NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.7%

Time: 11.8s

Alternatives: 7

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: 78.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. *-commutative 79.3%

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

   2. associate-*r* 79.3%

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

   3. associate-*r/ 79.4%

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

   4. associate-*r* 79.4%

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

   5. *-rgt-identity 79.4%

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

   6. sub-neg 79.4%

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

   7. distribute-neg-frac 79.4%

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

   8. metadata-eval 79.4%

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

3. Simplified 79.4%

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

   1. *-un-lft-identity 79.4%

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

   2. difference-of-squares 88.0%

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

   3. times-frac 99.5%

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

6. Applied egg-rr 67.8%

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

   1. associate-*l/ 67.9%

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

   2. *-lft-identity 67.9%

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

   3. associate-*r* 67.9%

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

   4. distribute-rgt-out 67.9%

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

   5. associate-*l/ 67.9%

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

   6. *-lft-identity 67.9%

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

   7. *-commutative 67.9%

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

   8. associate-*r/ 67.9%

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

   9. associate-*l/ 67.9%

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

   10. *-lft-identity 67.9%

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

   11. *-commutative 67.9%

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

   12. associate-*r/ 67.9%

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

   13. fma-define 67.9%

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

   14. associate-*r/ 67.9%

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

   15. +-commutative 67.9%

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

8. Simplified 67.9%

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

9. Taylor expanded in a around 0 99.7%

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

    1. associate-*r/ 99.7%

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

11. Simplified 99.7%

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

Alternative 2: 96.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot \left(a + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.3e+72)
   (* 0.5 (/ PI (* a (* b (+ a b)))))
   (/ (/ (* 0.5 PI) (* a b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.3e+72) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * (a + b))));
	} else {
		tmp = ((0.5 * ((double) M_PI)) / (a * b)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.3e+72) {
		tmp = 0.5 * (Math.PI / (a * (b * (a + b))));
	} else {
		tmp = ((0.5 * Math.PI) / (a * b)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.3e+72:
		tmp = 0.5 * (math.pi / (a * (b * (a + b))))
	else:
		tmp = ((0.5 * math.pi) / (a * b)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.3e+72)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * Float64(a + b)))));
	else
		tmp = Float64(Float64(Float64(0.5 * pi) / Float64(a * b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.3e+72)
		tmp = 0.5 * (pi / (a * (b * (a + b))));
	else
		tmp = ((0.5 * pi) / (a * b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.3e+72], N[(0.5 * N[(Pi / N[(a * N[(b * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.29999999999999991e72

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*r* 79.7%

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

      3. associate-*r/ 79.7%

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

      4. associate-*r* 79.7%

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

      5. *-rgt-identity 79.7%

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

      6. sub-neg 79.7%

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

      7. distribute-neg-frac 79.7%

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

      8. metadata-eval 79.7%

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

   3. Simplified 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. difference-of-squares 86.8%

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

      3. times-frac 99.5%

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

   6. Applied egg-rr 61.3%

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

      1. associate-*l/ 61.3%

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

      2. *-lft-identity 61.3%

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

      3. associate-*r* 61.3%

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

      4. distribute-rgt-out 61.3%

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

      5. associate-*l/ 61.3%

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

      6. *-lft-identity 61.3%

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

      7. *-commutative 61.3%

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

      8. associate-*r/ 61.3%

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

      9. associate-*l/ 61.3%

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

      10. *-lft-identity 61.3%

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

      11. *-commutative 61.3%

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

      12. associate-*r/ 61.3%

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

      13. fma-define 61.3%

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

      14. associate-*r/ 61.3%

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

      15. +-commutative 61.3%

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

   8. Simplified 61.3%

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

   9. Taylor expanded in a around 0 99.6%

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

       1. associate-*r/ 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in a around 0 99.6%

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

       1. associate-/r* 99.6%

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

   14. Simplified 99.6%

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

       1. associate-/r* 99.6%

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

       2. associate-/l* 99.6%

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

   16. Applied egg-rr 99.6%

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

       1. associate-/l/ 99.1%

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

       2. *-commutative 99.1%

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

       3. associate-*l* 95.1%

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

   18. Simplified 95.1%

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

   if 1.29999999999999991e72 < b

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*r* 77.5%

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

      3. associate-*r/ 77.6%

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

      4. associate-*r* 77.6%

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

      5. *-rgt-identity 77.6%

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

      6. sub-neg 77.6%

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

      7. distribute-neg-frac 77.6%

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

      8. metadata-eval 77.6%

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

   3. Simplified 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. difference-of-squares 93.5%

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

      3. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. associate-*r* 99.7%

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

      4. distribute-rgt-out 99.7%

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

      5. associate-*l/ 99.8%

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

      6. *-lft-identity 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.8%

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

      9. associate-*l/ 99.8%

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

      10. *-lft-identity 99.8%

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

      11. *-commutative 99.8%

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

      12. associate-*r/ 99.8%

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

      13. fma-define 99.8%

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

      14. associate-*r/ 99.8%

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

      15. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in a around 0 99.8%

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

       1. associate-*r/ 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in a around 0 99.8%

