NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.1% → 99.6%

Time: 12.8s

Alternatives: 12

Speedup: 1.2×

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: 77.1% 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, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\pi \cdot 0.5}{b + a}}{b - a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (* (+ (/ 1.0 a) (/ -1.0 b)) (/ (* PI 0.5) (+ b a))) (- b a)))

C

double code(double a, double b) {
	return (((1.0 / a) + (-1.0 / b)) * ((((double) M_PI) * 0.5) / (b + a))) / (b - a);
}

Java

public static double code(double a, double b) {
	return (((1.0 / a) + (-1.0 / b)) * ((Math.PI * 0.5) / (b + a))) / (b - a);
}

Python

def code(a, b):
	return (((1.0 / a) + (-1.0 / b)) * ((math.pi * 0.5) / (b + a))) / (b - a)

Julia

function code(a, b)
	return Float64(Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(pi * 0.5) / Float64(b + a))) / Float64(b - a))
end

MATLAB

function tmp = code(a, b)
	tmp = (((1.0 / a) + (-1.0 / b)) * ((pi * 0.5) / (b + a))) / (b - a);
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

   1. un-div-inv 81.4%

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

   2. difference-of-squares 89.6%

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

   3. associate-/r* 89.9%

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

   4. div-inv 89.9%

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

   5. metadata-eval 89.9%

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

4. Applied egg-rr 89.9%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \frac{\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\pi \cdot 0.5}{b + a}}{b - a} \]
8. Add Preprocessing

Alternative 2: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-170}:\\ \;\;\;\;\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b + a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+112)
   (/ (/ (* PI -0.5) (* b a)) (- b a))
   (if (<= a -1.1e-170)
     (* (* PI 0.5) (/ (+ (/ 1.0 a) (/ -1.0 b)) (- (* b b) (* a a))))
     (/ (/ (/ (* PI 0.5) a) (+ b a)) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+112) {
		tmp = ((((double) M_PI) * -0.5) / (b * a)) / (b - a);
	} else if (a <= -1.1e-170) {
		tmp = (((double) M_PI) * 0.5) * (((1.0 / a) + (-1.0 / b)) / ((b * b) - (a * a)));
	} else {
		tmp = (((((double) M_PI) * 0.5) / a) / (b + a)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+112) {
		tmp = ((Math.PI * -0.5) / (b * a)) / (b - a);
	} else if (a <= -1.1e-170) {
		tmp = (Math.PI * 0.5) * (((1.0 / a) + (-1.0 / b)) / ((b * b) - (a * a)));
	} else {
		tmp = (((Math.PI * 0.5) / a) / (b + a)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4e+112:
		tmp = ((math.pi * -0.5) / (b * a)) / (b - a)
	elif a <= -1.1e-170:
		tmp = (math.pi * 0.5) * (((1.0 / a) + (-1.0 / b)) / ((b * b) - (a * a)))
	else:
		tmp = (((math.pi * 0.5) / a) / (b + a)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+112)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * a)) / Float64(b - a));
	elseif (a <= -1.1e-170)
		tmp = Float64(Float64(pi * 0.5) * Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(Float64(pi * 0.5) / a) / Float64(b + a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4e+112)
		tmp = ((pi * -0.5) / (b * a)) / (b - a);
	elseif (a <= -1.1e-170)
		tmp = (pi * 0.5) * (((1.0 / a) + (-1.0 / b)) / ((b * b) - (a * a)));
	else
		tmp = (((pi * 0.5) / a) / (b + a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+112], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.1e-170], N[(N[(Pi * 0.5), $MachinePrecision] * N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / a), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.4e112

   1. Initial program 69.4%

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

      1. un-div-inv 69.4%

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

      2. difference-of-squares 90.6%

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

      3. associate-/r* 90.6%

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

      4. div-inv 90.6%

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

      5. metadata-eval 90.6%

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

   4. Applied egg-rr 90.6%

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

      1. associate-*l/ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in b around 0 99.9%

