NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 21.3s

Alternatives: 8

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: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

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(Float64(Float64(pi * 0.5) / Float64(b + a)) * 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[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

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

   1. un-div-inv 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 92.0%

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

   3. associate-/r* 92.1%

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

   4. div-inv 92.1%

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

   5. metadata-eval 92.1%

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

4. Applied egg-rr 92.1%

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

   1. associate-*l/ 99.7%

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

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. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(0.5 * N[(Pi / 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 83.3%

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

   1. un-div-inv 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 92.0%

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

   3. associate-/r* 92.1%

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

   4. div-inv 92.1%

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

   5. metadata-eval 92.1%

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

4. Applied egg-rr 92.1%

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

   1. associate-*l/ 99.7%

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

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%

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

   2. *-commutative 99.6%

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

   3. associate-*r/ 99.6%

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

   4. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{\frac{\pi}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot \frac{0.5}{a}}{b + a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.4e-70)
   (* (/ -0.5 a) (/ (/ PI b) (- b a)))
   (/ (/ (* PI (/ 0.5 a)) (+ b a)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.4e-70) {
		tmp = (-0.5 / a) * ((((double) M_PI) / b) / (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 <= -4.4e-70) {
		tmp = (-0.5 / a) * ((Math.PI / b) / (b - a));
	} else {
		tmp = ((Math.PI * (0.5 / a)) / (b + a)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.4e-70:
		tmp = (-0.5 / a) * ((math.pi / b) / (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 <= -4.4e-70)
		tmp = Float64(Float64(-0.5 / a) * Float64(Float64(pi / b) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(pi * Float64(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 <= -4.4e-70)
		tmp = (-0.5 / a) * ((pi / b) / (b - a));
	else
		tmp = ((pi * (0.5 / a)) / (b + a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.4e-70], N[(N[(-0.5 / a), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.3999999999999998e-70

   1. Initial program 86.2%

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

      1. un-div-inv 86.2%

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

      2. difference-of-squares 97.2%

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

      3. associate-/r* 97.2%

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

      4. div-inv 97.2%

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

      5. metadata-eval 97.2%

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

   4. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. Taylor expanded in b around 0 87.1%

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

      1. *-un-lft-identity 87.1%

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

      2. associate-*r/ 87.1%

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

      3. times-frac 87.1%

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

   9. Applied egg-rr 87.1%

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

       1. *-lft-identity 87.1%

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

       2. associate-/l* 87.2%

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

   11. Simplified 87.2%

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

   if -4.3999999999999998e-70 < a

   1. Initial program 82.2%

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

      1. associate-*l* 82.2%

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

      2. *-rgt-identity 82.2%

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

      3. associate-/l* 82.2%

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

      4. metadata-eval 82.2%

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

      5. associate-*l/ 82.2%

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

      6. *-lft-identity 82.2%

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

      7. sub-neg 82.2%

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

      8. distribute-neg-frac 82.2%

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

      9. metadata-eval 82.2%

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

   3. Simplified 82.2%

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

      1. metadata-eval 82.2%

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

      2. div-inv 82.2%

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

      3. associate-*r/ 82.2%

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

      4. *-commutative 82.2%

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

      5. difference-of-squares 89.9%

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

      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 71.9%

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

   7. Taylor expanded in a around 0 77.8%

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

Alternative 4: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \frac{\frac{\pi}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e-92)
   (* (/ -0.5 a) (/ (/ PI b) (- b a)))
   (/ (/ (* PI 0.5) (* b a)) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-92) {
		tmp = (-0.5 / a) * ((((double) M_PI) / b) / (b - a));
	} else {
		tmp = ((((double) M_PI) * 0.5) / (b * a)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-92) {
		tmp = (-0.5 / a) * ((Math.PI / b) / (b - a));
	} else {
		tmp = ((Math.PI * 0.5) / (b * a)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.7e-92:
		tmp = (-0.5 / a) * ((math.pi / b) / (b - a))
	else:
		tmp = ((math.pi * 0.5) / (b * a)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e-92)
		tmp = Float64(Float64(-0.5 / a) * Float64(Float64(pi / b) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b * a)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.7e-92)
		tmp = (-0.5 / a) * ((pi / b) / (b - a));
	else
		tmp = ((pi * 0.5) / (b * a)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e-92], N[(N[(-0.5 / a), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.69999999999999995e-92

