NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 10.5s

Alternatives: 6

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% 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: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. *-commutative 78.3%

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

   2. *-commutative 78.3%

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

   3. associate-*l/ 78.3%

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

   4. *-lft-identity 78.3%

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

   5. sub-neg 78.3%

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

   6. distribute-neg-frac 78.3%

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

   7. metadata-eval 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in a around 0 57.4%

   \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \]
6. Step-by-step derivation

   1. div-inv 57.4%

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

   2. metadata-eval 57.4%

      \[\leadsto \frac{1}{a} \cdot \frac{\pi \cdot \color{blue}{0.5}}{b \cdot b - a \cdot a} \]

   3. *-commutative 57.4%

      \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b \cdot b - a \cdot a} \]

   4. difference-of-squares 63.3%

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

   5. frac-times 63.3%

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

   6. associate-*r/ 63.3%

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

   7. +-commutative 63.3%

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

   8. *-commutative 63.3%

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

   9. frac-times 69.3%

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

   10. *-un-lft-identity 69.3%

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

   11. *-commutative 69.3%

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

7. Applied egg-rr 69.3%

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

8. Taylor expanded in a around 0 99.6%

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

   1. add0 99.6%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a + b} \cdot \pi}{a \cdot b} + 0} \]

   2. associate-/l* 99.6%

      \[\leadsto \color{blue}{\frac{0.5}{a + b} \cdot \frac{\pi}{a \cdot b}} + 0 \]

10. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\frac{0.5}{a + b} \cdot \frac{\pi}{a \cdot b} + 0} \]
11. Step-by-step derivation

    1. add0 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-114}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{-0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.1e-114)
   (* (/ PI (- b a)) (/ -0.5 (* a b)))
   (/ (/ (* 0.5 PI) b) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.1e-114) {
		tmp = (((double) M_PI) / (b - a)) * (-0.5 / (a * b));
	} else {
		tmp = ((0.5 * ((double) M_PI)) / b) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.1e-114) {
		tmp = (Math.PI / (b - a)) * (-0.5 / (a * b));
	} else {
		tmp = ((0.5 * Math.PI) / b) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.1e-114:
		tmp = (math.pi / (b - a)) * (-0.5 / (a * b))
	else:
		tmp = ((0.5 * math.pi) / b) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.1e-114)
		tmp = Float64(Float64(pi / Float64(b - a)) * Float64(-0.5 / Float64(a * b)));
	else
		tmp = Float64(Float64(Float64(0.5 * pi) / b) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.1e-114)
		tmp = (pi / (b - a)) * (-0.5 / (a * b));
	else
		tmp = ((0.5 * pi) / b) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.1e-114], N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * Pi), $MachinePrecision] / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.09999999999999993e-114

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. *-commutative 80.7%

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

      3. associate-*l/ 80.7%

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

      4. *-lft-identity 80.7%

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

      5. sub-neg 80.7%

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

      6. distribute-neg-frac 80.7%

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

      7. metadata-eval 80.7%

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

   3. Simplified 80.7%

      \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 80.7%

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

      2. metadata-eval 80.7%

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

      3. *-commutative 80.7%

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

      4. difference-of-squares 92.9%

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

      5. times-frac 93.6%

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

   6. Applied egg-rr 93.6%

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

      1. add0 93.6%

         \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\frac{0.5}{b + a} \cdot \frac{\pi}{b - a}\right) + 0} \]

      2. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{b + a}\right) \cdot \frac{\pi}{b - a}} + 0 \]

      3. fma-define 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{b + a}, \frac{\pi}{b - a}, 0\right)} \]

      4. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{\color{blue}{a + b}}, \frac{\pi}{b - a}, 0\right) \]

   8. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{a + b}, \frac{\pi}{b - a}, 0\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 99.6%

         \[\leadsto \color{blue}{\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{a + b}\right) \cdot \frac{\pi}{b - a} + 0} \]

      2. +-rgt-identity 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in a around inf 91.6%

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

   if -2.09999999999999993e-114 < a

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*l/ 77.2%

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

      4. *-lft-identity 77.2%

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

      5. sub-neg 77.2%

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

      6. distribute-neg-frac 77.2%

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

      7. metadata-eval 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in a around 0 62.4%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \]
   6. Step-by-step derivation

      1. div-inv 62.4%

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

      2. metadata-eval 62.4%

         \[\leadsto \frac{1}{a} \cdot \frac{\pi \cdot \color{blue}{0.5}}{b \cdot b - a \cdot a} \]

      3. *-commutative 62.4%

         \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b \cdot b - a \cdot a} \]

      4. difference-of-squares 65.3%

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

      5. frac-times 65.4%

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

      6. associate-*r/ 65.3%

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

      7. +-commutative 65.3%

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

      8. *-commutative 65.3%

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

      9. frac-times 74.1%

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

      10. *-un-lft-identity 74.1%

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

      11. *-commutative 74.1%

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

   7. Applied egg-rr 74.1%

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

   8. Taylor expanded in a around 0 99.6%

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

   9. Taylor expanded in a around 0 69.6%

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

       1. associate-*r/ 69.6%

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

   11. Simplified 69.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-114}:\\ \;\;\;\;\frac{\pi}{b - a} \cdot \frac{-0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{b}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.8e-82)
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ 0.5 (* a b)) (/ PI (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.8e-82) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8.8e-82) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8.8e-82:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 / (a * b)) * (math.pi / (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.8e-82)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.8e-82)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 / (a * b)) * (pi / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.8e-82], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.79999999999999943e-82