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

      1. associate-*r/ 99.9%

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

   9. Simplified 99.9%

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

   if -2.4e112 < a < -1.10000000000000007e-170

   1. Initial program 95.0%

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

      1. associate-*l* 94.8%

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

      2. *-rgt-identity 94.8%

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

      3. associate-/l* 94.8%

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

      4. metadata-eval 94.8%

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

      5. associate-*l/ 95.0%

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

      6. *-lft-identity 95.0%

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

      7. sub-neg 95.0%

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

      8. distribute-neg-frac 95.0%

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

      9. metadata-eval 95.0%

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

   3. Simplified 95.0%

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

   if -1.10000000000000007e-170 < a

   1. Initial program 78.8%

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

      1. associate-*l* 78.8%

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

      2. *-rgt-identity 78.8%

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

      3. associate-/l* 78.8%

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

      4. metadata-eval 78.8%

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

      5. associate-*l/ 78.8%

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

      6. *-lft-identity 78.8%

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

      7. sub-neg 78.8%

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

      8. distribute-neg-frac 78.8%

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

      9. metadata-eval 78.8%

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

   3. Simplified 78.8%

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

      1. metadata-eval 78.8%

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

      2. div-inv 78.8%

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

      3. associate-*r/ 78.7%

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

      4. *-commutative 78.7%

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

      5. difference-of-squares 87.3%

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

      6. associate-/r* 99.7%

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

   6. Applied egg-rr 74.9%

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

   7. Taylor expanded in a around inf 75.0%

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

      1. distribute-lft-out 75.0%

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

   9. Simplified 75.0%

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

   10. Taylor expanded in a around 0 77.3%

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

       1. *-commutative 77.3%

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

       2. associate-*l/ 77.3%

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

   12. Simplified 77.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-170}:\\ \;\;\;\;\left(\pi \cdot 0.5\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b + a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot -2}{b}\right)}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.65e-110)
   (/ (/ (* PI 0.5) (+ b a)) (* b (- a b)))
   (/ (/ (* 0.5 (+ (/ PI a) (/ (* PI -2.0) b))) b) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.65e-110) {
		tmp = ((((double) M_PI) * 0.5) / (b + a)) / (b * (a - b));
	} else {
		tmp = ((0.5 * ((((double) M_PI) / a) + ((((double) M_PI) * -2.0) / b))) / b) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.65e-110) {
		tmp = ((Math.PI * 0.5) / (b + a)) / (b * (a - b));
	} else {
		tmp = ((0.5 * ((Math.PI / a) + ((Math.PI * -2.0) / b))) / b) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.65e-110:
		tmp = ((math.pi * 0.5) / (b + a)) / (b * (a - b))
	else:
		tmp = ((0.5 * ((math.pi / a) + ((math.pi * -2.0) / b))) / b) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.65e-110)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b + a)) / Float64(b * Float64(a - b)));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(Float64(pi / a) + Float64(Float64(pi * -2.0) / b))) / b) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.65e-110)
		tmp = ((pi * 0.5) / (b + a)) / (b * (a - b));
	else
		tmp = ((0.5 * ((pi / a) + ((pi * -2.0) / b))) / b) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.65e-110], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(N[(Pi / a), $MachinePrecision] + N[(N[(Pi * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.65e-110

   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. un-div-inv 84.3%

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

      2. difference-of-squares 93.4%

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

      3. associate-/r* 94.4%

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

      4. div-inv 94.4%

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

      5. metadata-eval 94.4%

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

   4. Applied egg-rr 94.4%

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

   5. Taylor expanded in a around inf 81.1%

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

      1. frac-times 85.0%

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

      2. associate-/l* 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. *-commutative 85.0%

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

      2. mul-1-neg 85.0%

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

      3. associate-*r/ 85.0%

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

      4. +-commutative 85.0%

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

      5. *-commutative 85.0%

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

   9. Simplified 85.0%

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

   if -2.65e-110 < a

   1. Initial program 80.1%

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

      1. un-div-inv 80.2%

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

      2. difference-of-squares 88.0%

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

      3. associate-/r* 87.9%

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

      4. div-inv 87.9%

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

      5. metadata-eval 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around inf 68.8%

