NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 90.4%

Time: 3.4s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{+70}:\\ \;\;\;\;\frac{\pi \cdot 1}{2 \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a \cdot \left(b + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e+70)
   (* (/ (* PI 1.0) (* 2.0 (* (+ b a) (- b a)))) (- (/ 1.0 a) (/ 1.0 b)))
   (/ (/ PI b) (* a (+ b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e+70) {
		tmp = ((((double) M_PI) * 1.0) / (2.0 * ((b + a) * (b - a)))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = (((double) M_PI) / b) / (a * (b + b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.65e+70) {
		tmp = ((Math.PI * 1.0) / (2.0 * ((b + a) * (b - a)))) * ((1.0 / a) - (1.0 / b));
	} else {
		tmp = (Math.PI / b) / (a * (b + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.65e+70:
		tmp = ((math.pi * 1.0) / (2.0 * ((b + a) * (b - a)))) * ((1.0 / a) - (1.0 / b))
	else:
		tmp = (math.pi / b) / (a * (b + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e+70)
		tmp = Float64(Float64(Float64(pi * 1.0) / Float64(2.0 * Float64(Float64(b + a) * Float64(b - a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)));
	else
		tmp = Float64(Float64(pi / b) / Float64(a * Float64(b + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.65e+70)
		tmp = ((pi * 1.0) / (2.0 * ((b + a) * (b - a)))) * ((1.0 / a) - (1.0 / b));
	else
		tmp = (pi / b) / (a * (b + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e+70], N[(N[(N[(Pi * 1.0), $MachinePrecision] / N[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(a * N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65000000000000008e70

   1. Initial program 80.3%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 88.2

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

   3. Applied rewrites 88.2%

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

   if 1.65000000000000008e70 < b

   1. Initial program 69.8%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 85.9

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

   3. Applied rewrites 85.9%

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 2: 90.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(b + b\right)\\ \mathbf{if}\;b \leq 2.4 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\pi}{\left(b - a\right) \cdot \left(a + b\right)} \cdot \left(b - a\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ b b))))
   (if (<= b 2.4e+94)
     (/ (* (/ PI (* (- b a) (+ a b))) (- b a)) t_0)
     (/ (/ PI b) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (b + b);
	double tmp;
	if (b <= 2.4e+94) {
		tmp = ((((double) M_PI) / ((b - a) * (a + b))) * (b - a)) / t_0;
	} else {
		tmp = (((double) M_PI) / b) / t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = a * (b + b);
	double tmp;
	if (b <= 2.4e+94) {
		tmp = ((Math.PI / ((b - a) * (a + b))) * (b - a)) / t_0;
	} else {
		tmp = (Math.PI / b) / t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (b + b)
	tmp = 0
	if b <= 2.4e+94:
		tmp = ((math.pi / ((b - a) * (a + b))) * (b - a)) / t_0
	else:
		tmp = (math.pi / b) / t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(b + b))
	tmp = 0.0
	if (b <= 2.4e+94)
		tmp = Float64(Float64(Float64(pi / Float64(Float64(b - a) * Float64(a + b))) * Float64(b - a)) / t_0);
	else
		tmp = Float64(Float64(pi / b) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (b + b);
	tmp = 0.0;
	if (b <= 2.4e+94)
		tmp = ((pi / ((b - a) * (a + b))) * (b - a)) / t_0;
	else
		tmp = (pi / b) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(b + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.4e+94], N[(N[(N[(Pi / N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.39999999999999983e94

   1. Initial program 80.8%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 88.5

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

   3. Applied rewrites 88.5%

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

   5. Applied rewrites 88.5%

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

   if 2.39999999999999983e94 < b

   1. Initial program 66.2%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 84.3

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

   3. Applied rewrites 84.3%

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 99.7

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

   8. Applied rewrites 99.7%

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

Alternative 3: 89.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b + b\right) \cdot a\\ \mathbf{if}\;a \leq -2.35 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b} \cdot 0.5}{a}}{a}\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\pi}{\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot t\_0} \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (+ b b) a)))
   (if (<= a -2.35e+129)
     (/ (/ (* (/ PI b) 0.5) a) a)
     (if (<= a 2.5e+45)
       (* (/ PI (* (* (+ a b) (- b a)) t_0)) (- b a))
       (/ (/ PI a) t_0)))))

