NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.6%

Time: 10.7s

Alternatives: 12

Speedup: 1.9×

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: 78.1% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. associate-*r/ 73.3%

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

   2. *-rgt-identity 73.3%

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

   3. difference-of-squares 87.4%

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

   4. associate-/r* 88.0%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-/l/ 99.7%

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

3. Simplified 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. associate-*l/ 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-*l/ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. mul-1-neg 99.7%

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

   7. unsub-neg 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

    \[\leadsto \frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2 \cdot \left(a + b\right)}}{b - a} \]
11. Add Preprocessing

Alternative 2: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e-141)
   (* (/ PI (* a b)) (/ -0.5 (- b a)))
   (* (/ 1.0 (* a b)) (* PI (/ 0.5 b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-141) {
		tmp = (((double) M_PI) / (a * b)) * (-0.5 / (b - a));
	} else {
		tmp = (1.0 / (a * b)) * (((double) M_PI) * (0.5 / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-141) {
		tmp = (Math.PI / (a * b)) * (-0.5 / (b - a));
	} else {
		tmp = (1.0 / (a * b)) * (Math.PI * (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.5e-141:
		tmp = (math.pi / (a * b)) * (-0.5 / (b - a))
	else:
		tmp = (1.0 / (a * b)) * (math.pi * (0.5 / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e-141)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(-0.5 / Float64(b - a)));
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(pi * Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e-141)
		tmp = (pi / (a * b)) * (-0.5 / (b - a));
	else
		tmp = (1.0 / (a * b)) * (pi * (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e-141], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.50000000000000021e-141

   1. Initial program 82.1%

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

      1. associate-*r/ 82.2%

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

      2. *-rgt-identity 82.2%

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

      3. difference-of-squares 92.5%

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

      4. associate-/r* 93.2%

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

      5. associate-*l/ 99.6%

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

      6. *-commutative 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. associate-/l/ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-/r* 81.2%

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

   7. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. associate-/l* 81.2%

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

      3. associate-/l/ 81.2%

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

   9. Applied egg-rr 81.2%

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

       1. *-lft-identity 81.2%

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

       2. associate-/r/ 81.1%

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

   11. Simplified 81.1%

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

   if -8.50000000000000021e-141 < a

   1. Initial program 69.4%

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

      1. associate-*r/ 69.4%

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

      2. *-rgt-identity 69.4%

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

      3. difference-of-squares 85.1%

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

      4. associate-/r* 85.7%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l* 85.2%

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

      3. *-commutative 85.2%

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

   6. Applied egg-rr 85.2%

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

      1. *-lft-identity 85.2%

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

      2. associate-/r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-*r/ 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 99.7%

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

   10. Taylor expanded in a around 0 77.5%

       \[\leadsto \frac{1}{a \cdot b} \cdot \left(\pi \cdot \color{blue}{\frac{0.5}{b}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \frac{\pi \cdot 0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e-141)
   (* (/ PI (* a b)) (/ -0.5 (- b a)))
   (* (/ 1.0 (* a b)) (/ (* PI 0.5) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-141) {
		tmp = (((double) M_PI) / (a * b)) * (-0.5 / (b - a));
	} else {
		tmp = (1.0 / (a * b)) * ((((double) M_PI) * 0.5) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-141) {
		tmp = (Math.PI / (a * b)) * (-0.5 / (b - a));
	} else {
		tmp = (1.0 / (a * b)) * ((Math.PI * 0.5) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.5e-141:
		tmp = (math.pi / (a * b)) * (-0.5 / (b - a))
	else:
		tmp = (1.0 / (a * b)) * ((math.pi * 0.5) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e-141)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(-0.5 / Float64(b - a)));
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(Float64(pi * 0.5) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.5e-141)
		tmp = (pi / (a * b)) * (-0.5 / (b - a));
	else
		tmp = (1.0 / (a * b)) * ((pi * 0.5) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e-141], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.50000000000000021e-141

   1. Initial program 82.1%

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

      1. associate-*r/ 82.2%

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

      2. *-rgt-identity 82.2%

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

      3. difference-of-squares 92.5%

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

      4. associate-/r* 93.2%

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

      5. associate-*l/ 99.6%

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

      6. *-commutative 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. associate-/l/ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-/r* 81.2%