      \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \frac{\pi}{a} + 0.5 \cdot \frac{-1 \cdot \pi - \pi}{b}}{b}}}{b - a} \]
   8. Step-by-step derivation

      1. distribute-lft-out 68.8%

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

      2. sub-neg 68.8%

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

      3. mul-1-neg 68.8%

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

      4. distribute-rgt-out 68.8%

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

      5. metadata-eval 68.8%

         \[\leadsto \frac{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot \color{blue}{-2}}{b}\right)}{b}}{b - a} \]

   9. Simplified 68.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left(\frac{\pi}{a} + \frac{\pi \cdot -2}{b}\right)}{b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \frac{0.5}{b + a}\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* PI (/ 0.5 (+ b a))) (/ (+ (/ 1.0 a) (/ -1.0 b)) (- b a))))

C

double code(double a, double b) {
	return (((double) M_PI) * (0.5 / (b + a))) * (((1.0 / a) + (-1.0 / b)) / (b - a));
}

Java

public static double code(double a, double b) {
	return (Math.PI * (0.5 / (b + a))) * (((1.0 / a) + (-1.0 / b)) / (b - a));
}

Python

def code(a, b):
	return (math.pi * (0.5 / (b + a))) * (((1.0 / a) + (-1.0 / b)) / (b - a))

Julia

function code(a, b)
	return Float64(Float64(pi * Float64(0.5 / Float64(b + a))) * Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) / Float64(b - a)))
end

MATLAB

function tmp = code(a, b)
	tmp = (pi * (0.5 / (b + a))) * (((1.0 / a) + (-1.0 / b)) / (b - a));
end

Wolfram

code[a_, b_] := N[(N[(Pi * N[(0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

   1. un-div-inv 81.4%

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

   2. difference-of-squares 89.6%

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

   3. associate-/r* 89.9%

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

   4. div-inv 89.9%

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

   5. metadata-eval 89.9%

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

4. Applied egg-rr 89.9%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. +-commutative 99.6%

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

   4. sub-neg 99.6%

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

   5. distribute-neg-frac 99.6%

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

   6. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \left(\pi \cdot \frac{0.5}{b + a}\right) \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b - a} \]
10. Add Preprocessing

Alternative 5: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot -0.5}{b}}{b + a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.65e-110)
   (/ (/ (/ (* PI -0.5) b) (+ b a)) (- b a))
   (* (/ 1.0 b) (* 0.5 (/ (/ PI a) b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.65e-110) {
		tmp = (((((double) M_PI) * -0.5) / b) / (b + a)) / (b - a);
	} else {
		tmp = (1.0 / b) * (0.5 * ((((double) M_PI) / a) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.65e-110) {
		tmp = (((Math.PI * -0.5) / b) / (b + a)) / (b - a);
	} else {
		tmp = (1.0 / b) * (0.5 * ((Math.PI / a) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.65e-110:
		tmp = (((math.pi * -0.5) / b) / (b + a)) / (b - a)
	else:
		tmp = (1.0 / b) * (0.5 * ((math.pi / a) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.65e-110)
		tmp = Float64(Float64(Float64(Float64(pi * -0.5) / b) / Float64(b + a)) / Float64(b - a));
	else
		tmp = Float64(Float64(1.0 / b) * Float64(0.5 * Float64(Float64(pi / a) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.65e-110)
		tmp = (((pi * -0.5) / b) / (b + a)) / (b - a);
	else
		tmp = (1.0 / b) * (0.5 * ((pi / a) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.65e-110], N[(N[(N[(N[(Pi * -0.5), $MachinePrecision] / b), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] * N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.65e-110