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

Alternative 3: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \pi}{a \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.6e-54)
   (* (/ PI a) (/ 0.5 (* a b)))
   (/ (/ (* 0.5 PI) (* a b)) b)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-54) {
		tmp = (((double) M_PI) / a) * (0.5 / (a * b));
	} else {
		tmp = ((0.5 * ((double) M_PI)) / (a * b)) / b;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.6e-54) {
		tmp = (Math.PI / a) * (0.5 / (a * b));
	} else {
		tmp = ((0.5 * Math.PI) / (a * b)) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.6e-54:
		tmp = (math.pi / a) * (0.5 / (a * b))
	else:
		tmp = ((0.5 * math.pi) / (a * b)) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.6e-54)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(a * b)));
	else
		tmp = Float64(Float64(Float64(0.5 * pi) / Float64(a * b)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.6e-54)
		tmp = (pi / a) * (0.5 / (a * b));
	else
		tmp = ((0.5 * pi) / (a * b)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.6e-54], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * Pi), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5999999999999998e-54

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*r* 72.2%

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

      3. associate-*r/ 72.3%

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

      4. associate-*r* 72.3%

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

      5. *-rgt-identity 72.3%

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

      6. sub-neg 72.3%

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

      7. distribute-neg-frac 72.3%

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

      8. metadata-eval 72.3%

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

   3. Simplified 72.3%

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

      1. *-un-lft-identity 72.3%

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

      2. difference-of-squares 84.8%

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

      3. times-frac 99.6%

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

   6. Applied egg-rr 58.9%

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

      1. associate-*l/ 58.8%

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

      2. *-lft-identity 58.8%

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

      3. associate-*r* 58.8%

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

      4. distribute-rgt-out 58.8%

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

      5. associate-*l/ 58.8%

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

      6. *-lft-identity 58.8%

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

      7. *-commutative 58.8%

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

      8. associate-*r/ 58.8%

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

      9. associate-*l/ 58.8%

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

      10. *-lft-identity 58.8%

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

      11. *-commutative 58.8%

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

      12. associate-*r/ 58.8%

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

      13. fma-define 58.8%

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

      14. associate-*r/ 58.8%

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

      15. +-commutative 58.8%

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

   8. Simplified 58.8%

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

   9. Taylor expanded in a around 0 99.7%

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

       1. associate-*r/ 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in a around inf 88.3%

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

       1. associate-/l/ 88.5%

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

       2. *-commutative 88.5%

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

       3. times-frac 88.3%

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

   14. Applied egg-rr 88.3%

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

   if -4.5999999999999998e-54 < a

   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. *-commutative 82.1%

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

      2. associate-*r* 82.1%

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

      3. associate-*r/ 82.1%

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

      4. associate-*r* 82.1%

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

      5. *-rgt-identity 82.1%

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

      6. sub-neg 82.1%

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

      7. distribute-neg-frac 82.1%

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

      8. metadata-eval 82.1%

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

   3. Simplified 82.1%

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

      1. *-un-lft-identity 82.1%

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

      2. difference-of-squares 89.2%

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

      3. times-frac 99.5%

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

   6. Applied egg-rr 71.4%

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

      1. associate-*l/ 71.4%

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

      2. *-lft-identity 71.4%

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

      3. associate-*r* 71.4%

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

      4. distribute-rgt-out 71.4%

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

      5. associate-*l/ 71.5%

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

      6. *-lft-identity 71.5%

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

      7. *-commutative 71.5%

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

      8. associate-*r/ 71.5%

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

      9. associate-*l/ 71.5%

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

      10. *-lft-identity 71.5%

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

      11. *-commutative 71.5%

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

      12. associate-*r/ 71.5%

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

      13. fma-define 71.5%

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

      14. associate-*r/ 71.5%

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

      15. +-commutative 71.5%

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

   8. Simplified 71.5%

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

   9. Taylor expanded in a around 0 99.7%

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

       1. associate-*r/ 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in a around 0 70.2%