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

   7. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. associate-/l* 81.2%

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

      3. associate-/l/ 81.2%

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

   9. Applied egg-rr 81.2%

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

       1. *-lft-identity 81.2%

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

       2. associate-/r/ 81.1%

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

   11. Simplified 81.1%

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

   if -8.50000000000000021e-141 < a

   1. Initial program 69.4%

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

      1. associate-*r/ 69.4%

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

      2. *-rgt-identity 69.4%

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

      3. difference-of-squares 85.1%

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

      4. associate-/r* 85.7%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l* 85.2%

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

      3. *-commutative 85.2%

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

   6. Applied egg-rr 85.2%

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

      1. *-lft-identity 85.2%

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

      2. associate-/r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-*r/ 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 99.7%

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

   10. Taylor expanded in a around 0 77.6%

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

       1. associate-*r/ 77.6%

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

   12. Simplified 77.6%

       \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\frac{0.5 \cdot \pi}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \frac{\pi \cdot 0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \frac{\pi \cdot 0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2e+88)
   (* (/ PI a) (/ 0.5 (* b (+ a b))))
   (* (/ 1.0 (* a b)) (/ (* PI 0.5) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2e+88) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * (a + b)));
	} else {
		tmp = (1.0 / (a * b)) * ((((double) M_PI) * 0.5) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2e+88) {
		tmp = (Math.PI / a) * (0.5 / (b * (a + b)));
	} else {
		tmp = (1.0 / (a * b)) * ((Math.PI * 0.5) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2e+88:
		tmp = (math.pi / a) * (0.5 / (b * (a + b)))
	else:
		tmp = (1.0 / (a * b)) * ((math.pi * 0.5) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2e+88)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(Float64(pi * 0.5) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2e+88)
		tmp = (pi / a) * (0.5 / (b * (a + b)));
	else
		tmp = (1.0 / (a * b)) * ((pi * 0.5) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2e+88], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.99999999999999992e88

   1. Initial program 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-rgt-identity 75.6%

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

      3. difference-of-squares 88.7%

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

      4. associate-/r* 89.2%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l* 88.8%

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

      3. *-commutative 88.8%

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

   6. Applied egg-rr 88.8%

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

      1. *-lft-identity 88.8%

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

      2. associate-/r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-*r/ 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 99.6%

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

       1. associate-*l/ 99.6%

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

       2. *-un-lft-identity 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. times-frac 93.6%

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

       2. associate-/l/ 93.4%

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

   13. Simplified 93.4%

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

   if 1.99999999999999992e88 < b

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. *-rgt-identity 63.7%

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

      3. difference-of-squares 81.7%

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

      4. associate-/r* 82.8%

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

      5. associate-*l/ 99.6%

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

      6. *-commutative 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. associate-/l/ 99.6%

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

   3. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-/l* 81.6%

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

      3. *-commutative 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. *-lft-identity 81.6%

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

      2. associate-/r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-*r/ 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 99.8%

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

   10. Taylor expanded in a around 0 99.8%

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

       1. associate-*r/ 99.8%

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

   12. Simplified 99.8%

       \[\leadsto \frac{1}{a \cdot b} \cdot \color{blue}{\frac{0.5 \cdot \pi}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \frac{\pi \cdot 0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{+126}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e+126)
   (* (/ PI a) (/ 0.5 (* b (+ a b))))
   (/ (* 0.5 (/ PI (* a b))) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e+126) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * (a + b)));
	} else {
		tmp = (0.5 * (((double) M_PI) / (a * b))) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1e+126) {
		tmp = (Math.PI / a) * (0.5 / (b * (a + b)));
	} else {
		tmp = (0.5 * (Math.PI / (a * b))) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1e+126:
		tmp = (math.pi / a) * (0.5 / (b * (a + b)))
	else:
		tmp = (0.5 * (math.pi / (a * b))) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e+126)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(a * b))) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1e+126)
		tmp = (pi / a) * (0.5 / (b * (a + b)));
	else
		tmp = (0.5 * (pi / (a * b))) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e+126], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999925e125

