NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.7%

Time: 5.5s

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: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip-rev N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. difference-of-squares N/A

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

   8. associate-/r* N/A

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

   9. *-lft-identity N/A

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

   10. /-rgt-identity N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. div-flip N/A

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

   17. associate-/r/ N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. *-lft-identity N/A

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

   23. *-rgt-identity N/A

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

3. Applied rewrites 88.5%

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

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\left(b + a\right) \cdot \left(b \cdot a\right)}{\pi \cdot 0.5}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. associate-/r* N/A

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

   7. div-flip N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 99.7

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l/ N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lift-/.f64 N/A

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

   11. times-frac N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. associate-*l/ N/A

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

   15. lift-/.f64 N/A

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

   16. lower-/.f64 N/A

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

9. Applied rewrites 99.7%

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

Alternative 2: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\left(\left(b + a\right) \cdot a\right) \cdot \left(b + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e+154)
   (/ (/ PI a) (* (* a b) 2.0))
   (/ PI (* (* (+ b a) a) (+ b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e+154) {
		tmp = (((double) M_PI) / a) / ((a * b) * 2.0);
	} else {
		tmp = ((double) M_PI) / (((b + a) * a) * (b + b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.4e+154) {
		tmp = (Math.PI / a) / ((a * b) * 2.0);
	} else {
		tmp = Math.PI / (((b + a) * a) * (b + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.4e+154:
		tmp = (math.pi / a) / ((a * b) * 2.0)
	else:
		tmp = math.pi / (((b + a) * a) * (b + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.4e+154)
		tmp = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0));
	else
		tmp = Float64(pi / Float64(Float64(Float64(b + a) * a) * Float64(b + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.4e+154)
		tmp = (pi / a) / ((a * b) * 2.0);
	else
		tmp = pi / (((b + a) * a) * (b + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.4e+154], N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(Pi / N[(N[(N[(b + a), $MachinePrecision] * a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4e154

   1. Initial program 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.5%

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(b + a\right) \cdot \left(b \cdot a\right)}{\pi \cdot 0.5}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/r* N/A

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

      7. div-flip N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 64.3

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

   10. Applied rewrites 64.3%

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

   if -1.4e154 < a

   1. Initial program 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.5%

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(b + a\right) \cdot \left(b \cdot a\right)}{\pi \cdot 0.5}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/r* N/A

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

      7. div-flip N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. distribute-rgt-in N/A

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

      20. *-lft-identity N/A

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

      21. *-lft-identity N/A

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

      22. lower-+.f64 93.0

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

   9. Applied rewrites 93.0%

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

Alternative 3: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip-rev N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. difference-of-squares N/A

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

   8. associate-/r* N/A

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

   9. *-lft-identity N/A

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

   10. /-rgt-identity N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. div-flip N/A

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

   17. associate-/r/ N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. *-lft-identity N/A

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

   23. *-rgt-identity N/A

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

3. Applied rewrites 88.5%

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

5. Applied rewrites 99.1%

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

Alternative 4: 96.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip-rev N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. difference-of-squares N/A

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

   8. associate-/r* N/A

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

   9. *-lft-identity N/A

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

   10. /-rgt-identity N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. div-flip N/A

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

   17. associate-/r/ N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 N/A

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

   22. *-lft-identity N/A

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

   23. *-rgt-identity N/A

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

3. Applied rewrites 88.5%

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

5. Applied rewrites 99.0%

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

Alternative 5: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(a \cdot b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{\left(b + b\right) \cdot a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.3e-45) (/ (/ PI a) (* (* a b) 2.0)) (/ (/ PI (* (+ b b) a)) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.3e-45) {
		tmp = (((double) M_PI) / a) / ((a * b) * 2.0);
	} else {
		tmp = (((double) M_PI) / ((b + b) * a)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.3e-45) {
		tmp = (Math.PI / a) / ((a * b) * 2.0);
	} else {
		tmp = (Math.PI / ((b + b) * a)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.3e-45:
		tmp = (math.pi / a) / ((a * b) * 2.0)
	else:
		tmp = (math.pi / ((b + b) * a)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.3e-45)
		tmp = Float64(Float64(pi / a) / Float64(Float64(a * b) * 2.0));
	else
		tmp = Float64(Float64(pi / Float64(Float64(b + b) * a)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.3e-45)
		tmp = (pi / a) / ((a * b) * 2.0);
	else
		tmp = (pi / ((b + b) * a)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.3e-45], N[(N[(Pi / a), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(N[(b + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.2999999999999997e-45

   1. Initial program 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-squares N/A

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

      8. associate-/r* N/A

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

      9. *-lft-identity N/A

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

      10. /-rgt-identity N/A

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

      11. mult-flip N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. *-lft-identity N/A

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

      23. *-rgt-identity N/A

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

   3. Applied rewrites 88.5%

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(b + a\right) \cdot \left(b \cdot a\right)}{\pi \cdot 0.5}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-/r* N/A

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

      7. div-flip N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 99.7

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 64.3

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

   10. Applied rewrites 64.3%

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

   if -5.2999999999999997e-45 < a

   1. Initial program 78.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 56.8

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

   4. Applied rewrites 56.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 62.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 62.0

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

   6. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 61.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.9

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

   8. Applied rewrites 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-/.f64 N/A

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

   10. Applied rewrites 62.1%

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

Alternative 6: 62.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 56.8

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

4. Applied rewrites 56.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 62.0

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 62.0

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

6. Applied rewrites 62.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 61.9

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 61.9

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

8. Applied rewrites 61.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-/.f64 N/A

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

10. Applied rewrites 62.1%

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

Alternative 7: 61.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 56.8

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

4. Applied rewrites 56.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 62.0

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 62.0

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

6. Applied rewrites 62.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 61.9

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 61.9

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

8. Applied rewrites 61.9%

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

Alternative 8: 61.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

2. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 56.8

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

4. Applied rewrites 56.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 62.0

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 62.0

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

6. Applied rewrites 62.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 61.9

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 61.9

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

8. Applied rewrites 61.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. frac-times N/A

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

   8. div-flip N/A

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

   9. frac-times N/A

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

   10. *-commutative N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. distribute-rgt-in N/A

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

   18. *-lft-identity N/A

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

   19. *-lft-identity N/A

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

   20. lower-+.f64 61.9

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 61.9

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

10. Applied rewrites 61.9%

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

Reproduce

herbie shell --seed 2025142 
(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))))