NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.7%

Time: 10.9s

Alternatives: 9

Speedup: 2.4×

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: 78.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. un-div-inv N/A

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

   8. remove-double-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

4. Applied egg-rr 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{+138}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.32e+138)
   (/ (* PI 0.5) (* a (* a b)))
   (* PI (/ 0.5 (* b (* a (+ a b)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.32e+138) {
		tmp = (((double) M_PI) * 0.5) / (a * (a * b));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (a * (a + b))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.32e+138) {
		tmp = (Math.PI * 0.5) / (a * (a * b));
	} else {
		tmp = Math.PI * (0.5 / (b * (a * (a + b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.32e+138:
		tmp = (math.pi * 0.5) / (a * (a * b))
	else:
		tmp = math.pi * (0.5 / (b * (a * (a + b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.32e+138)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * Float64(a + b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.32e+138)
		tmp = (pi * 0.5) / (a * (a * b));
	else
		tmp = pi * (0.5 / (b * (a * (a + b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.32e+138], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(a * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.32000000000000001e138

   1. Initial program 46.2%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 99.9

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

   5. Simplified 99.9%

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

   if -1.32000000000000001e138 < 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. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. un-div-inv N/A

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

      8. remove-double-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

   4. Applied egg-rr 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r* N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r/ N/A

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

      9. div-inv N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

   8. Applied egg-rr 93.2%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 95.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 95.6

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

   10. Applied egg-rr 95.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{+138}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-60) (/ (* PI 0.5) (* a (* a b))) (/ (* PI 0.5) (* a (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-60) {
		tmp = (((double) M_PI) * 0.5) / (a * (a * b));
	} else {
		tmp = (((double) M_PI) * 0.5) / (a * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-60) {
		tmp = (Math.PI * 0.5) / (a * (a * b));
	} else {
		tmp = (Math.PI * 0.5) / (a * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.8e-60:
		tmp = (math.pi * 0.5) / (a * (a * b))
	else:
		tmp = (math.pi * 0.5) / (a * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-60)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-60)
		tmp = (pi * 0.5) / (a * (a * b));
	else
		tmp = (pi * 0.5) / (a * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.8e-60], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-60

   1. Initial program 78.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. 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 72.0

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

   5. Simplified 72.0%

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

   if 1.8e-60 < b

   1. Initial program 77.4%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 83.5

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

   5. Simplified 83.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-60) (/ (* PI 0.5) (* a (* a b))) (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-60) {
		tmp = (((double) M_PI) * 0.5) / (a * (a * b));
	} else {
		tmp = ((double) M_PI) * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-60) {
		tmp = (Math.PI * 0.5) / (a * (a * b));
	} else {
		tmp = Math.PI * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.8e-60:
		tmp = (math.pi * 0.5) / (a * (a * b))
	else:
		tmp = math.pi * (0.5 / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-60)
		tmp = Float64(Float64(pi * 0.5) / Float64(a * Float64(a * b)));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-60)
		tmp = (pi * 0.5) / (a * (a * b));
	else
		tmp = pi * (0.5 / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.8e-60], N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-60

   1. Initial program 78.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. 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 72.0

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

   5. Simplified 72.0%

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

   if 1.8e-60 < b

   1. Initial program 77.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. un-div-inv N/A

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

      8. remove-double-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

   4. Applied egg-rr 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 99.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r* N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r/ N/A

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

      9. div-inv N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

   8. Applied egg-rr 91.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 83.5

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

   11. Simplified 83.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e-60) (* PI (/ 0.5 (* a (* a b)))) (* PI (/ 0.5 (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-60) {
		tmp = ((double) M_PI) * (0.5 / (a * (a * b)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.8e-60) {
		tmp = Math.PI * (0.5 / (a * (a * b)));
	} else {
		tmp = Math.PI * (0.5 / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.8e-60:
		tmp = math.pi * (0.5 / (a * (a * b)))
	else:
		tmp = math.pi * (0.5 / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.8e-60)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(a * b))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.8e-60)
		tmp = pi * (0.5 / (a * (a * b)));
	else
		tmp = pi * (0.5 / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.8e-60], N[(Pi * N[(0.5 / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e-60

   1. Initial program 78.1%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. 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 72.0

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

   5. Simplified 72.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}} \]
   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 72.0

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

      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 63.1

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

   7. Applied egg-rr 63.1%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 72.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.0

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

   9. Applied egg-rr 72.0%

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

   if 1.8e-60 < b

   1. Initial program 77.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. un-div-inv N/A

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

      8. remove-double-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

   4. Applied egg-rr 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 99.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r* N/A

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

      7. associate-/l/ N/A

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

      8. associate-/r/ N/A

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

      9. div-inv N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

   8. Applied egg-rr 91.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 83.5

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

   11. Simplified 83.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. un-div-inv N/A

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

   8. remove-double-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

4. Applied egg-rr 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. clear-num N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. div-inv N/A

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

   9. associate-*r* N/A

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

   10. div-inv N/A

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

   11. clear-num N/A

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

   12. frac-times N/A

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 93.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. un-div-inv N/A

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

   8. remove-double-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

4. Applied egg-rr 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

6. Applied egg-rr 99.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. associate-/r/ N/A

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

   9. div-inv N/A

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

   10. associate-*l/ N/A

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

   11. lower-*.f64 N/A

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

8. Applied egg-rr 93.9%

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

9. Final simplification 93.9%

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

Alternative 8: 62.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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 61.5

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

5. Simplified 61.5%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}} \]
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 61.5

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

   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 55.3

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

7. Applied egg-rr 55.3%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. lower-*.f64 61.5

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 61.5

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

9. Applied egg-rr 61.5%

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

10. Final simplification 61.5%

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

Alternative 9: 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 77.8%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. 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 61.5

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

5. Simplified 61.5%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot \left(a \cdot b\right)}} \]
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 61.5

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

   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 55.3

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

7. Applied egg-rr 55.3%

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

Reproduce

herbie shell --seed 2024208 
(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))))