NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.6%

Time: 10.7s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. *-commutative 79.6%

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

   2. associate-/r/ 79.6%

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

   3. associate-*l/ 79.7%

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

   4. *-commutative 79.7%

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

   5. associate-/r/ 79.7%

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

   6. times-frac 79.7%

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

3. Simplified 79.7%

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

   1. clear-num 79.7%

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

   2. inv-pow 79.7%

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

5. Applied egg-rr 79.7%

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

   1. unpow-1 79.7%

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

   2. fma-def 79.7%

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

   3. +-commutative 79.7%

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

   4. associate-*r/ 79.7%

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

   5. *-commutative 79.7%

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

   6. associate-*r/ 79.7%

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

   7. mul-1-neg 79.7%

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

   8. unsub-neg 79.7%

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

7. Simplified 79.7%

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

   1. div-sub 68.0%

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

9. Applied egg-rr 68.0%

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

    1. div-sub 79.7%

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

    2. difference-of-squares 89.0%

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

    3. *-commutative 89.0%

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

    4. associate-/l* 99.6%

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

11. Simplified 99.6%

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

    1. expm1-log1p-u 77.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

    2. expm1-udef 51.8%

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

    3. associate-/r/ 51.8%

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

    4. +-commutative 51.8%

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

13. Applied egg-rr 51.8%

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

    1. expm1-def 77.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

    2. expm1-log1p 99.6%

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

    3. associate-*l/ 99.7%

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

    4. *-lft-identity 99.7%

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

15. Simplified 99.7%

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

16. Final simplification 99.7%

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

Alternative 2: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \frac{1}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \mathbf{elif}\;a \leq -4.4 \cdot 10^{-255}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+129)
   (* 0.5 (/ 1.0 (* a (/ a (/ PI b)))))
   (if (<= a -4.4e-255)
     (* (* 0.5 (/ (/ PI (+ a b)) (- b a))) (+ (/ 1.0 a) (/ -1.0 b)))
     (* 0.5 (* PI (/ (/ 1.0 (* a b)) b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+129) {
		tmp = 0.5 * (1.0 / (a * (a / (((double) M_PI) / b))));
	} else if (a <= -4.4e-255) {
		tmp = (0.5 * ((((double) M_PI) / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+129) {
		tmp = 0.5 * (1.0 / (a * (a / (Math.PI / b))));
	} else if (a <= -4.4e-255) {
		tmp = (0.5 * ((Math.PI / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+129:
		tmp = 0.5 * (1.0 / (a * (a / (math.pi / b))))
	elif a <= -4.4e-255:
		tmp = (0.5 * ((math.pi / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+129)
		tmp = Float64(0.5 * Float64(1.0 / Float64(a * Float64(a / Float64(pi / b)))));
	elseif (a <= -4.4e-255)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / Float64(a + b)) / Float64(b - a))) * Float64(Float64(1.0 / a) + Float64(-1.0 / b)));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+129)
		tmp = 0.5 * (1.0 / (a * (a / (pi / b))));
	elseif (a <= -4.4e-255)
		tmp = (0.5 * ((pi / (a + b)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+129], N[(0.5 * N[(1.0 / N[(a * N[(a / N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -4.4e-255], N[(N[(0.5 * N[(N[(Pi / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.0000000000000003e129

   1. Initial program 51.6%

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

      1. *-commutative 51.6%

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

      2. associate-/r/ 51.6%

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

      3. associate-*l/ 51.6%

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

      4. *-commutative 51.6%

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

      5. associate-/r/ 51.6%

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

      6. times-frac 51.6%

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

   3. Simplified 51.6%

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

      1. clear-num 51.6%

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

      2. inv-pow 51.6%

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

   5. Applied egg-rr 51.6%

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

      1. unpow-1 51.6%

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

      2. fma-def 51.6%

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

      3. +-commutative 51.6%

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

      4. associate-*r/ 51.6%

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

      5. *-commutative 51.6%

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

      6. associate-*r/ 51.6%

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

      7. mul-1-neg 51.6%

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

      8. unsub-neg 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in b around 0 76.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{{a}^{2} \cdot b}{\pi}}} \cdot 0.5 \]
   9. Step-by-step derivation

      1. unpow2 76.6%

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

      2. associate-/l* 76.6%

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

      3. *-lft-identity 76.6%

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

      4. times-frac 99.9%

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

      5. /-rgt-identity 99.9%

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

   10. Simplified 99.9%

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

   if -5.0000000000000003e129 < a < -4.3999999999999998e-255

   1. Initial program 89.3%

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

      1. times-frac 89.5%

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

      2. *-commutative 89.5%

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

      3. times-frac 89.5%

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

      4. difference-of-squares 93.0%

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

      5. associate-/r* 93.6%

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

      6. metadata-eval 93.6%

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

      7. sub-neg 93.6%

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

      8. distribute-neg-frac 93.6%

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

      9. metadata-eval 93.6%

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

   3. Simplified 93.6%

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

   if -4.3999999999999998e-255 < a

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-/r/ 81.0%

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

      3. associate-*l/ 81.0%

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

      4. *-commutative 81.0%

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

      5. associate-/r/ 81.0%

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

      6. times-frac 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in b around inf 66.2%