   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. un-div-inv 84.3%

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

      2. difference-of-squares 93.4%

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

      3. associate-/r* 94.4%

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

      4. div-inv 94.4%

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

      5. metadata-eval 94.4%

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

   4. Applied egg-rr 94.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in a around inf 83.8%

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

       1. associate-*r/ 85.1%

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

       2. *-commutative 85.1%

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

   11. Simplified 85.1%

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

   if -2.65e-110 < a

   1. Initial program 80.1%

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

      1. associate-*l* 80.1%

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

      2. *-rgt-identity 80.1%

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

      3. associate-/l* 80.1%

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

      4. metadata-eval 80.1%

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

      5. associate-*l/ 80.1%

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

      6. *-lft-identity 80.1%

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

      7. sub-neg 80.1%

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

      8. distribute-neg-frac 80.1%

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

      9. metadata-eval 80.1%

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

   3. Simplified 80.1%

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

      1. metadata-eval 80.1%

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

      2. div-inv 80.1%

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

      3. associate-*r/ 80.1%

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

      4. *-commutative 80.1%

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

      5. difference-of-squares 87.9%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in a around 0 75.5%

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

      1. div-inv 75.5%

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

      2. associate-/r* 75.4%

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

   9. Applied egg-rr 75.4%

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

   10. Taylor expanded in b around inf 68.8%

       \[\leadsto \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right) \cdot \color{blue}{\frac{1}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot -0.5}{b}}{b + a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot -0.5}{b}}{b + a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b + a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.7e-103)
   (/ (/ (/ (* PI -0.5) b) (+ b a)) (- b a))
   (/ (/ (/ (* PI 0.5) a) (+ b a)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-103) {
		tmp = (((((double) M_PI) * -0.5) / b) / (b + a)) / (b - a);
	} else {
		tmp = (((((double) M_PI) * 0.5) / a) / (b + a)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.7e-103) {
		tmp = (((Math.PI * -0.5) / b) / (b + a)) / (b - a);
	} else {
		tmp = (((Math.PI * 0.5) / a) / (b + a)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.7e-103:
		tmp = (((math.pi * -0.5) / b) / (b + a)) / (b - a)
	else:
		tmp = (((math.pi * 0.5) / a) / (b + a)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.7e-103)
		tmp = Float64(Float64(Float64(Float64(pi * -0.5) / b) / Float64(b + a)) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(Float64(pi * 0.5) / a) / Float64(b + a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.7e-103)
		tmp = (((pi * -0.5) / b) / (b + a)) / (b - a);
	else
		tmp = (((pi * 0.5) / a) / (b + a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.7e-103], N[(N[(N[(N[(Pi * -0.5), $MachinePrecision] / b), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / a), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.6999999999999999e-103

   1. Initial program 84.5%

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

      1. un-div-inv 84.5%

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

      2. difference-of-squares 94.3%

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

      3. associate-/r* 94.2%

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

      4. div-inv 94.2%

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

      5. metadata-eval 94.2%

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

   4. Applied egg-rr 94.2%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in a around inf 86.7%

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

       1. associate-*r/ 88.1%

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

       2. *-commutative 88.1%

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

   11. Simplified 88.1%

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

   if -3.6999999999999999e-103 < a

   1. Initial program 80.1%

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

      1. associate-*l* 80.1%

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

      2. *-rgt-identity 80.1%

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

      3. associate-/l* 80.1%

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

      4. metadata-eval 80.1%

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

      5. associate-*l/ 80.2%

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

      6. *-lft-identity 80.2%

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

      7. sub-neg 80.2%

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

      8. distribute-neg-frac 80.2%

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

      9. metadata-eval 80.2%

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

   3. Simplified 80.2%

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

      1. metadata-eval 80.2%

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

      2. div-inv 80.2%

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

      3. associate-*r/ 80.1%

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

      4. *-commutative 80.1%

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

      5. difference-of-squares 87.7%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 75.6%

