} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a \cdot \left(b \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+114}:\\ \;\;\;\;\frac{\pi}{2 \cdot \left(a \cdot \left(b \cdot \left(a + b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{1}{b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95e+114)
   (/ PI (* 2.0 (* a (* b (+ a b)))))
   (* (/ PI b) (/ (/ 1.0 b) (* a 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95e+114) {
		tmp = ((double) M_PI) / (2.0 * (a * (b * (a + b))));
	} else {
		tmp = (((double) M_PI) / b) * ((1.0 / b) / (a * 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.95e+114) {
		tmp = Math.PI / (2.0 * (a * (b * (a + b))));
	} else {
		tmp = (Math.PI / b) * ((1.0 / b) / (a * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.95e+114:
		tmp = math.pi / (2.0 * (a * (b * (a + b))))
	else:
		tmp = (math.pi / b) * ((1.0 / b) / (a * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95e+114)
		tmp = Float64(pi / Float64(2.0 * Float64(a * Float64(b * Float64(a + b)))));
	else
		tmp = Float64(Float64(pi / b) * Float64(Float64(1.0 / b) / Float64(a * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.95e+114)
		tmp = pi / (2.0 * (a * (b * (a + b))));
	else
		tmp = (pi / b) * ((1.0 / b) / (a * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95e+114], N[(Pi / N[(2.0 * N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(N[(1.0 / b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.95e114

   1. Initial program 82.4%

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

      1. associate-*l* 82.4%

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

      2. associate-*l/ 82.4%

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

      3. *-lft-identity 82.4%

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

      4. difference-of-squares 88.4%

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

      5. associate-/l/ 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around 0 99.6%

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

      1. div-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l/ 99.4%

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

      6. *-commutative 99.4%

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

      7. metadata-eval 99.4%

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

      8. div-inv 99.4%

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

      9. frac-times 99.5%

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

      10. *-un-lft-identity 99.5%

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

      11. +-commutative 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. expm1-log1p-u 70.4%

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

      2. expm1-udef 34.7%

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

      3. +-commutative 34.7%

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

      4. *-commutative 34.7%

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

   9. Applied egg-rr 34.7%

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

       1. expm1-def 70.4%

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

       2. expm1-log1p 99.5%

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

       3. associate-*r* 95.3%

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

       4. *-commutative 95.3%

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

       5. *-commutative 95.3%

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

       6. +-commutative 95.3%

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

   11. Simplified 95.3%

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

   if 1.95e114 < b

   1. Initial program 60.6%

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

      1. associate-*l* 60.5%

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

      2. associate-*l/ 60.6%

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

      3. *-lft-identity 60.6%

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

      4. difference-of-squares 79.0%

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

      5. associate-/l/ 99.8%

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

      6. sub-neg 99.8%

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

      7. distribute-neg-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around 0 99.6%

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

      1. div-inv 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l/ 96.4%

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

      6. *-commutative 96.4%

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

      7. metadata-eval 96.4%

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

      8. div-inv 96.4%

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

      9. frac-times 96.4%

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

      10. *-un-lft-identity 96.4%

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

      11. +-commutative 96.4%

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

   7. Applied egg-rr 96.4%

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

      1. associate-/r* 96.4%

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

      2. *-un-lft-identity 96.4%

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

      3. *-commutative 96.4%

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

      4. times-frac 99.7%

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

      5. associate-/l/ 99.8%

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

      6. times-frac 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 99.7%

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

      9. associate-*l* 99.7%

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

      10. *-commutative 99.7%

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

      11. times-frac 99.8%

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

      12. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in b around inf 99.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+114}:\\ \;\;\;\;\frac{\pi}{2 \cdot \left(a \cdot \left(b \cdot \left(a + b\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{1}{b}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. associate-*l* 79.2%

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

   2. associate-*l/ 79.2%

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

   3. *-lft-identity 79.2%

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

   4. difference-of-squares 87.0%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a around 0 99.6%