C

double code(double a, double b) {
	double t_0 = (b + b) * a;
	double tmp;
	if (a <= -2.35e+129) {
		tmp = (((((double) M_PI) / b) * 0.5) / a) / a;
	} else if (a <= 2.5e+45) {
		tmp = (((double) M_PI) / (((a + b) * (b - a)) * t_0)) * (b - a);
	} else {
		tmp = (((double) M_PI) / a) / t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (b + b) * a;
	double tmp;
	if (a <= -2.35e+129) {
		tmp = (((Math.PI / b) * 0.5) / a) / a;
	} else if (a <= 2.5e+45) {
		tmp = (Math.PI / (((a + b) * (b - a)) * t_0)) * (b - a);
	} else {
		tmp = (Math.PI / a) / t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b + b) * a
	tmp = 0
	if a <= -2.35e+129:
		tmp = (((math.pi / b) * 0.5) / a) / a
	elif a <= 2.5e+45:
		tmp = (math.pi / (((a + b) * (b - a)) * t_0)) * (b - a)
	else:
		tmp = (math.pi / a) / t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b + b) * a)
	tmp = 0.0
	if (a <= -2.35e+129)
		tmp = Float64(Float64(Float64(Float64(pi / b) * 0.5) / a) / a);
	elseif (a <= 2.5e+45)
		tmp = Float64(Float64(pi / Float64(Float64(Float64(a + b) * Float64(b - a)) * t_0)) * Float64(b - a));
	else
		tmp = Float64(Float64(pi / a) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b + b) * a;
	tmp = 0.0;
	if (a <= -2.35e+129)
		tmp = (((pi / b) * 0.5) / a) / a;
	elseif (a <= 2.5e+45)
		tmp = (pi / (((a + b) * (b - a)) * t_0)) * (b - a);
	else
		tmp = (pi / a) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b + b), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -2.35e+129], N[(N[(N[(N[(Pi / b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, 2.5e+45], N[(N[(Pi / N[(N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.35000000000000004e129

   1. Initial program 58.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{b}, \frac{1}{2}, \frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right)}{{a}^{2}} \]

      6. lift-PI.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}\right)}{{a}^{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      10. lift-PI.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\pi}{b}, \frac{1}{2}, \frac{\pi}{a} \cdot \frac{-1}{2}\right)}{{a}^{2}} \]

      11. pow2 N/A

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

      12. lift-*.f64 80.3

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

   4. Applied rewrites 80.3%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 80.3

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

   7. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -2.35000000000000004e129 < a < 2.5e45

   1. Initial program 85.7%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 90.6

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

   3. Applied rewrites 90.6%

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

   5. Applied rewrites 84.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   7. Applied rewrites 84.3%

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

   if 2.5e45 < a

   1. Initial program 70.1%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 84.5%

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

   4. Taylor expanded in a around inf

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 96.5

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

   6. Applied rewrites 96.5%

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

      1. *-commutative 96.5

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

      2. difference-of-squares-rev 96.5

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

      3. associate-*r/ 96.5

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

      4. *-rgt-identity 96.5

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

      5. difference-of-squares-rev 96.5

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

      6. +-commutative 96.5

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

      7. *-commutative 96.5

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

      8. *-lft-identity 96.5

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

      9. *-rgt-identity 96.5

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

      10. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(a \cdot b\right) \cdot 2\right)\right)} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(a \cdot b\right)\right) \cdot 2} \]

      12. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(*-commutative, \left(2 \cdot \left(a \cdot b\right)\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(count-2-rev, \left(a \cdot b + a \cdot b\right)\right)} \]

      14. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite<=}\left(distribute-lft-in, \left(a \cdot \left(b + b\right)\right)\right)} \]

      15. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(*-commutative, \left(\left(b + b\right) \cdot a\right)\right)} \]

      16. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(b + b\right) \cdot a\right)\right)} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite<=}\left(lift-+.f64, \left(b + b\right)\right) \cdot a} \]