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

      1. unpow2 66.2%

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

   6. Simplified 66.2%

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

      1. div-inv 66.1%

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

   8. Applied egg-rr 66.1%

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

      1. inv-pow 66.1%

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

      2. associate-*r* 73.1%

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

      3. unpow-prod-down 72.6%

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

      4. inv-pow 72.6%

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

   10. Applied egg-rr 72.6%

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

       1. associate-*r/ 72.6%

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

       2. *-rgt-identity 72.6%

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

       3. unpow-1 72.6%

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

   12. Simplified 72.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+129}:\\ \;\;\;\;0.5 \cdot \frac{1}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \mathbf{elif}\;a \leq -4.4 \cdot 10^{-255}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{a + b}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 3: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \frac{1}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+99)
   (* 0.5 (/ 1.0 (* a (/ a (/ PI b)))))
   (if (<= a -2.8e-162)
     (* (/ PI (- (* b b) (* a a))) (+ (/ 0.5 a) (/ -0.5 b)))
     (* 0.5 (* PI (/ (/ 1.0 (* a b)) b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+99) {
		tmp = 0.5 * (1.0 / (a * (a / (((double) M_PI) / b))));
	} else if (a <= -2.8e-162) {
		tmp = (((double) M_PI) / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+99) {
		tmp = 0.5 * (1.0 / (a * (a / (Math.PI / b))));
	} else if (a <= -2.8e-162) {
		tmp = (Math.PI / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e+99:
		tmp = 0.5 * (1.0 / (a * (a / (math.pi / b))))
	elif a <= -2.8e-162:
		tmp = (math.pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+99)
		tmp = Float64(0.5 * Float64(1.0 / Float64(a * Float64(a / Float64(pi / b)))));
	elseif (a <= -2.8e-162)
		tmp = Float64(Float64(pi / Float64(Float64(b * b) - Float64(a * a))) * Float64(Float64(0.5 / a) + Float64(-0.5 / b)));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+99)
		tmp = 0.5 * (1.0 / (a * (a / (pi / b))));
	elseif (a <= -2.8e-162)
		tmp = (pi / ((b * b) - (a * a))) * ((0.5 / a) + (-0.5 / b));
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+99], N[(0.5 * N[(1.0 / N[(a * N[(a / N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.8e-162], N[(N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.9999999999999997e98

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

      2. associate-/r/ 57.5%

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

      3. associate-*l/ 57.5%

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

      4. *-commutative 57.5%

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

      5. associate-/r/ 57.5%

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

      6. times-frac 57.5%

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

   3. Simplified 57.5%

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

      1. clear-num 57.5%

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

      2. inv-pow 57.5%

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

   5. Applied egg-rr 57.5%

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

      1. unpow-1 57.5%

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

      2. fma-def 57.5%

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

      3. +-commutative 57.5%

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

      4. associate-*r/ 57.5%

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

      5. *-commutative 57.5%

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

      6. associate-*r/ 57.5%

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

      7. mul-1-neg 57.5%

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

      8. unsub-neg 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in b around 0 79.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{a}^{2} \cdot b}{\pi}}} \cdot 0.5 \]
   9. Step-by-step derivation

      1. unpow2 79.4%

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

      2. associate-/l* 79.4%

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

      3. *-lft-identity 79.4%

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

      4. times-frac 99.9%

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

      5. /-rgt-identity 99.9%

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

   10. Simplified 99.9%

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

   if -9.9999999999999997e98 < a < -2.80000000000000022e-162

   1. Initial program 94.8%

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

      1. times-frac 95.0%

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

      2. *-commutative 95.0%

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

      3. times-frac 95.0%

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

      4. difference-of-squares 95.0%

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

      5. associate-/r* 95.7%

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

      6. metadata-eval 95.7%

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

      7. sub-neg 95.7%

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

      8. distribute-neg-frac 95.7%

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

      9. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

      1. distribute-lft-in 94.1%

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

      2. associate-/l/ 93.3%

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

      3. associate-/l/ 93.4%

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

   5. Applied egg-rr 93.4%

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

      1. distribute-lft-out 95.0%

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

      2. associate-*r* 95.0%

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

      3. associate-*l/ 94.9%

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

      4. *-commutative 94.9%

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

      5. difference-of-squares 94.9%

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

      6. associate-*l/ 95.0%

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

      7. distribute-lft-in 95.0%

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

      8. associate-*r/ 95.0%

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

      9. metadata-eval 95.0%

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

      10. associate-*r/ 95.0%

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

      11. metadata-eval 95.0%

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

   7. Simplified 95.0%

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

   if -2.80000000000000022e-162 < a

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-/r/ 79.4%

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

      3. associate-*l/ 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-/r/ 79.5%

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

      6. times-frac 79.5%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in b around inf 68.3%