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. *-commutative 80.0%

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

      3. associate-*l/ 79.9%

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

      4. *-lft-identity 79.9%

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

      5. sub-neg 79.9%

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

      6. distribute-neg-frac 79.9%

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

      7. metadata-eval 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in a around 0 53.7%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \]
   6. Step-by-step derivation

      1. div-inv 53.7%

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

      2. metadata-eval 53.7%

         \[\leadsto \frac{1}{a} \cdot \frac{\pi \cdot \color{blue}{0.5}}{b \cdot b - a \cdot a} \]

      3. *-commutative 53.7%

         \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b \cdot b - a \cdot a} \]

      4. difference-of-squares 57.5%

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

      5. frac-times 57.6%

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

      6. associate-*r/ 57.6%

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

      7. +-commutative 57.6%

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

      8. *-commutative 57.6%

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

      9. frac-times 61.0%

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

      10. *-un-lft-identity 61.0%

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

      11. *-commutative 61.0%

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

   7. Applied egg-rr 61.0%

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

   8. Taylor expanded in a around 0 99.6%

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

   9. Taylor expanded in a around inf 73.2%

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

   if 8.79999999999999943e-82 < b

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

      2. *-commutative 73.9%

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

      3. associate-*l/ 73.9%

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

      4. *-lft-identity 73.9%

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

      5. sub-neg 73.9%

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

      6. distribute-neg-frac 73.9%

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

      7. metadata-eval 73.9%

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

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 73.9%

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

      2. metadata-eval 73.9%

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

      3. *-commutative 73.9%

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

      4. difference-of-squares 85.5%

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

      5. times-frac 85.5%

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

   6. Applied egg-rr 85.5%

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

      1. add0 85.5%

         \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \left(\frac{0.5}{b + a} \cdot \frac{\pi}{b - a}\right) + 0} \]

      2. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{b + a}\right) \cdot \frac{\pi}{b - a}} + 0 \]

      3. fma-define 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{b + a}, \frac{\pi}{b - a}, 0\right)} \]

      4. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{\color{blue}{a + b}}, \frac{\pi}{b - a}, 0\right) \]

   8. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{a + b}, \frac{\pi}{b - a}, 0\right)} \]
   9. Step-by-step derivation

      1. fma-undefine 99.6%

         \[\leadsto \color{blue}{\left(\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{0.5}{a + b}\right) \cdot \frac{\pi}{b - a} + 0} \]

      2. +-rgt-identity 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in a around 0 91.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{-83}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.32e-83)
   (/ (* 0.5 (/ PI a)) (* a b))
   (/ (* PI (/ 0.5 b)) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.32e-83) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (((double) M_PI) * (0.5 / b)) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.32e-83) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (Math.PI * (0.5 / b)) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.32e-83:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (math.pi * (0.5 / b)) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.32e-83)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(pi * Float64(0.5 / b)) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.32e-83)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (pi * (0.5 / b)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.32e-83], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.31999999999999994e-83

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*l/ 79.7%

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

      4. *-lft-identity 79.7%

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

      5. sub-neg 79.7%

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

      6. distribute-neg-frac 79.7%

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

      7. metadata-eval 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in a around 0 45.4%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \]
   6. Step-by-step derivation

      1. div-inv 45.4%

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

      2. metadata-eval 45.4%

         \[\leadsto \frac{1}{a} \cdot \frac{\pi \cdot \color{blue}{0.5}}{b \cdot b - a \cdot a} \]

      3. *-commutative 45.4%

         \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b \cdot b - a \cdot a} \]

      4. difference-of-squares 58.2%

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

      5. frac-times 58.2%

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

      6. associate-*r/ 58.2%

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

      7. +-commutative 58.2%

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

      8. *-commutative 58.2%

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

      9. frac-times 58.2%

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

      10. *-un-lft-identity 58.2%

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

      11. *-commutative 58.2%

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

   7. Applied egg-rr 58.2%

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

   8. Taylor expanded in a around 0 99.5%

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

   9. Taylor expanded in a around inf 88.8%

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

   if -1.31999999999999994e-83 < a

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. *-commutative 77.7%

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

      3. associate-*l/ 77.7%

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

      4. *-lft-identity 77.7%

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

      5. sub-neg 77.7%

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

      6. distribute-neg-frac 77.7%

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

      7. metadata-eval 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in a around 0 62.7%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \]
   6. Step-by-step derivation

      1. div-inv 62.7%

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

      2. metadata-eval 62.7%

         \[\leadsto \frac{1}{a} \cdot \frac{\pi \cdot \color{blue}{0.5}}{b \cdot b - a \cdot a} \]