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

   7. Taylor expanded in a around inf 75.6%

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

      1. distribute-lft-out 75.6%

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

   9. Simplified 75.6%

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

   10. Taylor expanded in a around 0 78.8%

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

       1. *-commutative 78.8%

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

       2. associate-*l/ 78.8%

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

   12. Simplified 78.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot -0.5}{b}}{b + a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b + a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b + a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.4e-103)
   (/ (/ (* PI 0.5) (+ b a)) (* b (- a b)))
   (/ (/ (/ (* PI 0.5) a) (+ b a)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.4e-103) {
		tmp = ((((double) M_PI) * 0.5) / (b + a)) / (b * (a - b));
	} else {
		tmp = (((((double) M_PI) * 0.5) / a) / (b + a)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.4e-103) {
		tmp = ((Math.PI * 0.5) / (b + a)) / (b * (a - b));
	} else {
		tmp = (((Math.PI * 0.5) / a) / (b + a)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.4e-103:
		tmp = ((math.pi * 0.5) / (b + a)) / (b * (a - b))
	else:
		tmp = (((math.pi * 0.5) / a) / (b + a)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.4e-103)
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b + a)) / Float64(b * Float64(a - b)));
	else
		tmp = Float64(Float64(Float64(Float64(pi * 0.5) / a) / Float64(b + a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.4e-103)
		tmp = ((pi * 0.5) / (b + a)) / (b * (a - b));
	else
		tmp = (((pi * 0.5) / a) / (b + a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.4e-103], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / a), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.40000000000000019e-103

   1. Initial program 84.5%

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

      1. un-div-inv 84.5%

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

      2. difference-of-squares 94.3%

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

      3. associate-/r* 94.2%

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

      4. div-inv 94.2%

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

      5. metadata-eval 94.2%

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in a around inf 83.8%

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

      1. frac-times 88.0%

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

      2. associate-/l* 88.0%

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

   7. Applied egg-rr 88.0%

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

      1. *-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. associate-*r/ 88.0%

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

      4. +-commutative 88.0%

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

      5. *-commutative 88.0%

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

   9. Simplified 88.0%

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

   if -8.40000000000000019e-103 < a

   1. Initial program 80.1%

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

      1. associate-*l* 80.1%

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

      2. *-rgt-identity 80.1%

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

      3. associate-/l* 80.1%

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

      4. metadata-eval 80.1%

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

      5. associate-*l/ 80.2%

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

      6. *-lft-identity 80.2%

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

      7. sub-neg 80.2%

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

      8. distribute-neg-frac 80.2%

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

      9. metadata-eval 80.2%

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

   3. Simplified 80.2%

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

      1. metadata-eval 80.2%

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

      2. div-inv 80.2%

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

      3. associate-*r/ 80.1%

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

      4. *-commutative 80.1%

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

      5. difference-of-squares 87.7%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 75.6%

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

   7. Taylor expanded in a around inf 75.6%

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

      1. distribute-lft-out 75.6%

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

   9. Simplified 75.6%

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

   10. Taylor expanded in a around 0 78.8%

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

       1. *-commutative 78.8%

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

       2. associate-*l/ 78.8%

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

   12. Simplified 78.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b + a}}{b \cdot \left(a - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b + a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e+94)
   (* 0.5 (/ (/ PI a) (* b (- b a))))
   (* (/ 1.0 b) (* 0.5 (/ (/ PI a) b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e+94) {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * (b - a)));
	} else {
		tmp = (1.0 / b) * (0.5 * ((((double) M_PI) / a) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.4e+94) {
		tmp = 0.5 * ((Math.PI / a) / (b * (b - a)));
	} else {
		tmp = (1.0 / b) * (0.5 * ((Math.PI / a) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.4e+94:
		tmp = 0.5 * ((math.pi / a) / (b * (b - a)))
	else:
		tmp = (1.0 / b) * (0.5 * ((math.pi / a) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.4e+94)
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * Float64(b - a))));
	else
		tmp = Float64(Float64(1.0 / b) * Float64(0.5 * Float64(Float64(pi / a) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.4e+94)
		tmp = 0.5 * ((pi / a) / (b * (b - a)));
	else
		tmp = (1.0 / b) * (0.5 * ((pi / a) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.4e+94], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] * N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.39999999999999999e94