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

      1. unpow2 68.3%

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

   6. Simplified 68.3%

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

      1. div-inv 68.3%

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

   8. Applied egg-rr 68.3%

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

      1. inv-pow 68.3%

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

      2. associate-*r* 76.2%

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

      3. unpow-prod-down 75.8%

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

      4. inv-pow 75.8%

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

   10. Applied egg-rr 75.8%

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

       1. associate-*r/ 75.8%

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

       2. *-rgt-identity 75.8%

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

       3. unpow-1 75.8%

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

   12. Simplified 75.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \frac{1}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{\pi}{b \cdot b - a \cdot a} \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 4: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b - a}}{a + b}}{\frac{a}{0.5}}\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \frac{-\frac{\pi}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e-23)
   (* (/ (/ PI (* a b)) (- b a)) (- 0.5))
   (if (<= a -7.2e-59)
     (/ (/ (/ PI (- b a)) (+ a b)) (/ a 0.5))
     (if (<= a -7.5e-132)
       (* 0.5 (/ (- (/ PI b)) (- (* b b) (* a a))))
       (* 0.5 (* PI (/ (/ 1.0 (* a b)) b)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.4e-23) {
		tmp = ((((double) M_PI) / (a * b)) / (b - a)) * -0.5;
	} else if (a <= -7.2e-59) {
		tmp = ((((double) M_PI) / (b - a)) / (a + b)) / (a / 0.5);
	} else if (a <= -7.5e-132) {
		tmp = 0.5 * (-(((double) M_PI) / b) / ((b * b) - (a * a)));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.4e-23) {
		tmp = ((Math.PI / (a * b)) / (b - a)) * -0.5;
	} else if (a <= -7.2e-59) {
		tmp = ((Math.PI / (b - a)) / (a + b)) / (a / 0.5);
	} else if (a <= -7.5e-132) {
		tmp = 0.5 * (-(Math.PI / b) / ((b * b) - (a * a)));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.4e-23:
		tmp = ((math.pi / (a * b)) / (b - a)) * -0.5
	elif a <= -7.2e-59:
		tmp = ((math.pi / (b - a)) / (a + b)) / (a / 0.5)
	elif a <= -7.5e-132:
		tmp = 0.5 * (-(math.pi / b) / ((b * b) - (a * a)))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.4e-23)
		tmp = Float64(Float64(Float64(pi / Float64(a * b)) / Float64(b - a)) * Float64(-0.5));
	elseif (a <= -7.2e-59)
		tmp = Float64(Float64(Float64(pi / Float64(b - a)) / Float64(a + b)) / Float64(a / 0.5));
	elseif (a <= -7.5e-132)
		tmp = Float64(0.5 * Float64(Float64(-Float64(pi / b)) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.4e-23)
		tmp = ((pi / (a * b)) / (b - a)) * -0.5;
	elseif (a <= -7.2e-59)
		tmp = ((pi / (b - a)) / (a + b)) / (a / 0.5);
	elseif (a <= -7.5e-132)
		tmp = 0.5 * (-(pi / b) / ((b * b) - (a * a)));
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.4e-23], N[(N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * (-0.5)), $MachinePrecision], If[LessEqual[a, -7.2e-59], N[(N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(a / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -7.5e-132], N[(0.5 * N[((-N[(Pi / b), $MachinePrecision]) / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.3999999999999999e-23

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-/r/ 75.8%

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

      3. associate-*l/ 75.9%

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

      4. *-commutative 75.9%

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

      5. associate-/r/ 75.9%

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

      6. times-frac 75.9%

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

   3. Simplified 76.0%

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

      1. clear-num 75.9%

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

      2. inv-pow 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. unpow-1 75.9%

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

      2. fma-def 75.9%

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

      3. +-commutative 75.9%

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

      4. associate-*r/ 75.9%

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

      5. *-commutative 75.9%

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

      6. associate-*r/ 75.9%

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

      7. mul-1-neg 75.9%

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

      8. unsub-neg 75.9%

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

   7. Simplified 75.9%

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

      1. div-sub 69.1%

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

   9. Applied egg-rr 69.1%

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

       1. div-sub 75.9%

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

       2. difference-of-squares 88.3%

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

       3. *-commutative 88.3%

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

       4. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 89.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 62.1%

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

       3. associate-/r/ 62.1%

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

       4. +-commutative 62.1%

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

   13. Applied egg-rr 62.1%

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

       1. expm1-def 89.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

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

       3. associate-*l/ 99.8%

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

       4. *-lft-identity 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in a around inf 92.9%

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

       1. associate-*r/ 92.9%

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

       2. neg-mul-1 92.9%

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

   18. Simplified 92.9%

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

   if -1.3999999999999999e-23 < a < -7.20000000000000001e-59

   1. Initial program 98.9%

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

      1. times-frac 98.9%

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

      2. *-commutative 98.9%

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

      3. times-frac 98.9%

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

      4. difference-of-squares 98.9%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. sub-neg 99.1%

