NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 11.3s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.8% 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, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\pi \cdot 0.5}{b + a} \cdot \frac{1}{b \cdot a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. associate-*r/ N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. *-lft-identity N/A

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

   8. times-frac N/A

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

   9. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

   1. frac-times N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. div-inv N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. associate-*l/ N/A

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

   10. associate-/r/ N/A

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

   11. *-commutative N/A

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

   12. difference-of-squares N/A

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

   13. flip-+ N/A

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

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\pi \cdot 0.5}{b + a} \cdot \frac{1}{b \cdot a}} \]
7. Add Preprocessing

Alternative 2: 96.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e+88)
   (/ (/ (* PI 0.5) (* b a)) a)
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+88) {
		tmp = ((((double) M_PI) * 0.5) / (b * a)) / a;
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e+88) {
		tmp = ((Math.PI * 0.5) / (b * a)) / a;
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.4e+88:
		tmp = ((math.pi * 0.5) / (b * a)) / a
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.4e+88)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b * a)) / a);
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e+88)
		tmp = ((pi * 0.5) / (b * a)) / a;
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.4e+88], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.40000000000000031e88

   1. Initial program 71.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 99.7

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

   5. Simplified 99.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. PI-lowering-PI.f64 N/A

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

      9. *-lowering-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -5.40000000000000031e88 < a

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-*r/ N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. times-frac N/A

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

      9. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.6%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-inv N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. *-commutative N/A

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

      12. difference-of-squares N/A

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

      13. flip-+ N/A

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

   6. Applied egg-rr 99.7%

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

      1. frac-times N/A

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

      2. *-rgt-identity N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lowering-+.f64 97.1

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

   8. Applied egg-rr 97.1%

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

Alternative 3: 96.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.6e+94)
   (/ (* PI (/ 0.5 (* b a))) a)
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.6e+94) {
		tmp = (((double) M_PI) * (0.5 / (b * a))) / a;
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.6e+94) {
		tmp = (Math.PI * (0.5 / (b * a))) / a;
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.6e+94:
		tmp = (math.pi * (0.5 / (b * a))) / a
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.6e+94)
		tmp = Float64(Float64(pi * Float64(0.5 / Float64(b * a))) / a);
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.6e+94)
		tmp = (pi * (0.5 / (b * a))) / a;
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.6e+94], N[(N[(Pi * N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5999999999999999e94

   1. Initial program 70.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 99.8

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

   5. Simplified 99.8%

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

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 99.9

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

   7. Applied egg-rr 99.9%

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

      1. associate-/l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. PI-lowering-PI.f64 99.8

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

   9. Applied egg-rr 99.8%

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

   if -2.5999999999999999e94 < a

   1. Initial program 84.0%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-*r/ N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. times-frac N/A

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

      9. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-inv N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. *-commutative N/A

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

      12. difference-of-squares N/A

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

      13. flip-+ N/A

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

   6. Applied egg-rr 99.6%

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

      1. frac-times N/A

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

      2. *-rgt-identity N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lowering-+.f64 97.1

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

   8. Applied egg-rr 97.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.35e+154)
   (* PI (/ 0.5 (* a (* b a))))
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.35e+154) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.35e+154) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.35e+154:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.35e+154)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.35e+154)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.35e+154], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35000000000000003e154

   1. Initial program 59.4%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 99.8

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

   5. Simplified 99.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 84.4

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

   7. Applied egg-rr 84.4%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. PI-lowering-PI.f64 99.8

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

   9. Applied egg-rr 99.8%

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

   if -1.35000000000000003e154 < a

   1. Initial program 84.6%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-*r/ N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. times-frac N/A

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

      9. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-inv N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. *-commutative N/A

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

      12. difference-of-squares N/A

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

      13. flip-+ N/A

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

   6. Applied egg-rr 99.6%

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

      1. frac-times N/A

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

      2. *-rgt-identity N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-lowering-*.f64 N/A

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

      16. +-lowering-+.f64 97.2

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

   8. Applied egg-rr 97.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+109}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+109)
   (* PI (/ 0.5 (* a (* b a))))
   (* PI (/ 0.5 (* b (* a (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+109) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+109) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (b + a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+109:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi * (0.5 / (b * (a * (b + a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+109)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(b + a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+109)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi * (0.5 / (b * (a * (b + a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+109], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.0000000000000001e109

   1. Initial program 65.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 99.9

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

   5. Simplified 99.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 86.6

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

   7. Applied egg-rr 86.6%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. PI-lowering-PI.f64 99.9

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

   9. Applied egg-rr 99.9%

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

   if -5.0000000000000001e109 < a

   1. Initial program 84.3%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. associate-*r/ N/A

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

      4. difference-of-squares N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. times-frac N/A