   1. Initial program 86.5%

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

      1. un-div-inv 86.5%

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

      2. difference-of-squares 97.2%

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

      3. associate-/r* 97.2%

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

      4. div-inv 97.2%

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

      5. metadata-eval 97.2%

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

   4. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. Taylor expanded in b around 0 86.1%

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

      1. *-un-lft-identity 86.1%

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

      2. associate-*r/ 86.1%

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

      3. times-frac 86.1%

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

   9. Applied egg-rr 86.1%

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

       1. *-lft-identity 86.1%

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

       2. associate-/l* 86.2%

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

   11. Simplified 86.2%

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

   if -2.69999999999999995e-92 < a

   1. Initial program 82.0%

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

      1. associate-*l* 82.0%

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

      2. *-rgt-identity 82.0%

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

      3. associate-/l* 82.0%

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

      4. metadata-eval 82.0%

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

      5. associate-*l/ 82.0%

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

      6. *-lft-identity 82.0%

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

      7. sub-neg 82.0%

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

      8. distribute-neg-frac 82.0%

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

      9. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in a around 0 59.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 59.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

   7. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]
   8. Step-by-step derivation

      1. pow2 59.5%

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

      2. associate-/r* 59.1%

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

      3. associate-*r/ 59.1%

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

      4. *-un-lft-identity 59.1%

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

      5. times-frac 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. associate-*l/ 68.6%

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

       2. *-lft-identity 68.6%

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

       3. associate-/l* 68.6%

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

       4. associate-*l/ 68.5%

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

       5. associate-/r* 68.6%

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

   11. Simplified 68.6%

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

       1. *-commutative 68.6%

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

       2. frac-times 68.7%

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

       3. *-un-lft-identity 68.7%

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

       4. frac-times 68.5%

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

       5. *-commutative 68.5%

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

       6. associate-/l/ 68.5%

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

       7. associate-*l/ 68.6%

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

       8. frac-times 68.6%

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

       9. *-un-lft-identity 68.6%

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

       10. *-commutative 68.6%

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

   13. Applied egg-rr 68.6%

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

Alternative 5: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-92}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e-92)
   (* -0.5 (/ (/ (/ PI a) b) (- b a)))
   (/ (/ (* PI 0.5) (* b a)) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-92) {
		tmp = -0.5 * (((((double) M_PI) / a) / b) / (b - a));
	} else {
		tmp = ((((double) M_PI) * 0.5) / (b * a)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.7e-92) {
		tmp = -0.5 * (((Math.PI / a) / b) / (b - a));
	} else {
		tmp = ((Math.PI * 0.5) / (b * a)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.7e-92:
		tmp = -0.5 * (((math.pi / a) / b) / (b - a))
	else:
		tmp = ((math.pi * 0.5) / (b * a)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e-92)
		tmp = Float64(-0.5 * Float64(Float64(Float64(pi / a) / b) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b * a)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.7e-92)
		tmp = -0.5 * (((pi / a) / b) / (b - a));
	else
		tmp = ((pi * 0.5) / (b * a)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e-92], N[(-0.5 * N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.69999999999999995e-92

   1. Initial program 86.5%

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

      1. un-div-inv 86.5%

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

      2. difference-of-squares 97.2%

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

      3. associate-/r* 97.2%

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

      4. div-inv 97.2%

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

      5. metadata-eval 97.2%

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

   4. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. Taylor expanded in b around 0 86.1%

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

      1. associate-/r* 86.2%

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

      2. associate-/l* 86.2%

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

   9. Applied egg-rr 86.2%

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

   if -2.69999999999999995e-92 < a

   1. Initial program 82.0%

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

      1. associate-*l* 82.0%

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

      2. *-rgt-identity 82.0%

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

      3. associate-/l* 82.0%

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

      4. metadata-eval 82.0%

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

      5. associate-*l/ 82.0%

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

      6. *-lft-identity 82.0%

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

      7. sub-neg 82.0%

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

      8. distribute-neg-frac 82.0%

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

      9. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in a around 0 59.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 59.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

   7. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]
   8. Step-by-step derivation

      1. pow2 59.5%

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

      2. associate-/r* 59.1%

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

      3. associate-*r/ 59.1%

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

      4. *-un-lft-identity 59.1%

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

      5. times-frac 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. associate-*l/ 68.6%