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

      8. distribute-neg-frac 99.1%

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

      9. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 83.5%

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

      1. expm1-log1p-u 60.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{1}{a}\right)\right)} \]

      2. expm1-udef 12.0%

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

      3. un-div-inv 12.0%

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

      4. associate-/l/ 12.0%

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

      5. +-commutative 12.0%

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

   6. Applied egg-rr 12.0%

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

      1. expm1-def 60.1%

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

      2. expm1-log1p 83.4%

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

      3. associate-/l* 83.4%

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

      4. associate-/r* 83.6%

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

   8. Simplified 83.6%

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

   if -7.20000000000000001e-59 < a < -7.49999999999999989e-132

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-/r/ 85.0%

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

      3. associate-*l/ 85.1%

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

      4. *-commutative 85.1%

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

      5. associate-/r/ 85.1%

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

      6. times-frac 85.1%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in b around 0 55.4%

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

      1. associate-*r/ 55.4%

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

      2. mul-1-neg 55.4%

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

   6. Simplified 55.4%

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

   if -7.49999999999999989e-132 < a

   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%

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

      2. associate-/r/ 79.7%

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

      3. associate-*l/ 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-/r/ 79.8%

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

      6. times-frac 79.8%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in b around inf 67.9%

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

      1. unpow2 67.9%

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

   6. Simplified 67.9%

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

      1. div-inv 67.8%

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

   8. Applied egg-rr 67.8%

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

      1. inv-pow 67.8%

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

      2. associate-*r* 76.0%

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

      3. unpow-prod-down 75.6%

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

      4. inv-pow 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. associate-*r/ 75.6%

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

       2. *-rgt-identity 75.6%

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

       3. unpow-1 75.6%

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

   12. Simplified 75.6%

       \[\leadsto \left(\pi \cdot \color{blue}{\frac{\frac{1}{a \cdot b}}{b}}\right) \cdot 0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b - a}}{a + b}}{\frac{a}{0.5}}\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{-132}:\\ \;\;\;\;0.5 \cdot \frac{-\frac{\pi}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 5: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{elif}\;a \leq -3.4 \cdot 10^{-59}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b \cdot b}\right) \cdot \frac{b - a}{a \cdot b}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \frac{-\frac{\pi}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.8e-33)
   (* (/ (/ PI (* a b)) (- b a)) (- 0.5))
   (if (<= a -3.4e-59)
     (* (* 0.5 (/ PI (* b b))) (/ (- b a) (* a b)))
     (if (<= a -1.02e-131)
       (* 0.5 (/ (- (/ PI b)) (- (* b b) (* a a))))
       (* 0.5 (* PI (/ (/ 1.0 (* a b)) b)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.8e-33) {
		tmp = ((((double) M_PI) / (a * b)) / (b - a)) * -0.5;
	} else if (a <= -3.4e-59) {
		tmp = (0.5 * (((double) M_PI) / (b * b))) * ((b - a) / (a * b));
	} else if (a <= -1.02e-131) {
		tmp = 0.5 * (-(((double) M_PI) / b) / ((b * b) - (a * a)));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.8e-33) {
		tmp = ((Math.PI / (a * b)) / (b - a)) * -0.5;
	} else if (a <= -3.4e-59) {
		tmp = (0.5 * (Math.PI / (b * b))) * ((b - a) / (a * b));
	} else if (a <= -1.02e-131) {
		tmp = 0.5 * (-(Math.PI / b) / ((b * b) - (a * a)));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.8e-33:
		tmp = ((math.pi / (a * b)) / (b - a)) * -0.5
	elif a <= -3.4e-59:
		tmp = (0.5 * (math.pi / (b * b))) * ((b - a) / (a * b))
	elif a <= -1.02e-131:
		tmp = 0.5 * (-(math.pi / b) / ((b * b) - (a * a)))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.8e-33)
		tmp = Float64(Float64(Float64(pi / Float64(a * b)) / Float64(b - a)) * Float64(-0.5));
	elseif (a <= -3.4e-59)
		tmp = Float64(Float64(0.5 * Float64(pi / Float64(b * b))) * Float64(Float64(b - a) / Float64(a * b)));
	elseif (a <= -1.02e-131)
		tmp = Float64(0.5 * Float64(Float64(-Float64(pi / b)) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.8e-33)
		tmp = ((pi / (a * b)) / (b - a)) * -0.5;
	elseif (a <= -3.4e-59)
		tmp = (0.5 * (pi / (b * b))) * ((b - a) / (a * b));
	elseif (a <= -1.02e-131)
		tmp = 0.5 * (-(pi / b) / ((b * b) - (a * a)));
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.8e-33], N[(N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * (-0.5)), $MachinePrecision], If[LessEqual[a, -3.4e-59], N[(N[(0.5 * N[(Pi / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.02e-131], N[(0.5 * N[((-N[(Pi / b), $MachinePrecision]) / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.80000000000000005e-33