   8. Applied rewrites 96.5%

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

Alternative 4: 87.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(b + b\right)\\ \mathbf{if}\;b \leq 1.65 \cdot 10^{+70}:\\ \;\;\;\;\left(b - a\right) \cdot \frac{\frac{\pi}{\left(b - a\right) \cdot \left(a + b\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ b b))))
   (if (<= b 1.65e+70)
     (* (- b a) (/ (/ PI (* (- b a) (+ a b))) t_0))
     (/ (/ PI b) t_0))))

C

double code(double a, double b) {
	double t_0 = a * (b + b);
	double tmp;
	if (b <= 1.65e+70) {
		tmp = (b - a) * ((((double) M_PI) / ((b - a) * (a + b))) / t_0);
	} else {
		tmp = (((double) M_PI) / b) / t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = a * (b + b);
	double tmp;
	if (b <= 1.65e+70) {
		tmp = (b - a) * ((Math.PI / ((b - a) * (a + b))) / t_0);
	} else {
		tmp = (Math.PI / b) / t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (b + b)
	tmp = 0
	if b <= 1.65e+70:
		tmp = (b - a) * ((math.pi / ((b - a) * (a + b))) / t_0)
	else:
		tmp = (math.pi / b) / t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(b + b))
	tmp = 0.0
	if (b <= 1.65e+70)
		tmp = Float64(Float64(b - a) * Float64(Float64(pi / Float64(Float64(b - a) * Float64(a + b))) / t_0));
	else
		tmp = Float64(Float64(pi / b) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (b + b);
	tmp = 0.0;
	if (b <= 1.65e+70)
		tmp = (b - a) * ((pi / ((b - a) * (a + b))) / t_0);
	else
		tmp = (pi / b) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(b + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.65e+70], N[(N[(b - a), $MachinePrecision] * N[(N[(Pi / N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65000000000000008e70

   1. Initial program 80.3%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 88.2

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

   3. Applied rewrites 88.2%

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

   5. Applied rewrites 82.8%

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

   if 1.65000000000000008e70 < b

   1. Initial program 69.8%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 85.9

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

   3. Applied rewrites 85.9%

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 5: 87.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{\pi}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;b \leq 0.6:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(b + b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a \cdot \left(b + b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -5.7e-18)
   (* (/ PI (* b (* a b))) 0.5)
   (if (<= b 0.6) (/ (/ PI a) (* (+ b b) a)) (/ (/ PI b) (* a (+ b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -5.7e-18) {
		tmp = (((double) M_PI) / (b * (a * b))) * 0.5;
	} else if (b <= 0.6) {
		tmp = (((double) M_PI) / a) / ((b + b) * a);
	} else {
		tmp = (((double) M_PI) / b) / (a * (b + b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -5.7e-18) {
		tmp = (Math.PI / (b * (a * b))) * 0.5;
	} else if (b <= 0.6) {
		tmp = (Math.PI / a) / ((b + b) * a);
	} else {
		tmp = (Math.PI / b) / (a * (b + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -5.7e-18:
		tmp = (math.pi / (b * (a * b))) * 0.5
	elif b <= 0.6:
		tmp = (math.pi / a) / ((b + b) * a)
	else:
		tmp = (math.pi / b) / (a * (b + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -5.7e-18)
		tmp = Float64(Float64(pi / Float64(b * Float64(a * b))) * 0.5);
	elseif (b <= 0.6)
		tmp = Float64(Float64(pi / a) / Float64(Float64(b + b) * a));
	else
		tmp = Float64(Float64(pi / b) / Float64(a * Float64(b + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -5.7e-18)
		tmp = (pi / (b * (a * b))) * 0.5;
	elseif (b <= 0.6)
		tmp = (pi / a) / ((b + b) * a);
	else
		tmp = (pi / b) / (a * (b + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -5.7e-18], N[(N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[b, 0.6], N[(N[(Pi / a), $MachinePrecision] / N[(N[(b + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] / N[(a * N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.69999999999999971e-18

