NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.6%

Time: 8.7s

Alternatives: 7

Speedup: 1.1×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. times-frac 78.2%

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

   2. *-commutative 78.2%

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

   3. times-frac 78.2%

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

   4. difference-of-squares 88.7%

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

   5. associate-/r* 88.9%

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

   6. metadata-eval 88.9%

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

   7. sub-neg 88.9%

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

   8. distribute-neg-frac 88.9%

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

   9. metadata-eval 88.9%

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

3. Simplified 88.9%

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

   1. distribute-lft-in 81.1%

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

   2. associate-/l/ 80.9%

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

   3. associate-/l/ 80.9%

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

5. Applied egg-rr 80.9%

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

   1. distribute-lft-out 88.7%

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

   2. associate-*r* 88.7%

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

   3. associate-*l/ 88.7%

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

   4. *-commutative 88.7%

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

   5. difference-of-squares 78.2%

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

   6. associate-*l/ 78.2%

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

   7. distribute-lft-in 78.2%

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

   8. associate-*r/ 78.2%

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

   9. metadata-eval 78.2%

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

   10. associate-*r/ 78.2%

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

   11. metadata-eval 78.2%

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

7. Simplified 78.2%

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

   1. distribute-lft-in 73.1%

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

9. Applied egg-rr 73.1%

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

    1. distribute-lft-in 78.2%

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

    2. associate-*l/ 78.2%

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

    3. difference-of-squares 88.7%

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

    4. times-frac 99.7%

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

    5. +-commutative 99.7%

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

    6. metadata-eval 99.7%

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

    7. distribute-neg-frac 99.7%

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

    8. metadata-eval 99.7%

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

    9. associate-*r/ 99.7%

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

    10. sub-neg 99.7%

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

    11. associate-*r/ 99.7%

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

    12. metadata-eval 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-44} \lor \neg \left(a \leq 6.2 \cdot 10^{-23}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.35e-44) (not (<= a 6.2e-23)))
   (* 0.5 (/ (/ PI b) (* a a)))
   (* (/ 0.5 (* a b)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.35e-44) || !(a <= 6.2e-23)) {
		tmp = 0.5 * ((((double) M_PI) / b) / (a * a));
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.35e-44) || !(a <= 6.2e-23)) {
		tmp = 0.5 * ((Math.PI / b) / (a * a));
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.35e-44) or not (a <= 6.2e-23):
		tmp = 0.5 * ((math.pi / b) / (a * a))
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.35e-44) || !(a <= 6.2e-23))
		tmp = Float64(0.5 * Float64(Float64(pi / b) / Float64(a * a)));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.35e-44) || ~((a <= 6.2e-23)))
		tmp = 0.5 * ((pi / b) / (a * a));
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.35e-44], N[Not[LessEqual[a, 6.2e-23]], $MachinePrecision]], N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35e-44 or 6.1999999999999998e-23 < a

   1. Initial program 75.8%

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

      1. times-frac 75.8%

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

      2. *-commutative 75.8%

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

      3. times-frac 75.8%

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

      4. difference-of-squares 90.0%

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

      5. associate-/r* 90.0%

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

      6. metadata-eval 90.0%

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

      7. sub-neg 90.0%

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

      8. distribute-neg-frac 90.0%

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

      9. metadata-eval 90.0%

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

   3. Simplified 90.0%

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

      1. frac-add 90.0%

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

      2. *-un-lft-identity 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. *-commutative 90.0%

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

      2. neg-mul-1 90.0%

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

      3. sub-neg 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in b around 0 80.3%

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

      1. unpow2 80.3%

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

      2. *-commutative 80.3%

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

      3. associate-/r* 79.7%

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

   10. Simplified 79.7%

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

   if -1.35e-44 < a < 6.1999999999999998e-23

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-/r/ 80.5%

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

      3. associate-*l/ 80.4%

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

      4. *-commutative 80.4%

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

      5. associate-/r/ 80.4%

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

      6. times-frac 80.4%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in b around inf 76.4%