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-/r/ 76.4%

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

      3. associate-*l/ 76.5%

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

      4. *-commutative 76.5%

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

      5. associate-/r/ 76.5%

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

      6. times-frac 76.5%

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

   3. Simplified 76.6%

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

      1. clear-num 76.6%

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

      2. inv-pow 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. unpow-1 76.6%

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

      2. fma-def 76.5%

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

      3. +-commutative 76.5%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-*r/ 76.6%

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

      7. mul-1-neg 76.6%

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

      8. unsub-neg 76.6%

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

   7. Simplified 76.6%

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

      1. div-sub 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. div-sub 76.6%

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

       2. difference-of-squares 88.5%

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

       3. *-commutative 88.5%

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

       4. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 88.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 61.6%

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

       3. associate-/r/ 61.6%

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

       4. +-commutative 61.6%

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

   13. Applied egg-rr 61.6%

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

       1. expm1-def 88.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

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

       3. associate-*l/ 99.8%

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

       4. *-lft-identity 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in a around inf 91.8%

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

       1. associate-*r/ 91.8%

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

       2. neg-mul-1 91.8%

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

   18. Simplified 91.8%

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

   if -5.80000000000000005e-33 < a < -3.40000000000000018e-59

   1. Initial program 99.0%

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

      1. times-frac 99.0%

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

      2. *-commutative 99.0%

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

      3. times-frac 99.0%

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

      4. difference-of-squares 99.0%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-frac 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. frac-add 99.2%

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

      2. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. neg-mul-1 99.2%

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

      3. sub-neg 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in b around inf 93.1%

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

      1. unpow2 93.1%

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

   10. Simplified 93.1%

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

   if -3.40000000000000018e-59 < a < -1.02000000000000001e-131

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-/r/ 85.0%

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

      3. associate-*l/ 85.1%

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

      4. *-commutative 85.1%

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

      5. associate-/r/ 85.1%

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

      6. times-frac 85.1%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in b around 0 55.4%

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

      1. associate-*r/ 55.4%

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

      2. mul-1-neg 55.4%

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

   6. Simplified 55.4%

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

   if -1.02000000000000001e-131 < a

   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%

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

      2. associate-/r/ 79.7%

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

      3. associate-*l/ 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-/r/ 79.8%

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

      6. times-frac 79.8%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in b around inf 67.9%

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

      1. unpow2 67.9%

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

   6. Simplified 67.9%

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

      1. div-inv 67.8%

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

   8. Applied egg-rr 67.8%

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

      1. inv-pow 67.8%

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

      2. associate-*r* 76.0%

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

      3. unpow-prod-down 75.6%

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

      4. inv-pow 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. associate-*r/ 75.6%

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

       2. *-rgt-identity 75.6%

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

       3. unpow-1 75.6%

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

   12. Simplified 75.6%

       \[\leadsto \left(\pi \cdot \color{blue}{\frac{\frac{1}{a \cdot b}}{b}}\right) \cdot 0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{elif}\;a \leq -3.4 \cdot 10^{-59}:\\ \;\;\;\;\left(0.5 \cdot \frac{\pi}{b \cdot b}\right) \cdot \frac{b - a}{a \cdot b}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \frac{-\frac{\pi}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 6: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{-60}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \frac{-\frac{\pi}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.02e-26)
   (* (/ (/ PI (* a b)) (- b a)) (- 0.5))
   (if (<= a -8.8e-60)
     (* (+ (/ 1.0 a) (/ -1.0 b)) (/ (/ PI 2.0) (* b b)))
     (if (<= a -1.02e-131)
       (* 0.5 (/ (- (/ PI b)) (- (* b b) (* a a))))
       (* 0.5 (* PI (/ (/ 1.0 (* a b)) b)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.02e-26) {
		tmp = ((((double) M_PI) / (a * b)) / (b - a)) * -0.5;
	} else if (a <= -8.8e-60) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((((double) M_PI) / 2.0) / (b * b));
	} else if (a <= -1.02e-131) {
		tmp = 0.5 * (-(((double) M_PI) / b) / ((b * b) - (a * a)));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.02e-26) {
		tmp = ((Math.PI / (a * b)) / (b - a)) * -0.5;
	} else if (a <= -8.8e-60) {
		tmp = ((1.0 / a) + (-1.0 / b)) * ((Math.PI / 2.0) / (b * b));
	} else if (a <= -1.02e-131) {
		tmp = 0.5 * (-(Math.PI / b) / ((b * b) - (a * a)));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.02e-26:
		tmp = ((math.pi / (a * b)) / (b - a)) * -0.5
	elif a <= -8.8e-60:
		tmp = ((1.0 / a) + (-1.0 / b)) * ((math.pi / 2.0) / (b * b))
	elif a <= -1.02e-131:
		tmp = 0.5 * (-(math.pi / b) / ((b * b) - (a * a)))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.02e-26)
		tmp = Float64(Float64(Float64(pi / Float64(a * b)) / Float64(b - a)) * Float64(-0.5));
	elseif (a <= -8.8e-60)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(-1.0 / b)) * Float64(Float64(pi / 2.0) / Float64(b * b)));
	elseif (a <= -1.02e-131)
		tmp = Float64(0.5 * Float64(Float64(-Float64(pi / b)) / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.02e-26)
		tmp = ((pi / (a * b)) / (b - a)) * -0.5;
	elseif (a <= -8.8e-60)
		tmp = ((1.0 / a) + (-1.0 / b)) * ((pi / 2.0) / (b * b));
	elseif (a <= -1.02e-131)
		tmp = 0.5 * (-(pi / b) / ((b * b) - (a * a)));
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.02e-26], N[(N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * (-0.5)), $MachinePrecision], If[LessEqual[a, -8.8e-60], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / 2.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.02e-131], N[(0.5 * N[((-N[(Pi / b), $MachinePrecision]) / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.02e-26

