NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.6%

Time: 14.4s

Alternatives: 6

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. associate-*l* 78.5%

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

   2. *-rgt-identity 78.5%

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

   3. associate-/l* 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.4%

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

   6. *-lft-identity 78.4%

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

   7. sub-neg 78.4%

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

   8. distribute-neg-frac 78.4%

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

   9. metadata-eval 78.4%

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

3. Simplified 78.4%

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

   1. metadata-eval 78.4%

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

   2. div-inv 78.4%

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

   3. associate-*r/ 78.5%

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

   4. *-commutative 78.5%

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

   5. difference-of-squares 84.7%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. add-sqr-sqrt 29.5%

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

   3. sqrt-unprod 76.1%

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

   4. frac-times 76.1%

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

   5. metadata-eval 76.1%

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

   6. metadata-eval 76.1%

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

   7. frac-times 76.1%

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

   8. sqrt-unprod 53.7%

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

   9. add-sqr-sqrt 99.6%

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

   10. div-inv 99.6%

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

   11. fma-define 99.6%

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

8. Applied egg-rr 99.6%

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

   1. fma-undefine 99.6%

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

   2. neg-mul-1 99.6%

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

   3. distribute-neg-frac 99.6%

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

   4. metadata-eval 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. un-div-inv 78.5%

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

   2. difference-of-squares 84.7%

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

   3. associate-/r* 87.2%

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

   4. div-inv 87.2%

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

   5. metadata-eval 87.2%

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

4. Applied egg-rr 87.2%

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

   1. associate-*l/ 99.6%

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

   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. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 75.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a - b} \cdot \frac{\frac{-0.5}{a}}{b + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e-170)
   (/ (/ (* PI -0.5) (* b a)) (- b a))
   (* (/ PI (- a b)) (/ (/ -0.5 a) (+ b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-170) {
		tmp = ((((double) M_PI) * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (((double) M_PI) / (a - b)) * ((-0.5 / a) / (b + a));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e-170) {
		tmp = ((Math.PI * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (Math.PI / (a - b)) * ((-0.5 / a) / (b + a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.15e-170:
		tmp = ((math.pi * -0.5) / (b * a)) / (b - a)
	else:
		tmp = (math.pi / (a - b)) * ((-0.5 / a) / (b + a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.15e-170)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * a)) / Float64(b - a));
	else
		tmp = Float64(Float64(pi / Float64(a - 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 <= 1.15e-170)
		tmp = ((pi * -0.5) / (b * a)) / (b - a);
	else
		tmp = (pi / (a - b)) * ((-0.5 / a) / (b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.15e-170], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 / a), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.14999999999999993e-170

   1. Initial program 74.6%

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

      1. associate-*l* 74.7%

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

      2. *-rgt-identity 74.7%

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

      3. associate-/l* 74.7%

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

      4. metadata-eval 74.7%

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

      5. associate-*l/ 74.6%

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

      6. *-lft-identity 74.6%

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

      7. sub-neg 74.6%

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

      8. distribute-neg-frac 74.6%

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

      9. metadata-eval 74.6%

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

   3. Simplified 74.6%

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

      1. metadata-eval 74.6%

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

      2. div-inv 74.6%

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

      3. associate-*r/ 74.6%

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

      4. *-commutative 74.6%

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

      5. difference-of-squares 84.0%

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

      6. associate-/r* 99.7%

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

   6. Applied egg-rr 58.4%

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

      1. +-commutative 58.4%

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

      2. add-sqr-sqrt 1.4%

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

      3. sqrt-unprod 75.9%

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

      4. frac-times 75.9%

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

      5. metadata-eval 75.9%

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

      6. metadata-eval 75.9%

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

      7. frac-times 75.9%

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

      8. sqrt-unprod 85.9%

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

      9. add-sqr-sqrt 99.7%

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

      10. div-inv 99.7%

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

      11. fma-define 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. fma-undefine 99.7%

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

      2. neg-mul-1 99.7%

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

      3. distribute-neg-frac 99.7%

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

      4. metadata-eval 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in b around 0 68.6%