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

Alternative 4: 68.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.2e-54)
   (* (/ PI a) (/ 0.5 (* a b)))
   (/ (* 0.5 (/ PI a)) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.2e-54) {
		tmp = (((double) M_PI) / a) * (0.5 / (a * b));
	} else {
		tmp = (0.5 * (((double) M_PI) / a)) / (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.2e-54) {
		tmp = (Math.PI / a) * (0.5 / (a * b));
	} else {
		tmp = (0.5 * (Math.PI / a)) / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.2e-54:
		tmp = (math.pi / a) * (0.5 / (a * b))
	else:
		tmp = (0.5 * (math.pi / a)) / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.2e-54)
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(a * b)));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.2e-54)
		tmp = (pi / a) * (0.5 / (a * b));
	else
		tmp = (0.5 * (pi / a)) / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.2e-54], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.2e-54

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*r* 72.2%

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

      3. associate-*r/ 72.3%

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

      4. associate-*r* 72.3%

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

      5. *-rgt-identity 72.3%

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

      6. sub-neg 72.3%

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

      7. distribute-neg-frac 72.3%

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

      8. metadata-eval 72.3%

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

   3. Simplified 72.3%

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

      1. *-un-lft-identity 72.3%

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

      2. difference-of-squares 84.8%

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

      3. times-frac 99.6%

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

   6. Applied egg-rr 58.9%

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

      1. associate-*l/ 58.8%

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

      2. *-lft-identity 58.8%

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

      3. associate-*r* 58.8%

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

      4. distribute-rgt-out 58.8%

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

      5. associate-*l/ 58.8%

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

      6. *-lft-identity 58.8%

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

      7. *-commutative 58.8%

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

      8. associate-*r/ 58.8%

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

      9. associate-*l/ 58.8%

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

      10. *-lft-identity 58.8%

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

      11. *-commutative 58.8%

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

      12. associate-*r/ 58.8%

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

      13. fma-define 58.8%

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

      14. associate-*r/ 58.8%

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

      15. +-commutative 58.8%

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

   8. Simplified 58.8%

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

   9. Taylor expanded in a around 0 99.7%

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

       1. associate-*r/ 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in a around inf 88.3%

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

       1. associate-/l/ 88.5%

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

       2. *-commutative 88.5%

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

       3. times-frac 88.3%

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

   14. Applied egg-rr 88.3%

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

   if -4.2e-54 < a

   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. *-commutative 82.1%

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

      2. associate-*r* 82.1%

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

      3. associate-*r/ 82.1%

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

      4. associate-*r* 82.1%

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

      5. *-rgt-identity 82.1%

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

      6. sub-neg 82.1%

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

      7. distribute-neg-frac 82.1%

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

      8. metadata-eval 82.1%

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

   3. Simplified 82.1%

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

      1. difference-of-squares 89.2%

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

      2. times-frac 99.6%

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

      3. add-sqr-sqrt 51.9%

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

      4. sqrt-unprod 81.6%

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

      5. frac-times 81.6%

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

      6. metadata-eval 81.6%

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

      7. metadata-eval 81.6%

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

      8. frac-times 81.6%

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

      9. sqrt-unprod 35.5%

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

      10. add-sqr-sqrt 71.5%

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

      11. div-inv 71.5%

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

      12. metadata-eval 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. +-commutative 71.5%

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

      2. +-commutative 71.5%

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

      3. *-commutative 71.5%

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

   8. Simplified 71.5%

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

   9. Taylor expanded in b around 0 71.4%

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

       1. frac-times 70.9%

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

       2. *-un-lft-identity 70.9%

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

   11. Applied egg-rr 70.9%

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

       1. *-commutative 70.9%

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

       2. *-rgt-identity 70.9%

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

       3. times-frac 71.4%

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

       4. associate-/r* 71.5%

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

       5. times-frac 65.1%

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

       6. associate-*r/ 65.2%

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

       7. *-rgt-identity 65.2%

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

       8. associate-*r/ 65.2%

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

       9. *-commutative 65.2%

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

   13. Simplified 65.2%

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

   14. Taylor expanded in b around inf 63.9%

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

Alternative 5: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. *-commutative 79.3%