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-/r/ 76.4%

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

      3. associate-*l/ 76.5%

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

      4. *-commutative 76.5%

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

      5. associate-/r/ 76.5%

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

      6. times-frac 76.5%

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

   3. Simplified 76.6%

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

      1. clear-num 76.6%

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

      2. inv-pow 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. unpow-1 76.6%

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

      2. fma-def 76.5%

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

      3. +-commutative 76.5%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-*r/ 76.6%

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

      7. mul-1-neg 76.6%

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

      8. unsub-neg 76.6%

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

   7. Simplified 76.6%

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

      1. div-sub 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. div-sub 76.6%

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

       2. difference-of-squares 88.5%

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

       3. *-commutative 88.5%

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

       4. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 88.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 61.6%

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

       3. associate-/r/ 61.6%

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

       4. +-commutative 61.6%

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

   13. Applied egg-rr 61.6%

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

       1. expm1-def 88.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

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

       3. associate-*l/ 99.8%

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

       4. *-lft-identity 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in a around inf 91.8%

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

       1. associate-*r/ 91.8%

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

       2. neg-mul-1 91.8%

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

   18. Simplified 91.8%

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

   if -1.02e-26 < a < -8.7999999999999995e-60

   1. Initial program 99.0%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

      3. sub-neg 99.0%

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

      4. distribute-neg-frac 99.0%

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

      5. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in b around inf 93.3%

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

      1. unpow2 93.3%

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

   6. Simplified 93.3%

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

   if -8.7999999999999995e-60 < a < -1.02000000000000001e-131

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-/r/ 85.0%

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

      3. associate-*l/ 85.1%

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

      4. *-commutative 85.1%

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

      5. associate-/r/ 85.1%

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

      6. times-frac 85.1%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in b around 0 55.4%

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

      1. associate-*r/ 55.4%

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

      2. mul-1-neg 55.4%

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

   6. Simplified 55.4%

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

   if -1.02000000000000001e-131 < a

   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%

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

      2. associate-/r/ 79.7%

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

      3. associate-*l/ 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-/r/ 79.8%

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

      6. times-frac 79.8%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in b around inf 67.9%

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

      1. unpow2 67.9%

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

   6. Simplified 67.9%

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

      1. div-inv 67.8%

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

   8. Applied egg-rr 67.8%

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

      1. inv-pow 67.8%

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

      2. associate-*r* 76.0%

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

      3. unpow-prod-down 75.6%

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

      4. inv-pow 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. associate-*r/ 75.6%