   1. Initial program 75.9%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 88.0

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

   3. Applied rewrites 88.0%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 78.3

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

   6. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 89.2

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

   8. Applied rewrites 89.2%

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

   if -5.69999999999999971e-18 < b < 0.599999999999999978

   1. Initial program 80.4%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 86.9%

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

   4. Taylor expanded in a around inf

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 84.0

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

   6. Applied rewrites 84.0%

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

      1. *-commutative 84.0

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

      2. difference-of-squares-rev 84.0

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

      3. associate-*r/ 84.0

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

      4. *-rgt-identity 84.0

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

      5. difference-of-squares-rev 84.0

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

      6. +-commutative 84.0

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

      7. *-commutative 84.0

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

      8. *-lft-identity 84.0

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

      9. *-rgt-identity 84.0

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

      10. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(a \cdot b\right) \cdot 2\right)\right)} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(a \cdot b\right)\right) \cdot 2} \]

      12. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(*-commutative, \left(2 \cdot \left(a \cdot b\right)\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(count-2-rev, \left(a \cdot b + a \cdot b\right)\right)} \]

      14. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite<=}\left(distribute-lft-in, \left(a \cdot \left(b + b\right)\right)\right)} \]

      15. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(*-commutative, \left(\left(b + b\right) \cdot a\right)\right)} \]

      16. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(b + b\right) \cdot a\right)\right)} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite<=}\left(lift-+.f64, \left(b + b\right)\right) \cdot a} \]

   8. Applied rewrites 84.0%

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

   if 0.599999999999999978 < b

   1. Initial program 76.8%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 89.1

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

   3. Applied rewrites 89.1%

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lift-PI.f64 91.2

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

   8. Applied rewrites 91.2%

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

Alternative 6: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{if}\;b \leq -5.7 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.6:\\ \;\;\;\;\frac{\frac{\pi}{a}}{\left(b + b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* b (* a b))) 0.5)))
   (if (<= b -5.7e-18) t_0 (if (<= b 0.6) (/ (/ PI a) (* (+ b b) a)) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / (b * (a * b))) * 0.5;
	double tmp;
	if (b <= -5.7e-18) {
		tmp = t_0;
	} else if (b <= 0.6) {
		tmp = (((double) M_PI) / a) / ((b + b) * a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / (b * (a * b))) * 0.5;
	double tmp;
	if (b <= -5.7e-18) {
		tmp = t_0;
	} else if (b <= 0.6) {
		tmp = (Math.PI / a) / ((b + b) * a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / (b * (a * b))) * 0.5
	tmp = 0
	if b <= -5.7e-18:
		tmp = t_0
	elif b <= 0.6:
		tmp = (math.pi / a) / ((b + b) * a)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(b * Float64(a * b))) * 0.5)
	tmp = 0.0
	if (b <= -5.7e-18)
		tmp = t_0;
	elseif (b <= 0.6)
		tmp = Float64(Float64(pi / a) / Float64(Float64(b + b) * a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / (b * (a * b))) * 0.5;
	tmp = 0.0;
	if (b <= -5.7e-18)
		tmp = t_0;
	elseif (b <= 0.6)
		tmp = (pi / a) / ((b + b) * a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b, -5.7e-18], t$95$0, If[LessEqual[b, 0.6], N[(N[(Pi / a), $MachinePrecision] / N[(N[(b + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.69999999999999971e-18 or 0.599999999999999978 < b

   1. Initial program 76.4%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 90.0

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

   8. Applied rewrites 90.0%

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

   if -5.69999999999999971e-18 < b < 0.599999999999999978

   1. Initial program 80.4%

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

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. associate-*l/ N/A

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

      15. frac-times N/A

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

   3. Applied rewrites 86.9%

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

   4. Taylor expanded in a around inf

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

      1. lift-/.f64 N/A

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

      2. lift-PI.f64 84.0

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

   6. Applied rewrites 84.0%

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

      1. *-commutative 84.0

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

      2. difference-of-squares-rev 84.0

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

      3. associate-*r/ 84.0

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

      4. *-rgt-identity 84.0

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

      5. difference-of-squares-rev 84.0

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

      6. +-commutative 84.0

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

      7. *-commutative 84.0

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

      8. *-lft-identity 84.0

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

      9. *-rgt-identity 84.0

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

      10. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(a \cdot b\right) \cdot 2\right)\right)} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(a \cdot b\right)\right) \cdot 2} \]

      12. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(*-commutative, \left(2 \cdot \left(a \cdot b\right)\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(count-2-rev, \left(a \cdot b + a \cdot b\right)\right)} \]