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

   2. associate-*r* 79.3%

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

   3. associate-*r/ 79.4%

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

   4. associate-*r* 79.4%

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

   5. *-rgt-identity 79.4%

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

   6. sub-neg 79.4%

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

   7. distribute-neg-frac 79.4%

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

   8. metadata-eval 79.4%

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

3. Simplified 79.4%

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

   1. *-un-lft-identity 79.4%

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

   2. difference-of-squares 88.0%

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

   3. times-frac 99.5%

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

6. Applied egg-rr 67.8%

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

   1. associate-*l/ 67.9%

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

   2. *-lft-identity 67.9%

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

   3. associate-*r* 67.9%

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

   4. distribute-rgt-out 67.9%

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

   5. associate-*l/ 67.9%

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

   6. *-lft-identity 67.9%

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

   7. *-commutative 67.9%

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

   8. associate-*r/ 67.9%

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

   9. associate-*l/ 67.9%

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

   10. *-lft-identity 67.9%

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

   11. *-commutative 67.9%

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

   12. associate-*r/ 67.9%

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

   13. fma-define 67.9%

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

   14. associate-*r/ 67.9%

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

   15. +-commutative 67.9%

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

8. Simplified 67.9%

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

9. Taylor expanded in a around 0 99.7%

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

    1. associate-*r/ 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in a around 0 99.7%

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

    1. associate-/r* 99.6%

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

14. Simplified 99.6%

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

Alternative 6: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. *-commutative 79.3%

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

   2. associate-*r* 79.3%

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

   3. associate-*r/ 79.4%

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

   4. associate-*r* 79.4%

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

   5. *-rgt-identity 79.4%

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

   6. sub-neg 79.4%

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

   7. distribute-neg-frac 79.4%

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

   8. metadata-eval 79.4%

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

3. Simplified 79.4%

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

   1. *-un-lft-identity 79.4%

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

   2. difference-of-squares 88.0%

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

   3. times-frac 99.5%

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

6. Applied egg-rr 67.8%

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

   1. associate-*l/ 67.9%

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

   2. *-lft-identity 67.9%

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

   3. associate-*r* 67.9%

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

   4. distribute-rgt-out 67.9%

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

   5. associate-*l/ 67.9%

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

   6. *-lft-identity 67.9%

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

   7. *-commutative 67.9%

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

   8. associate-*r/ 67.9%

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

   9. associate-*l/ 67.9%

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

   10. *-lft-identity 67.9%

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

   11. *-commutative 67.9%

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

   12. associate-*r/ 67.9%

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

   13. fma-define 67.9%

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

   14. associate-*r/ 67.9%

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

   15. +-commutative 67.9%

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

8. Simplified 67.9%

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

9. Taylor expanded in a around 0 99.7%

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

    1. associate-*r/ 99.7%

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

11. Simplified 99.7%

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

    1. *-un-lft-identity 99.7%

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

    2. associate-/l/ 99.0%

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

13. Applied egg-rr 99.0%

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

    1. *-lft-identity 99.0%

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

    2. associate-/l* 99.0%

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

    3. *-commutative 99.0%

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

15. Simplified 99.0%

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

Alternative 7: 62.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. *-commutative 79.3%

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

   2. associate-*r* 79.3%

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

   3. associate-*r/ 79.4%

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

   4. associate-*r* 79.4%

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

   5. *-rgt-identity 79.4%

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

   6. sub-neg 79.4%

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

   7. distribute-neg-frac 79.4%

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

   8. metadata-eval 79.4%

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

3. Simplified 79.4%

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

   1. *-un-lft-identity 79.4%

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

   2. difference-of-squares 88.0%

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

   3. times-frac 99.5%

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

6. Applied egg-rr 67.8%

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

   1. associate-*l/ 67.9%

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

   2. *-lft-identity 67.9%

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

   3. associate-*r* 67.9%

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

   4. distribute-rgt-out 67.9%

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

   5. associate-*l/ 67.9%

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

   6. *-lft-identity 67.9%

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

   7. *-commutative 67.9%

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

   8. associate-*r/ 67.9%

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

   9. associate-*l/ 67.9%

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

   10. *-lft-identity 67.9%

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

   11. *-commutative 67.9%

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

   12. associate-*r/ 67.9%

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

   13. fma-define 67.9%

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

   14. associate-*r/ 67.9%

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

   15. +-commutative 67.9%

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

8. Simplified 67.9%

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

9. Taylor expanded in a around 0 99.7%

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

    1. associate-*r/ 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in a around inf 58.8%

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

    1. associate-/l/ 58.8%

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

    2. *-commutative 58.8%

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

    3. times-frac 58.8%

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

14. Applied egg-rr 58.8%

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

Reproduce

herbie shell --seed 2024145 
(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))))