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

       2. *-rgt-identity 75.6%

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

       3. unpow-1 75.6%

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

   12. Simplified 75.6%

       \[\leadsto \left(\pi \cdot \color{blue}{\frac{\frac{1}{a \cdot b}}{b}}\right) \cdot 0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{-60}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{-1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b \cdot b}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;0.5 \cdot \frac{-\frac{\pi}{b}}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 7: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{if}\;a \leq -1 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{\pi}{\left(a + b\right) \cdot \left(b - a\right)}}{\frac{a}{0.5}}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI (* a b)) (- b a)) (- 0.5))))
   (if (<= a -1e-23)
     t_0
     (if (<= a -4.5e-58)
       (/ (/ PI (* (+ a b) (- b a))) (/ a 0.5))
       (if (<= a -1.02e-131) t_0 (* 0.5 (* PI (/ (/ 1.0 (* a b)) b))))))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / (a * b)) / (b - a)) * -0.5;
	double tmp;
	if (a <= -1e-23) {
		tmp = t_0;
	} else if (a <= -4.5e-58) {
		tmp = (((double) M_PI) / ((a + b) * (b - a))) / (a / 0.5);
	} else if (a <= -1.02e-131) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / (a * b)) / (b - a)) * -0.5;
	double tmp;
	if (a <= -1e-23) {
		tmp = t_0;
	} else if (a <= -4.5e-58) {
		tmp = (Math.PI / ((a + b) * (b - a))) / (a / 0.5);
	} else if (a <= -1.02e-131) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / (a * b)) / (b - a)) * -0.5
	tmp = 0
	if a <= -1e-23:
		tmp = t_0
	elif a <= -4.5e-58:
		tmp = (math.pi / ((a + b) * (b - a))) / (a / 0.5)
	elif a <= -1.02e-131:
		tmp = t_0
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / Float64(a * b)) / Float64(b - a)) * Float64(-0.5))
	tmp = 0.0
	if (a <= -1e-23)
		tmp = t_0;
	elseif (a <= -4.5e-58)
		tmp = Float64(Float64(pi / Float64(Float64(a + b) * Float64(b - a))) / Float64(a / 0.5));
	elseif (a <= -1.02e-131)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / (a * b)) / (b - a)) * -0.5;
	tmp = 0.0;
	if (a <= -1e-23)
		tmp = t_0;
	elseif (a <= -4.5e-58)
		tmp = (pi / ((a + b) * (b - a))) / (a / 0.5);
	elseif (a <= -1.02e-131)
		tmp = t_0;
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * (-0.5)), $MachinePrecision]}, If[LessEqual[a, -1e-23], t$95$0, If[LessEqual[a, -4.5e-58], N[(N[(Pi / N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.02e-131], t$95$0, N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.9999999999999996e-24 or -4.5000000000000003e-58 < a < -1.02000000000000001e-131

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-/r/ 77.2%

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

      3. associate-*l/ 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-/r/ 77.3%

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

      6. times-frac 77.3%

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

   3. Simplified 77.4%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. unpow-1 77.3%

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

      2. fma-def 77.3%

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

      3. +-commutative 77.3%

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

      4. associate-*r/ 77.3%

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

      5. *-commutative 77.3%

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

      6. associate-*r/ 77.3%

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

      7. mul-1-neg 77.3%

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

      8. unsub-neg 77.3%

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

   7. Simplified 77.3%

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

      1. div-sub 68.0%

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

   9. Applied egg-rr 68.0%

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

       1. div-sub 77.3%

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

       2. difference-of-squares 87.8%

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

       3. *-commutative 87.8%

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

       4. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 83.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 57.5%

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

       3. associate-/r/ 57.5%

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

       4. +-commutative 57.5%

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

   13. Applied egg-rr 57.5%

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

       1. expm1-def 82.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

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

       3. associate-*l/ 99.7%

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

       4. *-lft-identity 99.7%

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

   15. Simplified 99.7%

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

   16. Taylor expanded in a around inf 87.1%

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

       1. associate-*r/ 87.1%

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

       2. neg-mul-1 87.1%

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

   18. Simplified 87.1%

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

   if -9.9999999999999996e-24 < a < -4.5000000000000003e-58

   1. Initial program 98.9%

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

      1. times-frac 98.9%

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

      2. *-commutative 98.9%

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

      3. times-frac 98.9%

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

      4. difference-of-squares 98.9%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. sub-neg 99.1%

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

      8. distribute-neg-frac 99.1%

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

      9. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 83.5%

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

      1. expm1-log1p-u 60.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{1}{a}\right)\right)} \]

      2. expm1-udef 12.0%

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

      3. un-div-inv 12.0%

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

      4. associate-/l/ 12.0%

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

      5. +-commutative 12.0%

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

   6. Applied egg-rr 12.0%

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

      1. expm1-def 60.1%

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

      2. expm1-log1p 83.4%

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

      3. associate-/l* 83.4%

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

      4. *-commutative 83.4%

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

   8. Simplified 83.4%

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

   if -1.02000000000000001e-131 < a

   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%

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

      2. associate-/r/ 79.7%

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

      3. associate-*l/ 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-/r/ 79.8%

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

      6. times-frac 79.8%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in b around inf 67.9%

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

      1. unpow2 67.9%

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

   6. Simplified 67.9%

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

      1. div-inv 67.8%

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

   8. Applied egg-rr 67.8%

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

      1. inv-pow 67.8%

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

      2. associate-*r* 76.0%

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

      3. unpow-prod-down 75.6%

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

      4. inv-pow 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. associate-*r/ 75.6%

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

       2. *-rgt-identity 75.6%

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

       3. unpow-1 75.6%

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

   12. Simplified 75.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{\pi}{\left(a + b\right) \cdot \left(b - a\right)}}{\frac{a}{0.5}}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 8: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{if}\;a \leq -3.3 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b - a}}{a + b}}{\frac{a}{0.5}}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ PI (* a b)) (- b a)) (- 0.5))))
   (if (<= a -3.3e-24)
     t_0
     (if (<= a -2.8e-58)
       (/ (/ (/ PI (- b a)) (+ a b)) (/ a 0.5))
       (if (<= a -1.02e-131) t_0 (* 0.5 (* PI (/ (/ 1.0 (* a b)) b))))))))