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

   1. div-inv 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-commutative 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-/l/ 99.0%

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

   6. *-commutative 99.0%

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

   7. metadata-eval 99.0%

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

   8. div-inv 99.0%

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

   9. frac-times 99.0%

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

   10. *-un-lft-identity 99.0%

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

   11. +-commutative 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

   \[\leadsto \frac{\pi}{2 \cdot \left(\left(a \cdot b\right) \cdot \left(a + b\right)\right)} \]
9. Add Preprocessing

Alternative 7: 31.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. associate-*l* 79.2%

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

   2. associate-*l/ 79.2%

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

   3. *-lft-identity 79.2%

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

   4. difference-of-squares 87.0%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a around 0 99.6%

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

   1. div-inv 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-commutative 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-/l/ 99.6%

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

   6. associate-/l/ 93.7%

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

   7. *-commutative 93.7%

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

   8. metadata-eval 93.7%

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

   9. div-inv 93.7%

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

   10. frac-2neg 93.7%

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

   11. frac-times 93.7%

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

   12. +-commutative 93.7%

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

   13. metadata-eval 93.7%

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

7. Applied egg-rr 93.7%

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

   1. *-commutative 93.7%

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

   2. associate-/r* 93.7%

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

   3. associate-*l/ 93.7%

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

   4. *-lft-identity 93.7%

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

   5. *-commutative 93.7%

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

9. Simplified 93.7%

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

10. Taylor expanded in a around 0 59.3%

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

    1. expm1-log1p-u 45.7%

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

    2. expm1-udef 41.4%

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

    3. div-inv 41.4%

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

    4. times-frac 41.4%

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

    5. add-sqr-sqrt 0.0%

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

    6. sqrt-unprod 31.5%

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

    7. sqr-neg 31.5%

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

    8. sqrt-unprod 31.5%

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

    9. add-sqr-sqrt 31.5%

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

    10. metadata-eval 31.5%

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

12. Applied egg-rr 31.5%

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

    1. expm1-def 23.8%

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

    2. expm1-log1p 27.3%

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

    3. associate-/l/ 27.3%

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

14. Simplified 27.3%

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

15. Final simplification 27.3%

    \[\leadsto \frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b} \]
16. Add Preprocessing

Alternative 8: 31.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. associate-*l* 79.2%

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

   2. associate-*l/ 79.2%

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

   3. *-lft-identity 79.2%

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

   4. difference-of-squares 87.0%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a around 0 99.6%

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

   1. div-inv 99.6%

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

   2. metadata-eval 99.6%

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

   3. *-commutative 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-/l/ 99.6%

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

   6. associate-/l/ 93.7%

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

   7. *-commutative 93.7%

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

   8. metadata-eval 93.7%

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

   9. div-inv 93.7%

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

   10. frac-2neg 93.7%

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

   11. frac-times 93.7%

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

   12. +-commutative 93.7%

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

   13. metadata-eval 93.7%

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

7. Applied egg-rr 93.7%

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

   1. *-commutative 93.7%

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

   2. associate-/r* 93.7%

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

   3. associate-*l/ 93.7%

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

   4. *-lft-identity 93.7%

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

   5. *-commutative 93.7%

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

9. Simplified 93.7%

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

10. Taylor expanded in a around 0 59.3%

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

    1. div-inv 59.3%

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

    2. *-commutative 59.3%

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

    3. times-frac 53.5%

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

    4. add-sqr-sqrt 0.0%

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

    5. sqrt-unprod 27.4%

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

    6. sqr-neg 27.4%

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

    7. sqrt-unprod 27.4%

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

    8. add-sqr-sqrt 27.4%

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

    9. metadata-eval 27.4%

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

12. Applied egg-rr 27.4%

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

13. Final simplification 27.4%

    \[\leadsto \frac{\frac{\pi}{b}}{b} \cdot \frac{-0.5}{a} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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))))