NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.7%

Time: 11.3s

Alternatives: 9

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. un-div-inv N/A

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

   8. remove-double-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. associate-*r/ N/A

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

   9. associate-*r/ N/A

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

6. Applied rewrites 99.7%

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

Alternative 2: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+87}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.9e+87)
   (* PI (/ 0.5 (* a (* b a))))
   (/ (* PI 0.5) (* b (* a (+ b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.9e+87) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.9e+87) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = (Math.PI * 0.5) / (b * (a * (b + a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.9e+87:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = (math.pi * 0.5) / (b * (a * (b + a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.9e+87)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.9e+87)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = (pi * 0.5) / (b * (a * (b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.9e+87], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.90000000000000006e87

   1. Initial program 74.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.4

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.3

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

   7. Applied rewrites 86.3%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if -1.90000000000000006e87 < a

   1. Initial program 84.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. un-div-inv N/A

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

      8. remove-double-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

   6. Applied rewrites 95.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+87}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. un-div-inv N/A

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

   8. remove-double-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-inv N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. div-inv N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. un-div-inv N/A

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

   8. remove-double-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. inv-pow N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{b + a} \cdot \frac{\frac{1}{2} \cdot {\left(b - a\right)}^{\color{blue}{0}}}{b \cdot a} \]

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 5: 75.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-24}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e-24)
   (* PI (/ 0.5 (* a (* b a))))
   (/ (* PI 0.5) (* b (* b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e-24) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = (((double) M_PI) * 0.5) / (b * (b * a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.4e-24) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = (Math.PI * 0.5) / (b * (b * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.4e-24:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = (math.pi * 0.5) / (b * (b * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.4e-24)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.4e-24)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = (pi * 0.5) / (b * (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.4e-24], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4000000000000001e-24

   1. Initial program 82.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.6

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.0

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

   7. Applied rewrites 82.0%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 90.6

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

   9. Applied rewrites 90.6%

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

   if -1.4000000000000001e-24 < a

   1. Initial program 81.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. un-div-inv N/A

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

      8. remove-double-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 66.8

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

   9. Applied rewrites 66.8%

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

       1. lift-PI.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. remove-double-neg N/A

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

       5. distribute-frac-neg N/A

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

       6. distribute-frac-neg2 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l/ N/A

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

       9. distribute-rgt-neg-in N/A

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

       10. frac-2neg N/A

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

       11. lower-/.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 66.8

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

   11. Applied rewrites 66.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-24}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-24}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e-24)
   (* PI (/ 0.5 (* a (* b a))))
   (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e-24) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.4e-24) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.4e-24:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.4e-24)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.4e-24)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.4e-24], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4000000000000001e-24

   1. Initial program 82.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.6

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.0

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

   7. Applied rewrites 82.0%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 90.6

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

   9. Applied rewrites 90.6%

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

   if -1.4000000000000001e-24 < a

   1. Initial program 81.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. un-div-inv N/A

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

      8. remove-double-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 66.8

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

   9. Applied rewrites 66.8%

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

       1. lift-PI.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. remove-double-neg N/A

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

       5. distribute-frac-neg N/A

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

       6. distribute-frac-neg2 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l/ N/A

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

       9. distribute-rgt-neg-in N/A

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

       10. frac-2neg N/A

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

       11. lift-*.f64 N/A

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

       12. associate-/l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 66.8

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

   11. Applied rewrites 66.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-24}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-24}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e-24)
   (* PI (/ 0.5 (* b (* a a))))
   (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e-24) {
		tmp = ((double) M_PI) * (0.5 / (b * (a * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.4e-24) {
		tmp = Math.PI * (0.5 / (b * (a * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.4e-24:
		tmp = math.pi * (0.5 / (b * (a * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.4e-24)
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.4e-24)
		tmp = pi * (0.5 / (b * (a * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.4e-24], N[(Pi * N[(0.5 / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.4000000000000001e-24

   1. Initial program 82.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.6

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.0

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

   7. Applied rewrites 82.0%

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

   if -1.4000000000000001e-24 < a

   1. Initial program 81.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. un-div-inv N/A

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

      8. remove-double-neg N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 66.8

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

   9. Applied rewrites 66.8%

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

       1. lift-PI.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. remove-double-neg N/A

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

       5. distribute-frac-neg N/A

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

       6. distribute-frac-neg2 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l/ N/A

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

       9. distribute-rgt-neg-in N/A

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

       10. frac-2neg N/A

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

       11. lift-*.f64 N/A

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

       12. associate-/l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-/.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 66.8

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

   11. Applied rewrites 66.8%

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

Alternative 8: 98.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. un-div-inv N/A

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

   8. remove-double-neg N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift--.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. associate-*r/ N/A

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

   9. associate-*r/ N/A

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

6. Applied rewrites 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. frac-2neg N/A

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

   4. frac-2neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lift-/.f64 N/A

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

   9. frac-times N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 9: 57.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 68.7

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

5. Applied rewrites 68.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 68.7

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 62.6

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

7. Applied rewrites 62.6%

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

Reproduce

herbie shell --seed 2024219 
(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))))