NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.6%

Time: 14.3s

Alternatives: 7

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. un-div-inv 78.2%

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

   2. difference-of-squares 85.7%

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

   3. associate-/r* 86.5%

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

   4. div-inv 86.5%

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

   5. metadata-eval 86.5%

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

4. Applied egg-rr 86.5%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. +-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. 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. Step-by-step derivation

   1. associate-*r/ 99.6%

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

   2. associate-*r/ 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(Pi * N[(0.5 / N[(a + b), $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 78.2%

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

   1. un-div-inv 78.2%

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

   2. difference-of-squares 85.7%

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

   3. associate-/r* 86.5%

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

   4. div-inv 86.5%

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

   5. metadata-eval 86.5%

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

4. Applied egg-rr 86.5%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. +-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. 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%

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

Alternative 3: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-148}:\\ \;\;\;\;\pi \cdot \frac{\frac{-0.5}{a + b}}{b \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.9e-148)
   (* PI (/ (/ -0.5 (+ a b)) (* b (- b a))))
   (* (* 0.5 (/ PI b)) (/ (/ 1.0 a) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.9e-148) {
		tmp = ((double) M_PI) * ((-0.5 / (a + b)) / (b * (b - a)));
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.9e-148) {
		tmp = Math.PI * ((-0.5 / (a + b)) / (b * (b - a)));
	} else {
		tmp = (0.5 * (Math.PI / b)) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.9e-148:
		tmp = math.pi * ((-0.5 / (a + b)) / (b * (b - a)))
	else:
		tmp = (0.5 * (math.pi / b)) * ((1.0 / a) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.9e-148)
		tmp = Float64(pi * Float64(Float64(-0.5 / Float64(a + b)) / Float64(b * Float64(b - a))));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.9e-148)
		tmp = pi * ((-0.5 / (a + b)) / (b * (b - a)));
	else
		tmp = (0.5 * (pi / b)) * ((1.0 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.9e-148], N[(Pi * N[(N[(-0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.9e-148

   1. Initial program 79.4%

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

      1. un-div-inv 79.4%

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

      2. difference-of-squares 84.2%

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

      3. associate-/r* 85.7%

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

      4. div-inv 85.7%

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

      5. metadata-eval 85.7%

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

   4. Applied egg-rr 85.7%

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

      1. associate-*l/ 99.6%

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

      2. associate-/l* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-frac 99.5%

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

      5. metadata-eval 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in a around inf 82.5%

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

       1. associate-*r/ 82.5%

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

   11. Applied egg-rr 82.5%

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

       1. associate-/l* 82.5%

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

       2. associate-/r* 82.5%

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

       3. associate-*r/ 82.5%

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

       4. *-commutative 82.5%

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

       5. times-frac 82.6%

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

       6. associate-*l* 82.6%

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

       7. metadata-eval 82.6%

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

       8. associate-*r/ 82.5%

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

       9. associate-/r* 82.5%

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

       10. *-commutative 82.5%

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

   13. Simplified 82.5%

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

   if -4.9e-148 < a

   1. Initial program 77.5%

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

      1. un-div-inv 77.4%

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

      2. difference-of-squares 86.6%

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

      3. associate-/r* 87.1%

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

      4. div-inv 87.1%

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

      5. metadata-eval 87.1%

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

   4. Applied egg-rr 87.1%

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

      1. associate-*l/ 99.6%

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

      2. associate-/l* 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/l* 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-neg-frac 99.6%

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

      5. 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. Taylor expanded in a around 0 71.4%