   1. Initial program 71.9%

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

      1. associate-*l* 71.9%

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

      2. *-rgt-identity 71.9%

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

      3. associate-/l* 71.9%

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

      4. metadata-eval 71.9%

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

      5. associate-*l/ 71.9%

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

      6. *-lft-identity 71.9%

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

      7. sub-neg 71.9%

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

      8. distribute-neg-frac 71.9%

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

      9. metadata-eval 71.9%

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

   3. Simplified 71.9%

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

      1. metadata-eval 71.9%

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

      2. div-inv 71.9%

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

      3. associate-*r/ 71.9%

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

      4. *-commutative 71.9%

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

      5. difference-of-squares 91.4%

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

      6. associate-/r* 99.8%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in a around 0 80.6%

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

      1. associate-/l* 80.6%

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

      2. associate-/r* 80.6%

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

   9. Applied egg-rr 80.6%

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

       1. associate-/l/ 80.6%

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

       2. *-commutative 80.6%

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

   11. Simplified 80.6%

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

   if -1.39999999999999999e94 < a

   1. Initial program 82.9%

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

      1. associate-*l* 82.9%

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

      2. *-rgt-identity 82.9%

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

      3. associate-/l* 82.9%

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

      4. metadata-eval 82.9%

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

      5. associate-*l/ 82.9%

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

      6. *-lft-identity 82.9%

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

      7. sub-neg 82.9%

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

      8. distribute-neg-frac 82.9%

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

      9. metadata-eval 82.9%

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

   3. Simplified 82.9%

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

      1. metadata-eval 82.9%

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

      2. div-inv 82.9%

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

      3. associate-*r/ 82.9%

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

      4. *-commutative 82.9%

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

      5. difference-of-squares 89.2%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in a around 0 71.4%

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

      1. div-inv 71.4%

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

      2. associate-/r* 71.3%

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

   9. Applied egg-rr 71.3%

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

   10. Taylor expanded in b around inf 66.0%

       \[\leadsto \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right) \cdot \color{blue}{\frac{1}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+94}:\\ \;\;\;\;\frac{0.5}{\left(b - a\right) \cdot \left(b \cdot \frac{a}{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e+94)
   (/ 0.5 (* (- b a) (* b (/ a PI))))
   (* (/ 1.0 b) (* 0.5 (/ (/ PI a) b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+94) {
		tmp = 0.5 / ((b - a) * (b * (a / ((double) M_PI))));
	} else {
		tmp = (1.0 / b) * (0.5 * ((((double) M_PI) / a) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+94) {
		tmp = 0.5 / ((b - a) * (b * (a / Math.PI)));
	} else {
		tmp = (1.0 / b) * (0.5 * ((Math.PI / a) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.7e+94:
		tmp = 0.5 / ((b - a) * (b * (a / math.pi)))
	else:
		tmp = (1.0 / b) * (0.5 * ((math.pi / a) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e+94)
		tmp = Float64(0.5 / Float64(Float64(b - a) * Float64(b * Float64(a / pi))));
	else
		tmp = Float64(Float64(1.0 / b) * Float64(0.5 * Float64(Float64(pi / a) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.7e+94)
		tmp = 0.5 / ((b - a) * (b * (a / pi)));
	else
		tmp = (1.0 / b) * (0.5 * ((pi / a) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e+94], N[(0.5 / N[(N[(b - a), $MachinePrecision] * N[(b * N[(a / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] * N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.7000000000000001e94

   1. Initial program 71.9%

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

      1. associate-*l* 71.9%

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

      2. *-rgt-identity 71.9%

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

      3. associate-/l* 71.9%

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

      4. metadata-eval 71.9%

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

      5. associate-*l/ 71.9%

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

      6. *-lft-identity 71.9%

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

      7. sub-neg 71.9%

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

      8. distribute-neg-frac 71.9%

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

      9. metadata-eval 71.9%

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

   3. Simplified 71.9%

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

      1. metadata-eval 71.9%

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

      2. div-inv 71.9%

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

      3. associate-*r/ 71.9%

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

      4. *-commutative 71.9%

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

      5. difference-of-squares 91.4%

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

      6. associate-/r* 99.8%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in a around 0 80.6%

