NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.6%

Time: 10.4s

Alternatives: 8

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: 78.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. un-div-inv 73.3%

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

   2. difference-of-squares 84.6%

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

   3. associate-/r* 85.0%

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

   4. div-inv 85.0%

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

   5. metadata-eval 85.0%

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

4. Applied egg-rr 85.0%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-/l* 99.7%

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

   2. *-commutative 99.7%

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

   3. +-commutative 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. un-div-inv 73.3%

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

   2. difference-of-squares 84.6%

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

   3. associate-/r* 85.0%

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

   4. div-inv 85.0%

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

   5. metadata-eval 85.0%

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

4. Applied egg-rr 85.0%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-/l* 99.7%

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

   2. *-commutative 99.7%

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

   3. +-commutative 99.7%

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

10. Simplified 99.7%

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

    1. associate-/l* 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 3: 73.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot -0.5}{a}}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 \cdot \pi}{b}}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e-200)
   (/ (/ (/ (* PI -0.5) a) b) (- b a))
   (/ (/ (/ (* 0.5 PI) b) a) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e-200) {
		tmp = (((((double) M_PI) * -0.5) / a) / b) / (b - a);
	} else {
		tmp = (((0.5 * ((double) M_PI)) / b) / a) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3e-200) {
		tmp = (((Math.PI * -0.5) / a) / b) / (b - a);
	} else {
		tmp = (((0.5 * Math.PI) / b) / a) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3e-200:
		tmp = (((math.pi * -0.5) / a) / b) / (b - a)
	else:
		tmp = (((0.5 * math.pi) / b) / a) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e-200)
		tmp = Float64(Float64(Float64(Float64(pi * -0.5) / a) / b) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * pi) / b) / a) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3e-200)
		tmp = (((pi * -0.5) / a) / b) / (b - a);
	else
		tmp = (((0.5 * pi) / b) / a) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e-200], N[(N[(N[(N[(Pi * -0.5), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999995e-200

   1. Initial program 72.4%

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

      1. un-div-inv 72.4%

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

      2. difference-of-squares 83.9%

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

      3. associate-/r* 84.7%

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

      4. div-inv 84.7%

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

      5. metadata-eval 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. *-commutative 70.4%

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

      3. times-frac 70.5%

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

   9. Simplified 70.5%

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

       1. frac-times 70.4%

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

       2. associate-/r* 70.5%

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

   11. Applied egg-rr 70.5%

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

   if 2.99999999999999995e-200 < b

   1. Initial program 74.3%

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

      1. un-div-inv 74.3%

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

      2. difference-of-squares 85.4%

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

      3. associate-/r* 85.4%

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

      4. div-inv 85.4%

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

      5. metadata-eval 85.4%

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

   4. Applied egg-rr 85.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in a around 0 84.8%

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

       1. associate-*r/ 84.8%

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

       2. *-commutative 84.8%

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

       3. metadata-eval 84.8%

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

       4. distribute-rgt-neg-in 84.8%

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

       5. distribute-frac-neg 84.8%

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

       6. times-frac 84.8%

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

       7. associate-*l/ 84.8%

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

       8. distribute-frac-neg 84.8%

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

       9. associate-*r/ 84.9%

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

       10. distribute-frac-neg 84.9%

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

       11. distribute-rgt-neg-in 84.9%

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

       12. metadata-eval 84.9%

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

       13. *-commutative 84.9%

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

   13. Simplified 84.9%

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

Alternative 4: 73.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-200}:\\ \;\;\;\;\frac{\pi \cdot \frac{-0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 \cdot \pi}{b}}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e-200)
   (/ (* PI (/ -0.5 (* a b))) (- b a))
   (/ (/ (/ (* 0.5 PI) b) a) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e-200) {
		tmp = (((double) M_PI) * (-0.5 / (a * b))) / (b - a);
	} else {
		tmp = (((0.5 * ((double) M_PI)) / b) / a) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3e-200) {
		tmp = (Math.PI * (-0.5 / (a * b))) / (b - a);
	} else {
		tmp = (((0.5 * Math.PI) / b) / a) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3e-200:
		tmp = (math.pi * (-0.5 / (a * b))) / (b - a)
	else:
		tmp = (((0.5 * math.pi) / b) / a) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e-200)
		tmp = Float64(Float64(pi * Float64(-0.5 / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 * pi) / b) / a) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3e-200)
		tmp = (pi * (-0.5 / (a * b))) / (b - a);
	else
		tmp = (((0.5 * pi) / b) / a) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e-200], N[(N[(Pi * N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * Pi), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999995e-200

   1. Initial program 72.4%

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

      1. un-div-inv 72.4%

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

      2. difference-of-squares 83.9%

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

      3. associate-/r* 84.7%

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

      4. div-inv 84.7%

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

      5. metadata-eval 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. *-commutative 70.4%