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

      1. unpow2 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in a around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. unpow2 76.4%

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

      3. associate-*l* 87.9%

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

   9. Simplified 87.9%

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

       1. expm1-log1p-u 61.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\right)\right)} \]

       2. expm1-udef 41.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\right)} - 1} \]

       3. *-commutative 41.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}}\right)} - 1 \]

       4. *-commutative 41.8%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \color{blue}{\left(a \cdot b\right)}}\right)} - 1 \]

   11. Applied egg-rr 41.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 61.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\right)\right)} \]

       2. expm1-log1p 87.9%

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

       3. *-commutative 87.9%

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

       4. associate-*l/ 87.9%

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

       5. times-frac 88.6%

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

       6. *-commutative 88.6%

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

   13. Simplified 88.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-44} \lor \neg \left(a \leq 6.2 \cdot 10^{-23}\right):\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 3: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-44} \lor \neg \left(a \leq 2.7 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.35e-44) (not (<= a 2.7e-23)))
   (/ (* PI 0.5) (* b (* a a)))
   (* (/ 0.5 (* a b)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.35e-44) || !(a <= 2.7e-23)) {
		tmp = (((double) M_PI) * 0.5) / (b * (a * a));
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.35e-44) || !(a <= 2.7e-23)) {
		tmp = (Math.PI * 0.5) / (b * (a * a));
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.35e-44) or not (a <= 2.7e-23):
		tmp = (math.pi * 0.5) / (b * (a * a))
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.35e-44) || !(a <= 2.7e-23))
		tmp = Float64(Float64(pi * 0.5) / Float64(b * Float64(a * a)));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.35e-44) || ~((a <= 2.7e-23)))
		tmp = (pi * 0.5) / (b * (a * a));
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.35e-44], N[Not[LessEqual[a, 2.7e-23]], $MachinePrecision]], N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35e-44 or 2.69999999999999985e-23 < a

   1. Initial program 75.8%

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

   2. Taylor expanded in b around 0 80.3%

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

      1. associate-*r/ 80.3%

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

      2. unpow2 80.3%

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

   4. Simplified 80.3%

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

   if -1.35e-44 < a < 2.69999999999999985e-23

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-/r/ 80.5%

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

      3. associate-*l/ 80.4%

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

      4. *-commutative 80.4%

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

      5. associate-/r/ 80.4%

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

      6. times-frac 80.4%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in b around inf 76.4%

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

      1. unpow2 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in a around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. unpow2 76.4%

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

      3. associate-*l* 87.9%

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

   9. Simplified 87.9%

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

       1. expm1-log1p-u 61.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\right)\right)} \]

       2. expm1-udef 41.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\right)} - 1} \]

       3. *-commutative 41.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}}\right)} - 1 \]

       4. *-commutative 41.8%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \color{blue}{\left(a \cdot b\right)}}\right)} - 1 \]

   11. Applied egg-rr 41.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 61.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\right)\right)} \]

       2. expm1-log1p 87.9%

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

       3. *-commutative 87.9%

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

       4. associate-*l/ 87.9%

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

       5. times-frac 88.6%

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

       6. *-commutative 88.6%

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

   13. Simplified 88.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-44} \lor \neg \left(a \leq 2.7 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\pi \cdot 0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 4: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-41} \lor \neg \left(a \leq 4.2 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.4e-41) (not (<= a 4.2e-23)))
   (/ (* 0.5 (/ PI a)) (* a b))
   (* (/ 0.5 (* a b)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.4e-41) || !(a <= 4.2e-23)) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.4e-41) || !(a <= 4.2e-23)) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = (0.5 / (a * b)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.4e-41) or not (a <= 4.2e-23):
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = (0.5 / (a * b)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.4e-41) || !(a <= 4.2e-23))
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.4e-41) || ~((a <= 4.2e-23)))
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = (0.5 / (a * b)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.4e-41], N[Not[LessEqual[a, 4.2e-23]], $MachinePrecision]], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.40000000000000022e-41 or 4.2000000000000002e-23 < a