   1. Initial program 77.0%

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

      1. associate-*r/ 77.1%

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

      2. *-rgt-identity 77.1%

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

      3. difference-of-squares 89.4%

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

      4. associate-/r* 89.9%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l* 89.5%

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

      3. *-commutative 89.5%

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

   6. Applied egg-rr 89.5%

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

      1. *-lft-identity 89.5%

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

      2. associate-/r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-*r/ 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 99.6%

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

       1. associate-*l/ 99.6%

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

       2. *-un-lft-identity 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. times-frac 93.9%

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

       2. associate-/l/ 93.8%

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

   13. Simplified 93.8%

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

   if 9.99999999999999925e125 < b

   1. Initial program 51.0%

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

      1. associate-*r/ 51.0%

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

      2. *-rgt-identity 51.0%

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

      3. difference-of-squares 75.3%

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

      4. associate-/r* 76.8%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around 0 99.8%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+126}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.92 \cdot 10^{+93}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{a}}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.92e+93)
   (* (/ PI a) (/ 0.5 (* b (+ a b))))
   (/ (* 0.5 (/ (/ PI a) b)) (- b a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.92e+93) {
		tmp = (((double) M_PI) / a) * (0.5 / (b * (a + b)));
	} else {
		tmp = (0.5 * ((((double) M_PI) / a) / b)) / (b - a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.92e+93) {
		tmp = (Math.PI / a) * (0.5 / (b * (a + b)));
	} else {
		tmp = (0.5 * ((Math.PI / a) / b)) / (b - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.92e+93:
		tmp = (math.pi / a) * (0.5 / (b * (a + b)))
	else:
		tmp = (0.5 * ((math.pi / a) / b)) / (b - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.92e+93)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * Float64(a + b))));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / a) / b)) / Float64(b - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.92e+93)
		tmp = (pi / a) * (0.5 / (b * (a + b)));
	else
		tmp = (0.5 * ((pi / a) / b)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.92e+93], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.92000000000000004e93

   1. Initial program 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-rgt-identity 75.6%

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

      3. difference-of-squares 88.7%

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

      4. associate-/r* 89.2%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. associate-/l* 88.8%

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

      3. *-commutative 88.8%

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

   6. Applied egg-rr 88.8%

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

      1. *-lft-identity 88.8%

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

      2. associate-/r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. associate-*r/ 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around 0 99.6%

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

       1. associate-*l/ 99.6%

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

       2. *-un-lft-identity 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. times-frac 93.6%

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

       2. associate-/l/ 93.4%

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

   13. Simplified 93.4%

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

   if 1.92000000000000004e93 < b

   1. Initial program 63.7%

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

      1. associate-*r/ 63.7%

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

      2. *-rgt-identity 63.7%

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

      3. difference-of-squares 81.7%

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

      4. associate-/r* 82.8%

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

      5. associate-*l/ 99.6%

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

      6. *-commutative 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. associate-/l/ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in a around 0 99.8%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l/ 99.8%

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

      5. associate-*r/ 99.8%

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

      6. mul-1-neg 99.8%

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

      7. unsub-neg 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in a around 0 99.8%

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

       1. associate-/r* 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.92 \cdot 10^{+93}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot \left(a + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\frac{\pi}{a}}{b}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot b}\\ \mathbf{if}\;b \leq 1.16 \cdot 10^{+120}:\\ \;\;\;\;t\_0 \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (* a b))))
   (if (<= b 1.16e+120) (* t_0 (/ 0.5 a)) (* t_0 (/ -0.5 b)))))

C

double code(double a, double b) {
	double t_0 = ((double) M_PI) / (a * b);
	double tmp;
	if (b <= 1.16e+120) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (-0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.PI / (a * b);
	double tmp;
	if (b <= 1.16e+120) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pi / (a * b)
	tmp = 0
	if b <= 1.16e+120:
		tmp = t_0 * (0.5 / a)
	else:
		tmp = t_0 * (-0.5 / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(pi / Float64(a * b))
	tmp = 0.0
	if (b <= 1.16e+120)
		tmp = Float64(t_0 * Float64(0.5 / a));
	else
		tmp = Float64(t_0 * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = pi / (a * b);
	tmp = 0.0;
	if (b <= 1.16e+120)
		tmp = t_0 * (0.5 / a);
	else
		tmp = t_0 * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.16e+120], N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.16000000000000003e120

