NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.6%

Time: 10.5s

Alternatives: 8

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: 78.9% 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, 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 76.5%

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

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. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. frac-2neg N/A

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

   9. frac-2neg N/A

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

   10. frac-times N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r/ N/A

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

   14. lift-/.f64 N/A

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

   15. times-frac N/A

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

6. Applied egg-rr 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. frac-times N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 99.7

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

8. Applied egg-rr 99.7%

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

Alternative 2: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.1e+141)
   (/ (/ (* PI 0.5) (* b a)) a)
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.1e+141) {
		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 <= -1.1e+141) {
		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 <= -1.1e+141:
		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 <= -1.1e+141)
		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 <= -1.1e+141)
		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, -1.1e+141], 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 < -1.1e141

   1. Initial program 51.3%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 98.8

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

   5. Simplified 98.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -1.1e141 < a

   1. Initial program 82.1%

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

   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. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. frac-2neg N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. times-frac N/A

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

   6. Applied egg-rr 99.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.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. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lower-*.f64 95.1

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

   8. Applied egg-rr 95.1%

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

Alternative 3: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.85 \cdot 10^{+137}:\\ \;\;\;\;\frac{\pi}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.85e+137)
   (* (/ PI (* b a)) (/ 0.5 a))
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.85e+137) {
		tmp = (((double) M_PI) / (b * a)) * (0.5 / 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 <= -2.85e+137) {
		tmp = (Math.PI / (b * a)) * (0.5 / a);
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.85e+137:
		tmp = (math.pi / (b * a)) * (0.5 / a)
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.85e+137)
		tmp = Float64(Float64(pi / Float64(b * a)) * Float64(0.5 / 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 <= -2.85e+137)
		tmp = (pi / (b * a)) * (0.5 / a);
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.85e+137], N[(N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $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 < -2.8499999999999999e137

   1. Initial program 52.3%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 98.8

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

   5. Simplified 98.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. times-frac N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -2.8499999999999999e137 < a

   1. Initial program 82.0%

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

   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. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. frac-2neg N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. times-frac N/A

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

   6. Applied egg-rr 99.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.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. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lower-*.f64 95.0

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

   8. Applied egg-rr 95.0%

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

Alternative 4: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e+67)
   (/ (* PI 0.5) (* a (* b a)))
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e+67) {
		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 <= -1.4e+67) {
		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 <= -1.4e+67:
		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 <= -1.4e+67)
		tmp = Float64(Float64(pi * 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 <= -1.4e+67)
		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, -1.4e+67], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $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 < -1.3999999999999999e67

   1. Initial program 62.6%

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

   3. 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if -1.3999999999999999e67 < a

   1. Initial program 80.8%

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

   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. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. frac-2neg N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. times-frac N/A

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

   6. Applied egg-rr 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.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. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-rgt-out N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lower-*.f64 94.7

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

   8. Applied egg-rr 94.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e+67)
   (/ (* PI 0.5) (* a (* b a)))
   (* PI (/ 0.5 (* b (* a (+ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e+67) {
		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 <= -1.4e+67) {
		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 <= -1.4e+67:
		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 <= -1.4e+67)
		tmp = Float64(Float64(pi * 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 <= -1.4e+67)
		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, -1.4e+67], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * a), $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 < -1.3999999999999999e67

   1. Initial program 62.6%

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

   3. 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 99.0

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

   5. Simplified 99.0%

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

   if -1.3999999999999999e67 < a

   1. Initial program 80.8%

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

   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. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. frac-2neg N/A

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

      10. frac-times N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-/.f64 N/A

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

      15. times-frac N/A

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

   6. Applied egg-rr 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   8. Applied egg-rr 94.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.2e-21) (/ (* PI 0.5) (* a (* b a))) (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.2e-21) {
		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 (b <= 7.2e-21) {
		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 b <= 7.2e-21:
		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 (b <= 7.2e-21)
		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 (b <= 7.2e-21)
		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[b, 7.2e-21], 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 b < 7.19999999999999979e-21

   1. Initial program 78.2%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 74.6

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

   5. Simplified 74.6%

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

   if 7.19999999999999979e-21 < b

   1. Initial program 71.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.8

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

   5. Simplified 74.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 74.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 87.9

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

   7. Applied egg-rr 87.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-21}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.2e-21) (* PI (/ 0.5 (* b (* a a)))) (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.2e-21) {
		tmp = ((double) M_PI) * (0.5 / (b * (a * 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 (b <= 7.2e-21) {
		tmp = Math.PI * (0.5 / (b * (a * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7.2e-21:
		tmp = math.pi * (0.5 / (b * (a * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.2e-21)
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * 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 (b <= 7.2e-21)
		tmp = pi * (0.5 / (b * (a * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.2e-21], N[(Pi * N[(0.5 / N[(b * N[(a * 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 b < 7.19999999999999979e-21

   1. Initial program 78.2%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 74.6

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

   5. Simplified 74.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 74.5

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 65.2

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

   7. Applied egg-rr 65.2%

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

   if 7.19999999999999979e-21 < b

   1. Initial program 71.4%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.8

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

   5. Simplified 74.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-/r/ N/A

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

      9. clear-num N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 74.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 87.9

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

   7. Applied egg-rr 87.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-21}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.4% 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 76.5%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

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

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

   8. lower-*.f64 67.4

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

5. Simplified 67.4%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 67.4

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 60.4

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

7. Applied egg-rr 60.4%

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

Reproduce

herbie shell --seed 2024207 
(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))))