NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.7%

Time: 4.0s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

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(Float64(pi * 0.5) / 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[(N[(Pi * 0.5), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

   1. 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

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift--.f64 N/A

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

   11. difference-of-squares N/A

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

   12. associate-/r/ N/A

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

   13. lift--.f64 N/A

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

   14. flip-+ N/A

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

   15. lift-+.f64 N/A

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

   16. lower-/.f64 99.7

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. 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

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 87.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in a around 0

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

   1. lower-PI.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 3: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+106}:\\ \;\;\;\;\frac{0.5 \cdot \pi}{\left(\left(b + a\right) \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.4e+106)
   (/ (* 0.5 PI) (* (* (+ b a) b) a))
   (/ (/ (* PI 0.5) (* a b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.4e+106) {
		tmp = (0.5 * ((double) M_PI)) / (((b + a) * b) * a);
	} 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 <= 2.4e+106) {
		tmp = (0.5 * Math.PI) / (((b + a) * b) * a);
	} else {
		tmp = ((Math.PI * 0.5) / (a * b)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.4e+106:
		tmp = (0.5 * math.pi) / (((b + a) * b) * a)
	else:
		tmp = ((math.pi * 0.5) / (a * b)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.4e+106)
		tmp = Float64(Float64(0.5 * pi) / Float64(Float64(Float64(b + a) * b) * a));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.4e+106)
		tmp = (0.5 * pi) / (((b + a) * b) * a);
	else
		tmp = ((pi * 0.5) / (a * b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.4e+106], N[(N[(0.5 * Pi), $MachinePrecision] / N[(N[(N[(b + a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4000000000000001e106

   1. Initial program 78.6%

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

      1. 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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-/r/ N/A

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

      13. lift--.f64 N/A

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

      14. flip-+ N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 93.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 93.1

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

   9. Applied rewrites 93.1%

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

   if 2.4000000000000001e106 < b

   1. Initial program 78.6%

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

   2. 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 56.9

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

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   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 62.7

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

      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 62.7

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

   6. Applied rewrites 62.7%

      \[\leadsto 0.5 \cdot \frac{\pi}{\left(b \cdot a\right) \cdot \color{blue}{b}} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 62.9

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

   8. Applied rewrites 62.9%

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

Alternative 4: 96.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{+106}:\\ \;\;\;\;\pi \cdot \frac{0.5}{\left(\left(b + a\right) \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.2e+106)
   (* PI (/ 0.5 (* (* (+ b a) b) a)))
   (/ (/ (* PI 0.5) (* a b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.2e+106) {
		tmp = ((double) M_PI) * (0.5 / (((b + a) * b) * a));
	} 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 <= 2.2e+106) {
		tmp = Math.PI * (0.5 / (((b + a) * b) * a));
	} else {
		tmp = ((Math.PI * 0.5) / (a * b)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.2e+106:
		tmp = math.pi * (0.5 / (((b + a) * b) * a))
	else:
		tmp = ((math.pi * 0.5) / (a * b)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.2e+106)
		tmp = Float64(pi * Float64(0.5 / Float64(Float64(Float64(b + a) * b) * a)));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.2e+106)
		tmp = pi * (0.5 / (((b + a) * b) * a));
	else
		tmp = ((pi * 0.5) / (a * b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.2e+106], N[(Pi * N[(0.5 / N[(N[(N[(b + a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.19999999999999992e106

   1. Initial program 78.6%

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

      1. 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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift--.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-/r/ N/A

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

      13. lift--.f64 N/A

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

      14. flip-+ N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 93.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 93.1

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

   9. Applied rewrites 93.1%

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

   if 2.19999999999999992e106 < b

   1. Initial program 78.6%

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

   2. 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 56.9

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

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   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 62.7

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

      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 62.7

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

   6. Applied rewrites 62.7%

      \[\leadsto 0.5 \cdot \frac{\pi}{\left(b \cdot a\right) \cdot \color{blue}{b}} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 62.9

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

   8. Applied rewrites 62.9%

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

Alternative 5: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-87}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e-87) (/ (* 0.5 (/ PI a)) (* a b)) (/ (/ (* PI 0.5) (* a b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e-87) {
		tmp = (0.5 * (((double) M_PI) / 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 <= 3e-87) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = ((Math.PI * 0.5) / (a * b)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3e-87:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = ((math.pi * 0.5) / (a * b)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e-87)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(a * b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3e-87)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = ((pi * 0.5) / (a * b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e-87], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.00000000000000016e-87

   1. Initial program 78.6%

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

      1. 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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-PI.f64 62.8

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

   8. Applied rewrites 62.8%

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

   if 3.00000000000000016e-87 < b

   1. Initial program 78.6%

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

   2. 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 56.9

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

   4. Applied rewrites 56.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   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 62.7

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

      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 62.7

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

   6. Applied rewrites 62.7%

      \[\leadsto 0.5 \cdot \frac{\pi}{\left(b \cdot a\right) \cdot \color{blue}{b}} \]
   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. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 62.9

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

   8. Applied rewrites 62.9%

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

Alternative 6: 62.9% accurate, 2.3× speedup

Math

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

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(Float64(Float64(pi * 0.5) / Float64(a * b)) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

2. 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 56.9

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

4. Applied rewrites 56.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
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 62.7

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

   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 62.7

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

6. Applied rewrites 62.7%

   \[\leadsto 0.5 \cdot \frac{\pi}{\left(b \cdot a\right) \cdot \color{blue}{b}} \]
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. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 62.9

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

8. Applied rewrites 62.9%

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

Alternative 7: 62.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

2. 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 56.9

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

4. Applied rewrites 56.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
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 62.7

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

   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 62.7

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

6. Applied rewrites 62.7%

   \[\leadsto 0.5 \cdot \frac{\pi}{\left(b \cdot a\right) \cdot \color{blue}{b}} \]
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. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 62.9

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

8. Applied rewrites 62.9%

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

Alternative 8: 62.7% 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 78.6%

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

2. 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 56.9

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

4. Applied rewrites 56.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
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 62.7

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

   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 62.7

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

6. Applied rewrites 62.7%

   \[\leadsto 0.5 \cdot \frac{\pi}{\left(b \cdot a\right) \cdot \color{blue}{b}} \]
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 62.7

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

   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. lift-*.f64 62.7

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

8. Applied rewrites 62.7%

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

Alternative 9: 62.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

2. 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 56.9

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

4. Applied rewrites 56.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
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 62.7

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

   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 62.7

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

6. Applied rewrites 62.7%

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

Reproduce

herbie shell --seed 2025150 
(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))))