   1. Initial program 76.8%

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

      1. associate-*r/ 76.9%

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

      2. *-rgt-identity 76.9%

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

      3. difference-of-squares 89.3%

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

      4. associate-/r* 89.8%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-/r* 64.5%

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

   7. Simplified 64.5%

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

      1. *-un-lft-identity 64.5%

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

      2. associate-/l* 64.3%

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

      3. associate-/l/ 64.3%

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

   9. Applied egg-rr 64.3%

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

       1. *-lft-identity 64.3%

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

       2. associate-/r/ 64.5%

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

   11. Simplified 64.5%

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

   12. Taylor expanded in b around 0 62.6%

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

   if 1.16000000000000003e120 < b

   1. Initial program 53.5%

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

      1. associate-*r/ 53.5%

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

      2. *-rgt-identity 53.5%

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

      3. difference-of-squares 76.5%

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

      4. associate-/r* 78.0%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-/r* 63.7%

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

   7. Simplified 63.7%

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

      1. *-un-lft-identity 63.7%

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

      2. associate-/l* 63.7%

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

      3. associate-/l/ 63.7%

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

   9. Applied egg-rr 63.7%

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

       1. *-lft-identity 63.7%

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

       2. associate-/r/ 63.7%

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

   11. Simplified 63.7%

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

   12. Taylor expanded in b around inf 63.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{+120}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{+120}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b \cdot \left(a \cdot \frac{b}{\pi}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.16e+120)
   (* (/ PI (* a b)) (/ 0.5 a))
   (/ -0.5 (* b (* a (/ b PI))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.16e+120) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = -0.5 / (b * (a * (b / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.16e+120) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = -0.5 / (b * (a * (b / Math.PI)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.16e+120:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = -0.5 / (b * (a * (b / math.pi)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.16e+120)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(-0.5 / Float64(b * Float64(a * Float64(b / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.16e+120)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = -0.5 / (b * (a * (b / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.16e+120], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b * N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.16000000000000003e120

   1. Initial program 76.8%

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

      1. associate-*r/ 76.9%

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

      2. *-rgt-identity 76.9%

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

      3. difference-of-squares 89.3%

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

      4. associate-/r* 89.8%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-/r* 64.5%

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

   7. Simplified 64.5%

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

      1. *-un-lft-identity 64.5%

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

      2. associate-/l* 64.3%

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

      3. associate-/l/ 64.3%

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

   9. Applied egg-rr 64.3%

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

       1. *-lft-identity 64.3%

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

       2. associate-/r/ 64.5%

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

   11. Simplified 64.5%

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

   12. Taylor expanded in b around 0 62.6%

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

   if 1.16000000000000003e120 < b

   1. Initial program 53.5%

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

      1. associate-*r/ 53.5%

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

      2. *-rgt-identity 53.5%

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

      3. difference-of-squares 76.5%

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

      4. associate-/r* 78.0%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-/r* 63.7%

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

   7. Simplified 63.7%

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

      1. *-un-lft-identity 63.7%

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

      2. associate-/l* 63.7%

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

      3. associate-/l/ 63.7%

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

   9. Applied egg-rr 63.7%

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

       1. *-lft-identity 63.7%

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

       2. associate-/r/ 63.7%

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

   11. Simplified 63.7%

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

   12. Taylor expanded in b around inf 63.7%

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

       1. *-commutative 63.7%

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

       2. clear-num 63.7%

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

       3. associate-*r/ 63.7%

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

       4. frac-times 63.7%

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

       5. metadata-eval 63.7%

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

   14. Applied egg-rr 63.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{+120}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b \cdot \left(a \cdot \frac{b}{\pi}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+120}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot \frac{b}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+120)
   (* (/ PI (* a b)) (/ 0.5 a))
   (/ (/ PI a) (* b (/ b -0.5)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+120) {
		tmp = (((double) M_PI) / (a * b)) * (0.5 / a);
	} else {
		tmp = (((double) M_PI) / a) / (b * (b / -0.5));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+120) {
		tmp = (Math.PI / (a * b)) * (0.5 / a);
	} else {
		tmp = (Math.PI / a) / (b * (b / -0.5));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.35e+120:
		tmp = (math.pi / (a * b)) * (0.5 / a)
	else:
		tmp = (math.pi / a) / (b * (b / -0.5))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+120)
		tmp = Float64(Float64(pi / Float64(a * b)) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(pi / a) / Float64(b * Float64(b / -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+120)
		tmp = (pi / (a * b)) * (0.5 / a);
	else
		tmp = (pi / a) / (b * (b / -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.35e+120], N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] / N[(b * N[(b / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e120

