NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.6%

Time: 11.4s

Alternatives: 7

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.2% 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

\[\begin{array}{l} \\ \frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{a + b}}{b - a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(Pi / 2.0), $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 76.5%

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

   1. associate-*l* 76.4%

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

3. Simplified 76.4%

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

   1. associate-*l/ 76.4%

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

   2. sub-neg 76.4%

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

   3. *-un-lft-identity 76.4%

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

   4. neg-mul-1 76.4%

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

   5. div-inv 76.4%

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

   6. difference-of-squares 85.0%

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

   7. associate-/r* 99.6%

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

   8. add-sqr-sqrt 52.1%

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

   9. sqrt-unprod 77.8%

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

   10. frac-times 77.7%

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

   11. metadata-eval 77.7%

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

   12. metadata-eval 77.7%

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

   13. frac-times 77.8%

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

   14. sqrt-unprod 33.3%

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

   15. add-sqr-sqrt 69.3%

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

6. Applied egg-rr 69.3%

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

   1. +-commutative 69.3%

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

   2. add-sqr-sqrt 33.3%

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

   3. sqrt-unprod 77.8%

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

   4. frac-times 77.7%

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

   5. metadata-eval 77.7%

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

   6. metadata-eval 77.7%

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

   7. frac-times 77.8%

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

   8. sqrt-unprod 52.1%

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

   9. add-sqr-sqrt 99.6%

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

   10. div-inv 99.6%

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

   11. fma-define 99.6%

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

8. Applied egg-rr 99.6%

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

   1. fma-undefine 99.6%

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

   2. neg-mul-1 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. metadata-eval 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

    \[\leadsto \frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{a + b}}{b - a} \]
12. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. un-div-inv 76.5%

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

   2. difference-of-squares 85.1%

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

   3. associate-/r* 86.4%

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

   4. div-inv 86.4%

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

   5. metadata-eval 86.4%

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

4. Applied egg-rr 86.4%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. +-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b}}{a - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.02e-135)
   (/ (/ (/ (* PI 0.5) a) b) (- a b))
   (* (/ PI b) (/ (/ 0.5 a) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.02e-135) {
		tmp = (((((double) M_PI) * 0.5) / a) / b) / (a - b);
	} else {
		tmp = (((double) M_PI) / b) * ((0.5 / a) / (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.02e-135) {
		tmp = (((Math.PI * 0.5) / a) / b) / (a - b);
	} else {
		tmp = (Math.PI / b) * ((0.5 / a) / (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.02e-135:
		tmp = (((math.pi * 0.5) / a) / b) / (a - b)
	else:
		tmp = (math.pi / b) * ((0.5 / a) / (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.02e-135)
		tmp = Float64(Float64(Float64(Float64(pi * 0.5) / a) / b) / Float64(a - b));
	else
		tmp = Float64(Float64(pi / b) * Float64(Float64(0.5 / a) / Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.02e-135)
		tmp = (((pi * 0.5) / a) / b) / (a - b);
	else
		tmp = (pi / b) * ((0.5 / a) / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.02e-135], N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.01999999999999994e-135

   1. Initial program 73.8%

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

      1. un-div-inv 73.8%

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

      2. difference-of-squares 84.0%

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

      3. associate-/r* 84.8%

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

      4. div-inv 84.8%

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

      5. metadata-eval 84.8%

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

   4. Applied egg-rr 84.8%

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

   5. Taylor expanded in b around 0 58.9%

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

   6. Taylor expanded in a around inf 61.0%

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

      1. associate-*l/ 71.2%

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

      2. associate-/l* 71.2%

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

   8. Applied egg-rr 71.2%

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

      1. associate-*r/ 71.2%

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

      2. *-commutative 71.2%

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

      3. associate-*r/ 71.2%

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

      4. *-commutative 71.2%

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

      5. associate-*r/ 71.2%

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

      6. mul-1-neg 71.2%

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

      7. distribute-rgt-neg-out 71.2%

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

      8. mul-1-neg 71.2%

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

      9. associate-*r/ 71.2%

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

      10. *-commutative 71.2%

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

      11. metadata-eval 71.2%

          \[\leadsto \frac{\frac{0.5 \cdot \frac{\pi \cdot \color{blue}{\left(-2 - -1\right)}}{a}}{b}}{b - a} \]