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

      1. associate-/l* 80.6%

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

      2. associate-/r* 80.6%

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

   9. Applied egg-rr 80.6%

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

       1. associate-/l/ 80.6%

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

       2. *-commutative 80.6%

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

   11. Simplified 80.6%

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

       1. associate-/r* 80.6%

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

       2. div-inv 80.6%

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

       3. clear-num 80.6%

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

       4. frac-times 80.6%

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

       5. metadata-eval 80.6%

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

       6. *-commutative 80.6%

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

       7. associate-/l* 80.6%

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

       8. div-inv 80.6%

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

       9. un-div-inv 80.6%

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

       10. frac-times 80.6%

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

       11. metadata-eval 80.6%

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

   13. Applied egg-rr 80.6%

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

   if -2.7000000000000001e94 < a

   1. Initial program 82.9%

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

      1. associate-*l* 82.9%

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

      2. *-rgt-identity 82.9%

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

      3. associate-/l* 82.9%

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

      4. metadata-eval 82.9%

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

      5. associate-*l/ 82.9%

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

      6. *-lft-identity 82.9%

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

      7. sub-neg 82.9%

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

      8. distribute-neg-frac 82.9%

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

      9. metadata-eval 82.9%

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

   3. Simplified 82.9%

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

      1. metadata-eval 82.9%

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

      2. div-inv 82.9%

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

      3. associate-*r/ 82.9%

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

      4. *-commutative 82.9%

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

      5. difference-of-squares 89.2%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in a around 0 71.4%

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

      1. div-inv 71.4%

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

      2. associate-/r* 71.3%

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

   9. Applied egg-rr 71.3%

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

   10. Taylor expanded in b around inf 66.0%

       \[\leadsto \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right) \cdot \color{blue}{\frac{1}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+94}:\\ \;\;\;\;\frac{0.5}{\left(b - a\right) \cdot \left(b \cdot \frac{a}{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{-0.5}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.65e-110)
   (/ (* (/ PI a) (/ -0.5 b)) (- b a))
   (* (/ 1.0 b) (* 0.5 (/ (/ PI a) b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.65e-110) {
		tmp = ((((double) M_PI) / a) * (-0.5 / b)) / (b - a);
	} else {
		tmp = (1.0 / b) * (0.5 * ((((double) M_PI) / a) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.65e-110) {
		tmp = ((Math.PI / a) * (-0.5 / b)) / (b - a);
	} else {
		tmp = (1.0 / b) * (0.5 * ((Math.PI / a) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.65e-110:
		tmp = ((math.pi / a) * (-0.5 / b)) / (b - a)
	else:
		tmp = (1.0 / b) * (0.5 * ((math.pi / a) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.65e-110)
		tmp = Float64(Float64(Float64(pi / a) * Float64(-0.5 / b)) / Float64(b - a));
	else
		tmp = Float64(Float64(1.0 / b) * Float64(0.5 * Float64(Float64(pi / a) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.65e-110)
		tmp = ((pi / a) * (-0.5 / b)) / (b - a);
	else
		tmp = (1.0 / b) * (0.5 * ((pi / a) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.65e-110], N[(N[(N[(Pi / a), $MachinePrecision] * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] * N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.65e-110

   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. un-div-inv 84.3%

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

      2. difference-of-squares 93.4%

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

      3. associate-/r* 94.4%

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

      4. div-inv 94.4%

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

      5. metadata-eval 94.4%

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

   4. Applied egg-rr 94.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in a around inf 84.7%

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

       1. associate-*r/ 84.7%

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

       2. *-commutative 84.7%

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

       3. times-frac 84.7%

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

   11. Simplified 84.7%

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

   if -2.65e-110 < a

   1. Initial program 80.1%

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

      1. associate-*l* 80.1%

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

      2. *-rgt-identity 80.1%

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

      3. associate-/l* 80.1%

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

      4. metadata-eval 80.1%

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

      5. associate-*l/ 80.1%

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

      6. *-lft-identity 80.1%

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

      7. sub-neg 80.1%

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

      8. distribute-neg-frac 80.1%

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

      9. metadata-eval 80.1%

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

   3. Simplified 80.1%

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

      1. metadata-eval 80.1%

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

      2. div-inv 80.1%

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

      3. associate-*r/ 80.1%

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

      4. *-commutative 80.1%

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

      5. difference-of-squares 87.9%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in a around 0 75.5%