   1. Initial program 76.8%

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

      1. associate-*r/ 76.9%

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

      2. *-rgt-identity 76.9%

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

      3. difference-of-squares 89.3%

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

      4. associate-/r* 89.8%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-/r* 64.5%

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

   7. Simplified 64.5%

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

      1. *-un-lft-identity 64.5%

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

      2. associate-/l* 64.3%

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

      3. associate-/l/ 64.3%

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

   9. Applied egg-rr 64.3%

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

       1. *-lft-identity 64.3%

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

       2. associate-/r/ 64.5%

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

   11. Simplified 64.5%

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

   12. Taylor expanded in b around 0 62.6%

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

   if 1.35e120 < b

   1. Initial program 53.5%

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

      1. associate-*r/ 53.5%

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

      2. *-rgt-identity 53.5%

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

      3. difference-of-squares 76.5%

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

      4. associate-/r* 78.0%

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

      5. associate-*l/ 99.7%

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

      6. *-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

      10. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-/r* 63.7%

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

   7. Simplified 63.7%

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

      1. *-un-lft-identity 63.7%

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

      2. associate-/l* 63.7%

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

      3. associate-/l/ 63.7%

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

   9. Applied egg-rr 63.7%

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

       1. *-lft-identity 63.7%

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

       2. associate-/r/ 63.7%

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

   11. Simplified 63.7%

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

   12. Taylor expanded in b around inf 63.7%

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

       1. clear-num 63.7%

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

       2. associate-/r* 63.7%

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

       3. frac-times 64.1%

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

       4. *-un-lft-identity 64.1%

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

   14. Applied egg-rr 64.1%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+120}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b \cdot \frac{b}{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. associate-*r/ 73.3%

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

   2. *-rgt-identity 73.3%

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

   3. difference-of-squares 87.4%

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

   4. associate-/r* 88.0%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-/l/ 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. associate-/l* 87.4%

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

   3. *-commutative 87.4%

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

6. Applied egg-rr 87.4%

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

   1. *-lft-identity 87.4%

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

   2. associate-/r/ 99.7%

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

   3. *-rgt-identity 99.7%

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

   4. associate-*r/ 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around 0 99.7%

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

10. Final simplification 99.7%

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

Alternative 11: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. associate-*r/ 73.3%

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

   2. *-rgt-identity 73.3%

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

   3. difference-of-squares 87.4%

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

   4. associate-/r* 88.0%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-/l/ 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. associate-/l* 87.4%

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

   3. *-commutative 87.4%

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

6. Applied egg-rr 87.4%

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

   1. *-lft-identity 87.4%

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

   2. associate-/r/ 99.7%

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

   3. *-rgt-identity 99.7%

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

   4. associate-*r/ 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around 0 99.7%

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

    1. associate-*l/ 99.6%

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

    2. *-un-lft-identity 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 12: 29.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. associate-*r/ 73.3%

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

   2. *-rgt-identity 73.3%

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

   3. difference-of-squares 87.4%

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

   4. associate-/r* 88.0%

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

   5. associate-*l/ 99.7%

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

   6. *-commutative 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-/l/ 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a around inf 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-/r* 64.3%

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

7. Simplified 64.3%

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

   1. *-un-lft-identity 64.3%

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

   2. associate-/l* 64.2%

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

   3. associate-/l/ 64.3%

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

9. Applied egg-rr 64.3%

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

    1. *-lft-identity 64.3%

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

    2. associate-/r/ 64.3%

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

11. Simplified 64.3%

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

12. Taylor expanded in b around inf 34.8%

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

13. Final simplification 34.8%

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

Reproduce

herbie shell --seed 2024030 
(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))))