   1. Initial program 75.8%

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

      1. times-frac 75.8%

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

      2. *-commutative 75.8%

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

      3. times-frac 75.8%

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

      4. difference-of-squares 90.0%

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

      5. associate-/r* 90.0%

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

      6. metadata-eval 90.0%

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

      7. sub-neg 90.0%

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

      8. distribute-neg-frac 90.0%

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

      9. metadata-eval 90.0%

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

   3. Simplified 90.0%

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

      1. frac-add 90.0%

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

      2. *-un-lft-identity 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. *-commutative 90.0%

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

      2. neg-mul-1 90.0%

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

      3. sub-neg 90.0%

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

   7. Simplified 90.0%

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

      1. associate-*r/ 90.1%

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

      2. *-commutative 90.1%

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

   9. Applied egg-rr 90.1%

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

   10. Taylor expanded in b around 0 89.4%

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

   if -2.40000000000000022e-41 < a < 4.2000000000000002e-23

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-/r/ 80.5%

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

      3. associate-*l/ 80.4%

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

      4. *-commutative 80.4%

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

      5. associate-/r/ 80.4%

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

      6. times-frac 80.4%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in b around inf 76.4%

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

      1. unpow2 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in a around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. unpow2 76.4%

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

      3. associate-*l* 87.9%

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

   9. Simplified 87.9%

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

       1. expm1-log1p-u 61.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\right)\right)} \]

       2. expm1-udef 41.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\pi}{b \cdot \left(b \cdot a\right)} \cdot 0.5\right)} - 1} \]

       3. *-commutative 41.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}}\right)} - 1 \]

       4. *-commutative 41.8%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \color{blue}{\left(a \cdot b\right)}}\right)} - 1 \]

   11. Applied egg-rr 41.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 61.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\right)\right)} \]

       2. expm1-log1p 87.9%

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

       3. *-commutative 87.9%

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

       4. associate-*l/ 87.9%

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

       5. times-frac 88.6%

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

       6. *-commutative 88.6%

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

   13. Simplified 88.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-41} \lor \neg \left(a \leq 4.2 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 5: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-43} \lor \neg \left(a \leq 3.3 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5.8e-43) (not (<= a 3.3e-23)))
   (/ (* 0.5 (/ PI a)) (* a b))
   (/ (/ (* PI 0.5) b) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -5.8e-43) || !(a <= 3.3e-23)) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((((double) M_PI) * 0.5) / b) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -5.8e-43) || !(a <= 3.3e-23)) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = ((Math.PI * 0.5) / b) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -5.8e-43) or not (a <= 3.3e-23):
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = ((math.pi * 0.5) / b) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -5.8e-43) || !(a <= 3.3e-23))
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / b) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -5.8e-43) || ~((a <= 3.3e-23)))
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = ((pi * 0.5) / b) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -5.8e-43], N[Not[LessEqual[a, 3.3e-23]], $MachinePrecision]], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8000000000000003e-43 or 3.30000000000000021e-23 < a

   1. Initial program 75.8%

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

      1. times-frac 75.8%

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

      2. *-commutative 75.8%

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

      3. times-frac 75.8%

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

      4. difference-of-squares 90.0%

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

      5. associate-/r* 90.0%

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

      6. metadata-eval 90.0%

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

      7. sub-neg 90.0%

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

      8. distribute-neg-frac 90.0%

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

      9. metadata-eval 90.0%

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

   3. Simplified 90.0%

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

      1. frac-add 90.0%

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

      2. *-un-lft-identity 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. *-commutative 90.0%