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

      1. div-inv 75.5%

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

      2. associate-/r* 75.4%

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

   9. Applied egg-rr 75.4%

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

   10. Taylor expanded in b around inf 68.8%

       \[\leadsto \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right) \cdot \color{blue}{\frac{1}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{-0.5}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.65e-110)
   (/ (/ (* PI -0.5) (* b a)) (- b a))
   (* (/ 1.0 b) (* 0.5 (/ (/ PI a) b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.65e-110) {
		tmp = ((((double) M_PI) * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (1.0 / b) * (0.5 * ((((double) M_PI) / a) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.65e-110) {
		tmp = ((Math.PI * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (1.0 / b) * (0.5 * ((Math.PI / a) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.65e-110:
		tmp = ((math.pi * -0.5) / (b * a)) / (b - a)
	else:
		tmp = (1.0 / b) * (0.5 * ((math.pi / a) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.65e-110)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * a)) / Float64(b - a));
	else
		tmp = Float64(Float64(1.0 / b) * Float64(0.5 * Float64(Float64(pi / a) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.65e-110)
		tmp = ((pi * -0.5) / (b * a)) / (b - a);
	else
		tmp = (1.0 / b) * (0.5 * ((pi / a) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.65e-110], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] * N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.65e-110

   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. un-div-inv 84.3%

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

      2. difference-of-squares 93.4%

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

      3. associate-/r* 94.4%

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

      4. div-inv 94.4%

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

      5. metadata-eval 94.4%

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

   4. Applied egg-rr 94.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 84.7%

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

      1. associate-*r/ 84.7%

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

   9. Simplified 84.7%

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

   if -2.65e-110 < a

   1. Initial program 80.1%

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

      1. associate-*l* 80.1%

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

      2. *-rgt-identity 80.1%

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

      3. associate-/l* 80.1%

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

      4. metadata-eval 80.1%

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

      5. associate-*l/ 80.1%

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

      6. *-lft-identity 80.1%

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

      7. sub-neg 80.1%

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

      8. distribute-neg-frac 80.1%

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

      9. metadata-eval 80.1%

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

   3. Simplified 80.1%

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

      1. metadata-eval 80.1%

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

      2. div-inv 80.1%

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

      3. associate-*r/ 80.1%

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

      4. *-commutative 80.1%

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

      5. difference-of-squares 87.9%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in a around 0 75.5%

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

      1. div-inv 75.5%

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

      2. associate-/r* 75.4%

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

   9. Applied egg-rr 75.4%

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

   10. Taylor expanded in b around inf 68.8%

       \[\leadsto \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right) \cdot \color{blue}{\frac{1}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} \cdot \left(0.5 \cdot \frac{\frac{\pi}{a}}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

   1. associate-*l* 81.3%

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

   2. *-rgt-identity 81.3%

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

   3. associate-/l* 81.3%

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

   4. metadata-eval 81.3%

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

   5. associate-*l/ 81.4%

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

   6. *-lft-identity 81.4%

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

   7. sub-neg 81.4%

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

   8. distribute-neg-frac 81.4%

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

   9. metadata-eval 81.4%

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

3. Simplified 81.4%

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

   1. metadata-eval 81.4%

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

   2. div-inv 81.4%

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

   3. associate-*r/ 81.3%

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

   4. *-commutative 81.3%

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

   5. difference-of-squares 89.5%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 72.6%

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

7. Taylor expanded in a around 0 72.7%

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

   1. associate-/l* 72.7%

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

   2. associate-/r* 72.7%

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

9. Applied egg-rr 72.7%

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

    1. associate-/l/ 67.1%

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

    2. *-commutative 67.1%

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

11. Simplified 67.1%

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

12. Final simplification 67.1%

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

Reproduce

herbie shell --seed 2024076 
(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))))