      12. distribute-rgt-out-- 71.2%

          \[\leadsto \frac{\frac{0.5 \cdot \frac{\color{blue}{-2 \cdot \pi - -1 \cdot \pi}}{a}}{b}}{b - a} \]

      13. associate-*r/ 71.2%

          \[\leadsto \frac{\frac{\color{blue}{\frac{0.5 \cdot \left(-2 \cdot \pi - -1 \cdot \pi\right)}{a}}}{b}}{b - a} \]

      14. distribute-rgt-out-- 71.2%

          \[\leadsto \frac{\frac{\frac{0.5 \cdot \color{blue}{\left(\pi \cdot \left(-2 - -1\right)\right)}}{a}}{b}}{b - a} \]

      15. metadata-eval 71.2%

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

      16. *-commutative 71.2%

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

      17. mul-1-neg 71.2%

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

   10. Simplified 71.2%

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

   if 1.01999999999999994e-135 < b

   1. Initial program 81.5%

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

      1. associate-*l* 81.4%

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

   3. Simplified 81.4%

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

      1. associate-*l/ 81.4%

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

      2. sub-neg 81.4%

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

      3. *-un-lft-identity 81.4%

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

      4. neg-mul-1 81.4%

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

      5. div-inv 81.4%

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

      6. difference-of-squares 87.0%

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

      7. associate-/r* 99.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 86.2%

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

      10. frac-times 86.2%

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

      11. metadata-eval 86.2%

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

      12. metadata-eval 86.2%

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

      13. frac-times 86.2%

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

      14. sqrt-unprod 86.2%

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

      15. add-sqr-sqrt 86.2%

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

   6. Applied egg-rr 86.2%

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

   7. Taylor expanded in a around 0 86.1%

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

      1. div-inv 86.1%

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

      2. metadata-eval 86.1%

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

      3. associate-*r/ 86.1%

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

   9. Applied egg-rr 86.1%

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

       1. associate-*r/ 86.2%

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

       2. *-commutative 86.2%

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

       3. *-rgt-identity 86.2%

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

       4. *-commutative 86.2%

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

       5. *-commutative 86.2%

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

       6. times-frac 86.1%

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

   11. Simplified 86.1%

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

       1. associate-/l* 86.2%

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

   13. Applied egg-rr 86.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\frac{\pi \cdot 0.5}{a}}{b}}{a - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{0.5}{a}}{a - b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.5e-137)
   (/ (* PI (/ (/ 0.5 a) (- a b))) b)
   (* (/ PI b) (/ (/ 0.5 a) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.5e-137) {
		tmp = (((double) M_PI) * ((0.5 / a) / (a - b))) / b;
	} else {
		tmp = (((double) M_PI) / b) * ((0.5 / a) / (b - a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.5e-137) {
		tmp = (Math.PI * ((0.5 / a) / (a - b))) / b;
	} else {
		tmp = (Math.PI / b) * ((0.5 / a) / (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.5e-137:
		tmp = (math.pi * ((0.5 / a) / (a - b))) / b
	else:
		tmp = (math.pi / b) * ((0.5 / a) / (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.5e-137)
		tmp = Float64(Float64(pi * Float64(Float64(0.5 / a) / Float64(a - b))) / b);
	else
		tmp = Float64(Float64(pi / b) * Float64(Float64(0.5 / a) / Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.5e-137)
		tmp = (pi * ((0.5 / a) / (a - b))) / b;
	else
		tmp = (pi / b) * ((0.5 / a) / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.5e-137], N[(N[(Pi * N[(N[(0.5 / a), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5000000000000007e-137