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

   10. Taylor expanded in a around 0 71.4%

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

       1. associate-/r* 71.4%

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

   12. Simplified 71.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.9 \cdot 10^{-148}:\\ \;\;\;\;\pi \cdot \frac{\frac{-0.5}{a + b}}{b \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.2e+109)
   (* (/ (/ PI a) b) (/ 0.5 (- b a)))
   (* (* 0.5 (/ PI b)) (/ (/ 1.0 a) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.2e+109) {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / (b - a));
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.2e+109) {
		tmp = ((Math.PI / a) / b) * (0.5 / (b - a));
	} else {
		tmp = (0.5 * (Math.PI / b)) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.2e+109:
		tmp = ((math.pi / a) / b) * (0.5 / (b - a))
	else:
		tmp = (0.5 * (math.pi / b)) * ((1.0 / a) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.2e+109)
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / Float64(b - a)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.2e+109)
		tmp = ((pi / a) / b) * (0.5 / (b - a));
	else
		tmp = (0.5 * (pi / b)) * ((1.0 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.2e+109], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.19999999999999939e109

   1. Initial program 51.3%

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

      1. associate-*l* 51.3%

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

   3. Simplified 51.3%

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

      1. *-commutative 51.3%

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

      2. div-inv 51.3%

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

      3. sub-neg 51.3%

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

      4. neg-mul-1 51.3%

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

      5. div-inv 51.3%

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

      6. difference-of-squares 63.5%

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

      7. associate-/r* 99.8%

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

      8. add-sqr-sqrt 55.8%

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

      9. sqrt-unprod 64.2%

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

      10. frac-times 64.1%

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

      11. metadata-eval 64.1%

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

      12. metadata-eval 64.1%

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

      13. frac-times 64.2%

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

      14. sqrt-unprod 25.3%

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

      15. add-sqr-sqrt 58.1%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in a around 0 58.1%

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

      1. add-cube-cbrt 58.1%

         \[\leadsto \frac{\pi}{2} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{1}{a \cdot b}}{b - a}} \cdot \sqrt[3]{\frac{\frac{1}{a \cdot b}}{b - a}}\right) \cdot \sqrt[3]{\frac{\frac{1}{a \cdot b}}{b - a}}\right)} \]

      2. pow3 58.1%

         \[\leadsto \frac{\pi}{2} \cdot \color{blue}{{\left(\sqrt[3]{\frac{\frac{1}{a \cdot b}}{b - a}}\right)}^{3}} \]

      3. associate-/l/ 58.1%

         \[\leadsto \frac{\pi}{2} \cdot {\left(\sqrt[3]{\color{blue}{\frac{1}{\left(b - a\right) \cdot \left(a \cdot b\right)}}}\right)}^{3} \]

      4. cbrt-div 58.1%

         \[\leadsto \frac{\pi}{2} \cdot {\color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\left(b - a\right) \cdot \left(a \cdot b\right)}}\right)}}^{3} \]

      5. metadata-eval 58.1%

         \[\leadsto \frac{\pi}{2} \cdot {\left(\frac{\color{blue}{1}}{\sqrt[3]{\left(b - a\right) \cdot \left(a \cdot b\right)}}\right)}^{3} \]

   9. Applied egg-rr 58.1%

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

       1. cube-div 58.1%

          \[\leadsto \frac{\pi}{2} \cdot \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{\left(b - a\right) \cdot \left(a \cdot b\right)}\right)}^{3}}} \]

       2. metadata-eval 58.1%

          \[\leadsto \frac{\pi}{2} \cdot \frac{\color{blue}{1}}{{\left(\sqrt[3]{\left(b - a\right) \cdot \left(a \cdot b\right)}\right)}^{3}} \]

       3. rem-cube-cbrt 58.1%

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

       4. *-commutative 58.1%

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

       5. *-commutative 58.1%

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

   11. Simplified 58.1%

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

       1. pow1 58.1%

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

       2. div-inv 58.1%

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

       3. metadata-eval 58.1%

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

       4. un-div-inv 58.1%

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

       5. times-frac 58.1%

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

       6. *-commutative 58.1%

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

   13. Applied egg-rr 58.1%

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

       1. unpow1 58.1%

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

       2. associate-/r* 58.1%

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

   15. Simplified 58.1%

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

   if -8.19999999999999939e109 < a

   1. Initial program 83.4%

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

      1. un-div-inv 83.4%

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

      2. difference-of-squares 89.9%

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

      3. associate-/r* 90.3%

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

      4. div-inv 90.3%

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

      5. metadata-eval 90.3%

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

   4. Applied egg-rr 90.3%

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

      1. associate-*l/ 99.6%

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

      2. associate-/l* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-frac 99.5%

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

      5. metadata-eval 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in a around 0 66.6%