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

       1. associate-*r/ 68.6%

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

       2. *-commutative 68.6%

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

   13. Simplified 68.6%

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

   if 1.14999999999999993e-170 < b

   1. Initial program 84.9%

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

      1. un-div-inv 84.9%

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

      2. difference-of-squares 85.9%

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

      3. associate-/r* 88.0%

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

      4. div-inv 88.0%

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

      5. metadata-eval 88.0%

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in a around 0 66.5%

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

      1. associate-/l/ 65.1%

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

      2. *-commutative 65.1%

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

      3. *-commutative 65.1%

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

      4. frac-2neg 65.1%

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

      5. metadata-eval 65.1%

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

      6. frac-times 65.1%

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

      7. *-commutative 65.1%

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

      8. +-commutative 65.1%

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

   7. Applied egg-rr 65.1%

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

      1. *-commutative 65.1%

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

      2. *-commutative 65.1%

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

      3. neg-mul-1 65.1%

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

      4. *-commutative 65.1%

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

      5. distribute-rgt-neg-in 65.1%

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

      6. metadata-eval 65.1%

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

      7. distribute-lft-neg-out 65.1%

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

      8. *-commutative 65.1%

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

      9. associate-*l* 76.8%

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

      10. distribute-lft-neg-in 76.8%

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

      11. sub-neg 76.8%

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

      12. distribute-neg-in 76.8%

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

      13. remove-double-neg 76.8%

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

      14. +-commutative 76.8%

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

      15. unsub-neg 76.8%

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

   9. Simplified 76.8%

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

       1. times-frac 76.8%

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

       2. +-commutative 76.8%

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

       3. *-commutative 76.8%

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

   11. Applied egg-rr 76.8%

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

       1. associate-/r* 76.7%

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

   13. Simplified 76.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a - b} \cdot \frac{\frac{-0.5}{a}}{b + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e-170)
   (/ (/ (* PI -0.5) (* b a)) (- b a))
   (/ (* 0.5 (/ PI (* b a))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-170) {
		tmp = ((((double) M_PI) * -0.5) / (b * a)) / (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 <= 1.65e-170) {
		tmp = ((Math.PI * -0.5) / (b * a)) / (b - a);
	} else {
		tmp = (0.5 * (Math.PI / (b * a))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.65e-170:
		tmp = ((math.pi * -0.5) / (b * a)) / (b - a)
	else:
		tmp = (0.5 * (math.pi / (b * a))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e-170)
		tmp = Float64(Float64(Float64(pi * -0.5) / Float64(b * a)) / Float64(b - a));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(b * a))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.65e-170)
		tmp = ((pi * -0.5) / (b * a)) / (b - a);
	else
		tmp = (0.5 * (pi / (b * a))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e-170], N[(N[(N[(Pi * -0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65000000000000002e-170

   1. Initial program 74.6%

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

      1. associate-*l* 74.7%

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

      2. *-rgt-identity 74.7%

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

      3. associate-/l* 74.7%

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

      4. metadata-eval 74.7%

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

      5. associate-*l/ 74.6%

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

      6. *-lft-identity 74.6%

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

      7. sub-neg 74.6%

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

      8. distribute-neg-frac 74.6%

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

      9. metadata-eval 74.6%

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

   3. Simplified 74.6%

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

      1. metadata-eval 74.6%

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

      2. div-inv 74.6%

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

      3. associate-*r/ 74.6%

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

      4. *-commutative 74.6%

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

      5. difference-of-squares 84.0%

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

      6. associate-/r* 99.7%

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

   6. Applied egg-rr 58.4%

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

      1. +-commutative 58.4%

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

      2. add-sqr-sqrt 1.4%

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

      3. sqrt-unprod 75.9%

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

      4. frac-times 75.9%

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

      5. metadata-eval 75.9%

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

      6. metadata-eval 75.9%

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

      7. frac-times 75.9%

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

      8. sqrt-unprod 85.9%

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

      9. add-sqr-sqrt 99.7%

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

      10. div-inv 99.7%

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

      11. fma-define 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. fma-undefine 99.7%

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

      2. neg-mul-1 99.7%

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

      3. distribute-neg-frac 99.7%

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

      4. metadata-eval 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in b around 0 68.6%

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

       1. associate-*r/ 68.6%

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

       2. *-commutative 68.6%

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

   13. Simplified 68.6%

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

   if 1.65000000000000002e-170 < b

   1. Initial program 84.9%

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

      1. associate-*l* 85.0%

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

      2. *-rgt-identity 85.0%

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

      3. associate-/l* 85.0%

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

      4. metadata-eval 85.0%

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

      5. associate-*l/ 84.8%

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

      6. *-lft-identity 84.8%

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

      7. sub-neg 84.8%

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

      8. distribute-neg-frac 84.8%

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

      9. metadata-eval 84.8%

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

   3. Simplified 84.8%

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

      1. metadata-eval 84.8%

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

      2. div-inv 84.8%

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

      3. associate-*r/ 84.8%

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

      4. *-commutative 84.8%

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

      5. difference-of-squares 85.9%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in a around 0 76.4%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{\pi \cdot -0.5}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{b \cdot a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. un-div-inv 78.5%

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

   2. difference-of-squares 84.7%

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

   3. associate-/r* 87.2%

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

   4. div-inv 87.2%

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

   5. metadata-eval 87.2%

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

4. Applied egg-rr 87.2%

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

   1. associate-*l/ 99.6%

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

   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. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around 0 99.6%

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

10. Final simplification 99.6%

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

Alternative 6: 66.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. associate-*l* 78.5%

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

   2. *-rgt-identity 78.5%

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

   3. associate-/l* 78.5%

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

   4. metadata-eval 78.5%

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

   5. associate-*l/ 78.4%

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

   6. *-lft-identity 78.4%

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

   7. sub-neg 78.4%

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

   8. distribute-neg-frac 78.4%

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

   9. metadata-eval 78.4%

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

3. Simplified 78.4%

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

   1. metadata-eval 78.4%

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

   2. div-inv 78.4%

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

   3. associate-*r/ 78.5%

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

   4. *-commutative 78.5%

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

   5. difference-of-squares 84.7%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 65.2%

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

7. Taylor expanded in a around 0 65.2%

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

8. Final simplification 65.2%

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

Reproduce

herbie shell --seed 2024044 
(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))))