   1. Initial program 73.8%

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

      1. un-div-inv 73.8%

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

      2. difference-of-squares 84.0%

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

      3. associate-/r* 84.8%

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

      4. div-inv 84.8%

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

      5. metadata-eval 84.8%

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

   4. Applied egg-rr 84.8%

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

   5. Taylor expanded in b around 0 58.9%

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

   6. Taylor expanded in a around inf 61.0%

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

      1. associate-*r/ 61.0%

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

      2. associate-/l/ 60.7%

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

   8. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. mul-1-neg 60.7%

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

      3. associate-/l* 60.7%

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

      4. associate-/l/ 61.0%

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

   10. Simplified 61.0%

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

   if 9.5000000000000007e-137 < b

   1. Initial program 81.5%

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

      1. associate-*l* 81.4%

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

   3. Simplified 81.4%

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

      1. associate-*l/ 81.4%

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

      2. sub-neg 81.4%

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

      3. *-un-lft-identity 81.4%

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

      4. neg-mul-1 81.4%

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

      5. div-inv 81.4%

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

      6. difference-of-squares 87.0%

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

      7. associate-/r* 99.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 86.2%

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

      10. frac-times 86.2%

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

      11. metadata-eval 86.2%

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

      12. metadata-eval 86.2%

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

      13. frac-times 86.2%

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

      14. sqrt-unprod 86.2%

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

      15. add-sqr-sqrt 86.2%

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

   6. Applied egg-rr 86.2%

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

   7. Taylor expanded in a around 0 86.1%

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

      1. div-inv 86.1%

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

      2. metadata-eval 86.1%

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

      3. associate-*r/ 86.1%

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

   9. Applied egg-rr 86.1%

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

       1. associate-*r/ 86.2%

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

       2. *-commutative 86.2%

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

       3. *-rgt-identity 86.2%

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

       4. *-commutative 86.2%

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

       5. *-commutative 86.2%

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

       6. times-frac 86.1%

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

   11. Simplified 86.1%

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

       1. associate-/l* 86.2%

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

   13. Applied egg-rr 86.2%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{\pi \cdot \frac{\frac{0.5}{a}}{a - b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{\frac{0.5}{a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 76.5%

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

   1. un-div-inv 76.5%

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

   2. difference-of-squares 85.1%

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

   3. associate-/r* 86.4%

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

   4. div-inv 86.4%

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

   5. metadata-eval 86.4%

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

4. Applied egg-rr 86.4%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. +-commutative 99.6%

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

   3. sub-neg 99.6%

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

   4. distribute-neg-frac 99.6%

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

   5. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around 0 99.6%

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

Alternative 6: 66.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. associate-*l* 76.4%

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

3. Simplified 76.4%

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

   1. associate-*l/ 76.4%

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

   2. sub-neg 76.4%

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

   3. *-un-lft-identity 76.4%

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

   4. neg-mul-1 76.4%

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

   5. div-inv 76.4%

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

   6. difference-of-squares 85.0%

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

   7. associate-/r* 99.6%

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

   8. add-sqr-sqrt 52.1%

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

   9. sqrt-unprod 77.8%

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

   10. frac-times 77.7%

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

   11. metadata-eval 77.7%

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

   12. metadata-eval 77.7%

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

   13. frac-times 77.8%

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

   14. sqrt-unprod 33.3%

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

   15. add-sqr-sqrt 69.3%

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

6. Applied egg-rr 69.3%

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

7. Taylor expanded in a around 0 69.3%

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

   1. div-inv 69.3%

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

   2. metadata-eval 69.3%

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

   3. associate-*r/ 69.3%

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

9. Applied egg-rr 69.3%

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

    1. associate-*r/ 69.3%

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

    2. *-commutative 69.3%

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

    3. *-rgt-identity 69.3%

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

    4. *-commutative 69.3%

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

    5. *-commutative 69.3%

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

    6. times-frac 69.3%

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

11. Simplified 69.3%

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

    1. associate-/l* 69.4%

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

13. Applied egg-rr 69.4%

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

Alternative 7: 31.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. un-div-inv 76.5%

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

   2. difference-of-squares 85.1%

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

   3. associate-/r* 86.4%

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

   4. div-inv 86.4%

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

   5. metadata-eval 86.4%

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

4. Applied egg-rr 86.4%

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

5. Taylor expanded in b around 0 50.7%

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

6. Taylor expanded in a around inf 54.1%

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

7. Taylor expanded in a around 0 27.6%

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

8. Final simplification 27.6%

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

Reproduce

herbie shell --seed 2024163 
(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))))