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

   10. Taylor expanded in a around 0 66.5%

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

       1. associate-/r* 66.6%

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

   12. Simplified 66.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-148}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7e-148)
   (/ (* -0.5 (/ PI (* a b))) (- b a))
   (* (* 0.5 (/ PI b)) (/ (/ 1.0 a) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7e-148) {
		tmp = (-0.5 * (((double) M_PI) / (a * b))) / (b - a);
	} else {
		tmp = (0.5 * (((double) M_PI) / b)) * ((1.0 / a) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7e-148) {
		tmp = (-0.5 * (Math.PI / (a * b))) / (b - a);
	} else {
		tmp = (0.5 * (Math.PI / b)) * ((1.0 / a) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7e-148:
		tmp = (-0.5 * (math.pi / (a * b))) / (b - a)
	else:
		tmp = (0.5 * (math.pi / b)) * ((1.0 / a) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7e-148)
		tmp = Float64(Float64(-0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / b)) * Float64(Float64(1.0 / a) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7e-148)
		tmp = (-0.5 * (pi / (a * b))) / (b - a);
	else
		tmp = (0.5 * (pi / b)) * ((1.0 / a) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7e-148], N[(N[(-0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / b), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.0000000000000001e-148

   1. Initial program 79.4%

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

      1. un-div-inv 79.4%

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

      2. difference-of-squares 84.2%

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

      3. associate-/r* 85.7%

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

      4. div-inv 85.7%

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

      5. metadata-eval 85.7%

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

   4. Applied egg-rr 85.7%

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

      1. associate-*l/ 99.6%

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

      2. associate-/l* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-frac 99.5%

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

      5. metadata-eval 99.5%

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

   8. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. associate-*r/ 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around inf 82.3%

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

   if -7.0000000000000001e-148 < a

   1. Initial program 77.5%

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

      1. un-div-inv 77.4%

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

      2. difference-of-squares 86.6%

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

      3. associate-/r* 87.1%

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

      4. div-inv 87.1%

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

      5. metadata-eval 87.1%

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

   4. Applied egg-rr 87.1%

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

      1. associate-*l/ 99.6%

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

      2. associate-/l* 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/l* 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-neg-frac 99.6%

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

      5. 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. Taylor expanded in a around 0 71.4%

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

   10. Taylor expanded in a around 0 71.4%

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

       1. associate-/r* 71.4%

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

   12. Simplified 71.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-148}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b}\right) \cdot \frac{\frac{1}{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. un-div-inv 78.2%

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

   2. difference-of-squares 85.7%

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

   3. associate-/r* 86.5%

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

   4. div-inv 86.5%

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

   5. metadata-eval 86.5%

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

4. Applied egg-rr 86.5%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. +-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. 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. Taylor expanded in a around 0 99.6%

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

10. Final simplification 99.6%

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

Alternative 7: 63.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. un-div-inv 78.2%

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

   2. difference-of-squares 85.7%

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

   3. associate-/r* 86.5%

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

   4. div-inv 86.5%

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

   5. metadata-eval 86.5%

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

4. Applied egg-rr 86.5%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. +-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. 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. Taylor expanded in a around 0 61.9%

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

10. Taylor expanded in a around 0 61.9%

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

    1. associate-/r* 61.9%

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

12. Simplified 61.9%

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

13. Final simplification 61.9%

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

Reproduce

herbie shell --seed 2024077 
(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))))