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

   9. Simplified 70.4%

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

       1. associate-/l* 70.4%

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

   11. Applied egg-rr 70.4%

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

   if 2.99999999999999995e-200 < b

   1. Initial program 74.3%

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

      1. un-div-inv 74.3%

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

      2. difference-of-squares 85.4%

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

      3. associate-/r* 85.4%

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

      4. div-inv 85.4%

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

      5. metadata-eval 85.4%

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

   4. Applied egg-rr 85.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

      3. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in a around 0 84.8%

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

       1. associate-*r/ 84.8%

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

       2. *-commutative 84.8%

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

       3. metadata-eval 84.8%

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

       4. distribute-rgt-neg-in 84.8%

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

       5. distribute-frac-neg 84.8%

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

       6. times-frac 84.8%

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

       7. associate-*l/ 84.8%

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

       8. distribute-frac-neg 84.8%

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

       9. associate-*r/ 84.9%

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

       10. distribute-frac-neg 84.9%

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

       11. distribute-rgt-neg-in 84.9%

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

       12. metadata-eval 84.9%

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

       13. *-commutative 84.9%

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

   13. Simplified 84.9%

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

Alternative 5: 73.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-200}:\\ \;\;\;\;\frac{\pi \cdot \frac{-0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e-200)
   (/ (* PI (/ -0.5 (* a b))) (- b a))
   (/ (/ (* 0.5 PI) (* a b)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e-200) {
		tmp = (((double) M_PI) * (-0.5 / (a * b))) / (b - a);
	} else {
		tmp = ((0.5 * ((double) M_PI)) / (a * b)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3e-200) {
		tmp = (Math.PI * (-0.5 / (a * b))) / (b - a);
	} else {
		tmp = ((0.5 * Math.PI) / (a * b)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3e-200:
		tmp = (math.pi * (-0.5 / (a * b))) / (b - a)
	else:
		tmp = ((0.5 * math.pi) / (a * b)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e-200)
		tmp = Float64(Float64(pi * Float64(-0.5 / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(0.5 * pi) / Float64(a * b)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3e-200)
		tmp = (pi * (-0.5 / (a * b))) / (b - a);
	else
		tmp = ((0.5 * pi) / (a * b)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e-200], N[(N[(Pi * N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999995e-200

   1. Initial program 72.4%

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

      1. un-div-inv 72.4%

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

      2. difference-of-squares 83.9%

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

      3. associate-/r* 84.7%

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

      4. div-inv 84.7%

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

      5. metadata-eval 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. *-commutative 70.4%

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

   9. Simplified 70.4%

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

       1. associate-/l* 70.4%

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

   11. Applied egg-rr 70.4%

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

   if 2.99999999999999995e-200 < b

   1. Initial program 74.3%

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

      1. un-div-inv 74.3%

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

      2. difference-of-squares 85.4%

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

      3. associate-/r* 85.4%

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

      4. div-inv 85.4%

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

      5. metadata-eval 85.4%

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

   4. Applied egg-rr 85.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf 84.8%

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

      1. associate-*r/ 84.8%

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

   9. Simplified 84.8%

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

Alternative 6: 73.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-200}:\\ \;\;\;\;\frac{\pi \cdot \frac{-0.5}{a \cdot b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e-200)
   (/ (* PI (/ -0.5 (* a b))) (- b a))
   (/ (* (/ PI a) (/ 0.5 b)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e-200) {
		tmp = (((double) M_PI) * (-0.5 / (a * b))) / (b - a);
	} else {
		tmp = ((((double) M_PI) / a) * (0.5 / b)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3e-200) {
		tmp = (Math.PI * (-0.5 / (a * b))) / (b - a);
	} else {
		tmp = ((Math.PI / a) * (0.5 / b)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3e-200:
		tmp = (math.pi * (-0.5 / (a * b))) / (b - a)
	else:
		tmp = ((math.pi / a) * (0.5 / b)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e-200)
		tmp = Float64(Float64(pi * Float64(-0.5 / Float64(a * b))) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi / a) * Float64(0.5 / b)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3e-200)
		tmp = (pi * (-0.5 / (a * b))) / (b - a);
	else
		tmp = ((pi / a) * (0.5 / b)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e-200], N[(N[(Pi * N[(-0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999995e-200

   1. Initial program 72.4%

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

      1. un-div-inv 72.4%

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

      2. difference-of-squares 83.9%

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

      3. associate-/r* 84.7%

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

      4. div-inv 84.7%

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

      5. metadata-eval 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. associate-*l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in b around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. *-commutative 70.4%

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

   9. Simplified 70.4%

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

       1. associate-/l* 70.4%

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

   11. Applied egg-rr 70.4%

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

   if 2.99999999999999995e-200 < b

   1. Initial program 74.3%

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

      1. un-div-inv 74.3%

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

      2. difference-of-squares 85.4%

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

      3. associate-/r* 85.4%

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

      4. div-inv 85.4%

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

      5. metadata-eval 85.4%

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

   4. Applied egg-rr 85.4%

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

      1. associate-*l/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. *-commutative 84.8%

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

      3. times-frac 84.8%

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

   9. Simplified 84.8%

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

Alternative 7: 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 73.3%

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

   1. un-div-inv 73.3%

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

   2. difference-of-squares 84.6%

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

   3. associate-/r* 85.0%

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

   4. div-inv 85.0%

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

   5. metadata-eval 85.0%

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

4. Applied egg-rr 85.0%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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* 99.7%

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

   2. associate-*r/ 99.6%

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

   3. +-commutative 99.6%

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

   4. sub-neg 99.6%

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

   5. distribute-neg-frac 99.6%

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

   6. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. 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. Add Preprocessing

Alternative 8: 66.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. un-div-inv 73.3%

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

   2. difference-of-squares 84.6%

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

   3. associate-/r* 85.0%

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

   4. div-inv 85.0%

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

   5. metadata-eval 85.0%

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

4. Applied egg-rr 85.0%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 66.1%

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

   1. associate-*r/ 66.1%

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

   2. *-commutative 66.1%

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

   3. times-frac 66.1%

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

9. Simplified 66.1%

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

    1. *-un-lft-identity 66.1%

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

    2. associate-/l* 66.2%

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

11. Applied egg-rr 66.2%

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

    1. *-lft-identity 66.2%

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

13. Simplified 66.2%

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

Reproduce

herbie shell --seed 2024165 
(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))))