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

      2. neg-mul-1 90.0%

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

      3. sub-neg 90.0%

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

   7. Simplified 90.0%

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

      1. associate-*r/ 90.1%

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

      2. *-commutative 90.1%

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

   9. Applied egg-rr 90.1%

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

   10. Taylor expanded in b around 0 89.4%

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

   if -5.8000000000000003e-43 < a < 3.30000000000000021e-23

   1. Initial program 80.4%

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

      1. times-frac 80.5%

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

      2. *-commutative 80.5%

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

      3. times-frac 80.5%

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

      4. difference-of-squares 87.5%

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

      5. associate-/r* 87.8%

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

      6. metadata-eval 87.8%

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

      7. sub-neg 87.8%

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

      8. distribute-neg-frac 87.8%

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

      9. metadata-eval 87.8%

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

   3. Simplified 87.8%

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

      1. frac-add 87.8%

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

      2. *-un-lft-identity 87.8%

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

   5. Applied egg-rr 87.8%

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

      1. *-commutative 87.8%

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

      2. neg-mul-1 87.8%

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

      3. sub-neg 87.8%

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

   7. Simplified 87.8%

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

      1. associate-*r/ 87.8%

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

      2. *-commutative 87.8%

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

   9. Applied egg-rr 87.8%

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

   10. Taylor expanded in b around inf 88.6%

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

       1. associate-*r/ 88.6%

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

   12. Simplified 88.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-43} \lor \neg \left(a \leq 3.3 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b}}{a \cdot b}\\ \end{array} \]

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. times-frac 78.2%

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

   2. *-commutative 78.2%

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

   3. times-frac 78.2%

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

   4. difference-of-squares 88.7%

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

   5. associate-/r* 88.9%

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

   6. metadata-eval 88.9%

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

   7. sub-neg 88.9%

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

   8. distribute-neg-frac 88.9%

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

   9. metadata-eval 88.9%

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

3. Simplified 88.9%

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

   1. distribute-lft-in 81.1%

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

   2. associate-/l/ 80.9%

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

   3. associate-/l/ 80.9%

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

5. Applied egg-rr 80.9%

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

   1. distribute-lft-out 88.7%

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

   2. associate-*r* 88.7%

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

   3. associate-*l/ 88.7%

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

   4. *-commutative 88.7%

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

   5. difference-of-squares 78.2%

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

   6. associate-*l/ 78.2%

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

   7. distribute-lft-in 78.2%

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

   8. associate-*r/ 78.2%

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

   9. metadata-eval 78.2%

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

   10. associate-*r/ 78.2%

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

   11. metadata-eval 78.2%

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

7. Simplified 78.2%

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

   1. distribute-lft-in 73.1%

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

9. Applied egg-rr 73.1%

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

    1. distribute-lft-in 78.2%

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

    2. associate-*l/ 78.2%

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

    3. difference-of-squares 88.7%

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

    4. times-frac 99.7%

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

    5. +-commutative 99.7%

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

    6. metadata-eval 99.7%

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

    7. distribute-neg-frac 99.7%

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

    8. metadata-eval 99.7%

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

    9. associate-*r/ 99.7%

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

    10. sub-neg 99.7%

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

    11. associate-*r/ 99.7%

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

    12. metadata-eval 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in a around 0 99.6%

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

    1. *-commutative 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 7: 56.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. times-frac 78.2%

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

   2. *-commutative 78.2%

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

   3. times-frac 78.2%

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

   4. difference-of-squares 88.7%

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

   5. associate-/r* 88.9%

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

   6. metadata-eval 88.9%

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

   7. sub-neg 88.9%

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

   8. distribute-neg-frac 88.9%

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

   9. metadata-eval 88.9%

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

3. Simplified 88.9%

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

   1. frac-add 88.9%

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

   2. *-un-lft-identity 88.9%

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

5. Applied egg-rr 88.9%

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

   1. *-commutative 88.9%

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

   2. neg-mul-1 88.9%

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

   3. sub-neg 88.9%

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

7. Simplified 88.9%

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

8. Taylor expanded in b around 0 55.5%

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

   1. unpow2 55.5%

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

   2. *-commutative 55.5%

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

   3. associate-/r* 55.2%

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

10. Simplified 55.2%

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

11. Final simplification 55.2%

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

Reproduce

herbie shell --seed 2023192 
(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))))