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

       2. *-lft-identity 68.6%

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

       3. associate-/l* 68.6%

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

       4. associate-*l/ 68.5%

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

       5. associate-/r* 68.6%

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

   11. Simplified 68.6%

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

       1. *-commutative 68.6%

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

       2. frac-times 68.7%

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

       3. *-un-lft-identity 68.7%

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

       4. frac-times 68.5%

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

       5. *-commutative 68.5%

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

       6. associate-/l/ 68.5%

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

       7. associate-*l/ 68.6%

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

       8. frac-times 68.6%

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

       9. *-un-lft-identity 68.6%

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

       10. *-commutative 68.6%

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

   13. Applied egg-rr 68.6%

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

Alternative 6: 63.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. associate-*l* 83.3%

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

   2. *-rgt-identity 83.3%

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

   3. associate-/l* 83.3%

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

   4. metadata-eval 83.3%

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

   5. associate-*l/ 83.4%

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

   6. *-lft-identity 83.4%

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

   7. sub-neg 83.4%

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

   8. distribute-neg-frac 83.4%

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

   9. metadata-eval 83.4%

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

3. Simplified 83.4%

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

5. Taylor expanded in a around 0 58.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
6. Step-by-step derivation

   1. associate-*r/ 58.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

7. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]
8. Step-by-step derivation

   1. pow2 58.3%

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

   2. associate-/r* 58.0%

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

   3. associate-*r/ 58.0%

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

   4. *-un-lft-identity 58.0%

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

   5. times-frac 64.7%

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

9. Applied egg-rr 64.7%

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

    1. associate-*l/ 64.7%

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

    2. *-lft-identity 64.7%

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

    3. associate-/l* 64.7%

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

    4. associate-*l/ 64.7%

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

    5. associate-/r* 64.7%

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

11. Simplified 64.7%

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

    1. *-commutative 64.7%

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

    2. frac-times 64.8%

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

    3. *-un-lft-identity 64.8%

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

    4. frac-times 64.7%

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

    5. *-commutative 64.7%

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

    6. associate-/l/ 64.6%

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

    7. associate-*l/ 64.7%

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

    8. frac-times 64.7%

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

    9. *-un-lft-identity 64.7%

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

    10. *-commutative 64.7%

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

13. Applied egg-rr 64.7%

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

Alternative 7: 63.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. associate-*l* 83.3%

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

   2. *-rgt-identity 83.3%

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

   3. associate-/l* 83.3%

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

   4. metadata-eval 83.3%

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

   5. associate-*l/ 83.4%

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

   6. *-lft-identity 83.4%

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

   7. sub-neg 83.4%

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

   8. distribute-neg-frac 83.4%

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

   9. metadata-eval 83.4%

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

3. Simplified 83.4%

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

5. Taylor expanded in a around 0 58.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
6. Step-by-step derivation

   1. associate-*r/ 58.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

7. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]
8. Step-by-step derivation

   1. pow2 58.3%

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

   2. associate-/r* 58.0%

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

   3. associate-*r/ 58.0%

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

   4. *-un-lft-identity 58.0%

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

   5. times-frac 64.7%

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

9. Applied egg-rr 64.7%

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

    1. associate-*l/ 64.7%

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

    2. *-lft-identity 64.7%

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

    3. associate-/l* 64.7%

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

    4. associate-*l/ 64.7%

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

    5. associate-/r* 64.7%

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

11. Simplified 64.7%

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

    1. associate-*l/ 64.7%

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

    2. associate-/r* 64.7%

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

13. Applied egg-rr 64.7%

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

Alternative 8: 63.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. associate-*l* 83.3%

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

   2. *-rgt-identity 83.3%

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

   3. associate-/l* 83.3%

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

   4. metadata-eval 83.3%

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

   5. associate-*l/ 83.4%

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

   6. *-lft-identity 83.4%

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

   7. sub-neg 83.4%

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

   8. distribute-neg-frac 83.4%

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

   9. metadata-eval 83.4%

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

3. Simplified 83.4%

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

5. Taylor expanded in a around 0 58.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
6. Step-by-step derivation

   1. associate-*r/ 58.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

7. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]
8. Step-by-step derivation

   1. pow2 58.3%

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

   2. associate-/r* 58.0%

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

   3. associate-*r/ 58.0%

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

   4. *-un-lft-identity 58.0%

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

   5. times-frac 64.7%

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

9. Applied egg-rr 64.7%

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

    1. associate-*l/ 64.7%

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

    2. *-lft-identity 64.7%

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

    3. associate-/l* 64.7%

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

    4. associate-*l/ 64.7%

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

    5. associate-/r* 64.7%

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

11. Simplified 64.7%

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

12. Final simplification 64.7%

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

Reproduce

herbie shell --seed 2024188 
(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))))