      3. *-commutative 62.7%

         \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b \cdot b - a \cdot a} \]

      4. difference-of-squares 65.5%

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

      5. frac-times 65.6%

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

      6. associate-*r/ 65.5%

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

      7. +-commutative 65.5%

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

      8. *-commutative 65.5%

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

      9. frac-times 74.1%

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

      10. *-un-lft-identity 74.1%

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

      11. *-commutative 74.1%

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

   7. Applied egg-rr 74.1%

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

   8. Taylor expanded in a around 0 99.6%

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

   9. Taylor expanded in a around 0 69.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{-83}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{-83}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.32e-83)
   (/ (* 0.5 (/ PI a)) (* a b))
   (/ (/ (* 0.5 PI) b) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.32e-83) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((0.5 * ((double) M_PI)) / b) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.32e-83) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = ((0.5 * Math.PI) / b) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.32e-83:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = ((0.5 * math.pi) / b) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.32e-83)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(Float64(0.5 * pi) / b) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.32e-83)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = ((0.5 * pi) / b) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.32e-83], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * Pi), $MachinePrecision] / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.31999999999999994e-83

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-*l/ 79.7%

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

      4. *-lft-identity 79.7%

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

      5. sub-neg 79.7%

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

      6. distribute-neg-frac 79.7%

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

      7. metadata-eval 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in a around 0 45.4%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \]
   6. Step-by-step derivation

      1. div-inv 45.4%

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

      2. metadata-eval 45.4%

         \[\leadsto \frac{1}{a} \cdot \frac{\pi \cdot \color{blue}{0.5}}{b \cdot b - a \cdot a} \]

      3. *-commutative 45.4%

         \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b \cdot b - a \cdot a} \]

      4. difference-of-squares 58.2%

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

      5. frac-times 58.2%

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

      6. associate-*r/ 58.2%

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

      7. +-commutative 58.2%

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

      8. *-commutative 58.2%

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

      9. frac-times 58.2%

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

      10. *-un-lft-identity 58.2%

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

      11. *-commutative 58.2%

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

   7. Applied egg-rr 58.2%

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

   8. Taylor expanded in a around 0 99.5%

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

   9. Taylor expanded in a around inf 88.8%

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

   if -1.31999999999999994e-83 < a

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. *-commutative 77.7%

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

      3. associate-*l/ 77.7%

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

      4. *-lft-identity 77.7%

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

      5. sub-neg 77.7%

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

      6. distribute-neg-frac 77.7%

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

      7. metadata-eval 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in a around 0 62.7%

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \]
   6. Step-by-step derivation

      1. div-inv 62.7%

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

      2. metadata-eval 62.7%

         \[\leadsto \frac{1}{a} \cdot \frac{\pi \cdot \color{blue}{0.5}}{b \cdot b - a \cdot a} \]

      3. *-commutative 62.7%

         \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b \cdot b - a \cdot a} \]

      4. difference-of-squares 65.5%

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

      5. frac-times 65.6%

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

      6. associate-*r/ 65.5%

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

      7. +-commutative 65.5%

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

      8. *-commutative 65.5%

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

      9. frac-times 74.1%

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

      10. *-un-lft-identity 74.1%

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

      11. *-commutative 74.1%

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

   7. Applied egg-rr 74.1%

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

   8. Taylor expanded in a around 0 99.6%

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

   9. Taylor expanded in a around 0 69.7%

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

       1. associate-*r/ 69.7%

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

   11. Simplified 69.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.32 \cdot 10^{-83}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{b}}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $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. Step-by-step derivation

   1. *-commutative 78.3%

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

   2. *-commutative 78.3%

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

   3. associate-*l/ 78.3%

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

   4. *-lft-identity 78.3%

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

   5. sub-neg 78.3%

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

   6. distribute-neg-frac 78.3%

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

   7. metadata-eval 78.3%

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

3. Simplified 78.3%

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

5. Taylor expanded in a around 0 57.4%

   \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\frac{\pi}{2}}{b \cdot b - a \cdot a} \]
6. Step-by-step derivation

   1. div-inv 57.4%

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

   2. metadata-eval 57.4%

      \[\leadsto \frac{1}{a} \cdot \frac{\pi \cdot \color{blue}{0.5}}{b \cdot b - a \cdot a} \]

   3. *-commutative 57.4%

      \[\leadsto \frac{1}{a} \cdot \frac{\color{blue}{0.5 \cdot \pi}}{b \cdot b - a \cdot a} \]

   4. difference-of-squares 63.3%

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

   5. frac-times 63.3%

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

   6. associate-*r/ 63.3%

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

   7. +-commutative 63.3%

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

   8. *-commutative 63.3%

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

   9. frac-times 69.3%

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

   10. *-un-lft-identity 69.3%

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

   11. *-commutative 69.3%

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

7. Applied egg-rr 69.3%

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

8. Taylor expanded in a around 0 99.6%

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

9. Taylor expanded in a around inf 66.0%

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

10. Final simplification 66.0%

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

Reproduce

herbie shell --seed 2024046 
(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))))