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

      9. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-inv N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. *-commutative N/A

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

      12. difference-of-squares N/A

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

      13. flip-+ N/A

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

   6. Applied egg-rr 99.6%

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

      1. un-div-inv N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. +-commutative N/A

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

      18. *-lowering-*.f64 N/A

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

      19. +-lowering-+.f64 N/A

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

      20. PI-lowering-PI.f64 97.1

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

   8. Applied egg-rr 97.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+109}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. associate-*r/ N/A

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

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. *-lft-identity N/A

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

   8. times-frac N/A

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

   9. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

   1. frac-times N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. div-inv N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. associate-*l/ N/A

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

   10. associate-/r/ N/A

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

   11. *-commutative N/A

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

   12. difference-of-squares N/A

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

   13. flip-+ N/A

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

6. Applied egg-rr 99.7%

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

   1. un-div-inv N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. +-lowering-+.f64 99.7

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

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\pi}{b \cdot a} \cdot \frac{0.5}{b + a}} \]
9. Add Preprocessing

Alternative 7: 74.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.55 \cdot 10^{-55}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.55e-55)
   (/ (* PI 0.5) (* a (* b a)))
   (/ (* PI 0.5) (* b (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.55e-55) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (b * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.55e-55) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = (Math.PI * 0.5) / (b * (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.55e-55:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = (math.pi * 0.5) / (b * (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.55e-55)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.55e-55)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = (pi * 0.5) / (b * (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.55e-55], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.54999999999999998e-55

   1. Initial program 84.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 79.3

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

   5. Simplified 79.3%

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

   if -2.54999999999999998e-55 < a

   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. Add Preprocessing
   3. Step-by-step derivation

      1. frac-sub N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 90.6%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. flip-+ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. PI-lowering-PI.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 68.6

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

   9. Simplified 68.6%

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

       1. associate-*l/ N/A

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

       2. *-commutative N/A

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

       3. associate-/l/ N/A

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

       4. /-lowering-/.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. PI-lowering-PI.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 68.6

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

   11. Applied egg-rr 68.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.55 \cdot 10^{-55}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-55)
   (/ (* PI 0.5) (* a (* b a)))
   (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-55) {
		tmp = (((double) M_PI) * 0.5) / (a * (b * a));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-55) {
		tmp = (Math.PI * 0.5) / (a * (b * a));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.8e-55:
		tmp = (math.pi * 0.5) / (a * (b * a))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-55)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * a)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.8e-55)
		tmp = (pi * 0.5) / (a * (b * a));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.8e-55], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.79999999999999946e-55

   1. Initial program 84.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 79.3

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

   5. Simplified 79.3%

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

   if -6.79999999999999946e-55 < a

   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. Add Preprocessing
   3. Step-by-step derivation

      1. frac-sub N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 90.6%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. flip-+ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. PI-lowering-PI.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 68.6

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

   9. Simplified 68.6%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. associate-/r/ N/A

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

       5. *-lowering-*.f64 N/A

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

       6. associate-/l/ N/A

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

       7. /-lowering-/.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. PI-lowering-PI.f64 68.5

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

   11. Applied egg-rr 68.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-55}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.1e-55)
   (* PI (/ 0.5 (* a (* b a))))
   (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.1e-55) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.1e-55) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.1e-55:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.1e-55)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.1e-55)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.1e-55], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.09999999999999997e-55

   1. Initial program 84.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 79.3

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

   5. Simplified 79.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. PI-lowering-PI.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.6

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

   7. Applied egg-rr 73.6%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. PI-lowering-PI.f64 79.2

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

   9. Applied egg-rr 79.2%

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

   if -3.09999999999999997e-55 < a

   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. Add Preprocessing
   3. Step-by-step derivation

      1. frac-sub N/A

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

      2. associate-*r/ N/A

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

      3. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 90.6%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. difference-of-squares N/A

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

      4. flip-+ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. PI-lowering-PI.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 68.6

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

   9. Simplified 68.6%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. associate-/r/ N/A

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

       5. *-lowering-*.f64 N/A

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

       6. associate-/l/ N/A

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

       7. /-lowering-/.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. PI-lowering-PI.f64 68.5

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

   11. Applied egg-rr 68.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-55}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 59.4

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

5. Simplified 59.4%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 56.4

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

7. Applied egg-rr 56.4%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. PI-lowering-PI.f64 59.3

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

9. Applied egg-rr 59.3%

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

10. Final simplification 59.3%

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

Alternative 11: 56.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 59.4

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

5. Simplified 59.4%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 56.4

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

7. Applied egg-rr 56.4%

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

Reproduce

herbie shell --seed 2024198 
(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))))