      14. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite<=}\left(distribute-lft-in, \left(a \cdot \left(b + b\right)\right)\right)} \]

      15. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(*-commutative, \left(\left(b + b\right) \cdot a\right)\right)} \]

      16. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(b + b\right) \cdot a\right)\right)} \]

      17. *-rgt-identity N/A

          \[\leadsto \frac{\frac{\pi}{a}}{\mathsf{Rewrite<=}\left(lift-+.f64, \left(b + b\right)\right) \cdot a} \]

   8. Applied rewrites 84.0%

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

Alternative 7: 85.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{if}\;b \leq -5.7 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.6:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* b (* a b))) 0.5)))
   (if (<= b -5.7e-18) t_0 (if (<= b 0.6) (* (/ PI (* a (* a b))) 0.5) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / (b * (a * b))) * 0.5;
	double tmp;
	if (b <= -5.7e-18) {
		tmp = t_0;
	} else if (b <= 0.6) {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / (b * (a * b))) * 0.5;
	double tmp;
	if (b <= -5.7e-18) {
		tmp = t_0;
	} else if (b <= 0.6) {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / (b * (a * b))) * 0.5
	tmp = 0
	if b <= -5.7e-18:
		tmp = t_0
	elif b <= 0.6:
		tmp = (math.pi / (a * (a * b))) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(b * Float64(a * b))) * 0.5)
	tmp = 0.0
	if (b <= -5.7e-18)
		tmp = t_0;
	elseif (b <= 0.6)
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / (b * (a * b))) * 0.5;
	tmp = 0.0;
	if (b <= -5.7e-18)
		tmp = t_0;
	elseif (b <= 0.6)
		tmp = (pi / (a * (a * b))) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b, -5.7e-18], t$95$0, If[LessEqual[b, 0.6], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.69999999999999971e-18 or 0.599999999999999978 < b

   1. Initial program 76.4%

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

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 88.6

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 79.5

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

   6. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 90.0

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

   8. Applied rewrites 90.0%

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

   if -5.69999999999999971e-18 < b < 0.599999999999999978

   1. Initial program 80.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 71.3

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

   4. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 83.7

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

   6. Applied rewrites 83.7%

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

Alternative 8: 81.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{if}\;b \leq -5.7 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.6:\\ \;\;\;\;\frac{\pi}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ PI (* (* b b) a)) 0.5)))
   (if (<= b -5.7e-18) t_0 (if (<= b 0.6) (* (/ PI (* a (* a b))) 0.5) t_0))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / ((b * b) * a)) * 0.5;
	double tmp;
	if (b <= -5.7e-18) {
		tmp = t_0;
	} else if (b <= 0.6) {
		tmp = (((double) M_PI) / (a * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / ((b * b) * a)) * 0.5;
	double tmp;
	if (b <= -5.7e-18) {
		tmp = t_0;
	} else if (b <= 0.6) {
		tmp = (Math.PI / (a * (a * b))) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / ((b * b) * a)) * 0.5
	tmp = 0
	if b <= -5.7e-18:
		tmp = t_0
	elif b <= 0.6:
		tmp = (math.pi / (a * (a * b))) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / Float64(Float64(b * b) * a)) * 0.5)
	tmp = 0.0
	if (b <= -5.7e-18)
		tmp = t_0;
	elseif (b <= 0.6)
		tmp = Float64(Float64(pi / Float64(a * Float64(a * b))) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / ((b * b) * a)) * 0.5;
	tmp = 0.0;
	if (b <= -5.7e-18)
		tmp = t_0;
	elseif (b <= 0.6)
		tmp = (pi / (a * (a * b))) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b, -5.7e-18], t$95$0, If[LessEqual[b, 0.6], N[(N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.69999999999999971e-18 or 0.599999999999999978 < b

   1. Initial program 76.4%

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

   2. 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

   if -5.69999999999999971e-18 < b < 0.599999999999999978

   1. Initial program 80.4%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 71.3

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

   4. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 83.7

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

   6. Applied rewrites 83.7%

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

Alternative 9: 62.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

2. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 56.7

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

4. Applied rewrites 56.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 62.7

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

6. Applied rewrites 62.7%

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

Reproduce

herbie shell --seed 2025101 
(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))))