C

double code(double a, double b) {
	double t_0 = ((((double) M_PI) / (a * b)) / (b - a)) * -0.5;
	double tmp;
	if (a <= -3.3e-24) {
		tmp = t_0;
	} else if (a <= -2.8e-58) {
		tmp = ((((double) M_PI) / (b - a)) / (a + b)) / (a / 0.5);
	} else if (a <= -1.02e-131) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = ((Math.PI / (a * b)) / (b - a)) * -0.5;
	double tmp;
	if (a <= -3.3e-24) {
		tmp = t_0;
	} else if (a <= -2.8e-58) {
		tmp = ((Math.PI / (b - a)) / (a + b)) / (a / 0.5);
	} else if (a <= -1.02e-131) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((math.pi / (a * b)) / (b - a)) * -0.5
	tmp = 0
	if a <= -3.3e-24:
		tmp = t_0
	elif a <= -2.8e-58:
		tmp = ((math.pi / (b - a)) / (a + b)) / (a / 0.5)
	elif a <= -1.02e-131:
		tmp = t_0
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(pi / Float64(a * b)) / Float64(b - a)) * Float64(-0.5))
	tmp = 0.0
	if (a <= -3.3e-24)
		tmp = t_0;
	elseif (a <= -2.8e-58)
		tmp = Float64(Float64(Float64(pi / Float64(b - a)) / Float64(a + b)) / Float64(a / 0.5));
	elseif (a <= -1.02e-131)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((pi / (a * b)) / (b - a)) * -0.5;
	tmp = 0.0;
	if (a <= -3.3e-24)
		tmp = t_0;
	elseif (a <= -2.8e-58)
		tmp = ((pi / (b - a)) / (a + b)) / (a / 0.5);
	elseif (a <= -1.02e-131)
		tmp = t_0;
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * (-0.5)), $MachinePrecision]}, If[LessEqual[a, -3.3e-24], t$95$0, If[LessEqual[a, -2.8e-58], N[(N[(N[(Pi / N[(b - a), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / N[(a / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.02e-131], t$95$0, N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.29999999999999984e-24 or -2.8000000000000001e-58 < a < -1.02000000000000001e-131

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-/r/ 77.2%

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

      3. associate-*l/ 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-/r/ 77.3%

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

      6. times-frac 77.3%

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

   3. Simplified 77.4%

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

      1. clear-num 77.3%

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

      2. inv-pow 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. unpow-1 77.3%

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

      2. fma-def 77.3%

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

      3. +-commutative 77.3%

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

      4. associate-*r/ 77.3%

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

      5. *-commutative 77.3%

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

      6. associate-*r/ 77.3%

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

      7. mul-1-neg 77.3%

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

      8. unsub-neg 77.3%

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

   7. Simplified 77.3%

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

      1. div-sub 68.0%

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

   9. Applied egg-rr 68.0%

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

       1. div-sub 77.3%

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

       2. difference-of-squares 87.8%

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

       3. *-commutative 87.8%

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

       4. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 83.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 57.5%

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

       3. associate-/r/ 57.5%

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

       4. +-commutative 57.5%

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

   13. Applied egg-rr 57.5%

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

       1. expm1-def 82.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

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

       3. associate-*l/ 99.7%

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

       4. *-lft-identity 99.7%

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

   15. Simplified 99.7%

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

   16. Taylor expanded in a around inf 87.1%

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

       1. associate-*r/ 87.1%

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

       2. neg-mul-1 87.1%

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

   18. Simplified 87.1%

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

   if -3.29999999999999984e-24 < a < -2.8000000000000001e-58

   1. Initial program 98.9%

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

      1. times-frac 98.9%

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

      2. *-commutative 98.9%

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

      3. times-frac 98.9%

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

      4. difference-of-squares 98.9%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. sub-neg 99.1%

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

      8. distribute-neg-frac 99.1%

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

      9. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 83.5%

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

      1. expm1-log1p-u 60.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{1}{a}\right)\right)} \]

      2. expm1-udef 12.0%

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

      3. un-div-inv 12.0%

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

      4. associate-/l/ 12.0%

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

      5. +-commutative 12.0%

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

   6. Applied egg-rr 12.0%

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

      1. expm1-def 60.1%

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

      2. expm1-log1p 83.4%

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

      3. associate-/l* 83.4%

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

      4. associate-/r* 83.6%

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

   8. Simplified 83.6%

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

   if -1.02000000000000001e-131 < a

   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%

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

      2. associate-/r/ 79.7%

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

      3. associate-*l/ 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-/r/ 79.8%

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

      6. times-frac 79.8%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in b around inf 67.9%

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

      1. unpow2 67.9%

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

   6. Simplified 67.9%

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

      1. div-inv 67.8%

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

   8. Applied egg-rr 67.8%

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

      1. inv-pow 67.8%

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

      2. associate-*r* 76.0%

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

      3. unpow-prod-down 75.6%

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

      4. inv-pow 75.6%

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

   10. Applied egg-rr 75.6%

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

       1. associate-*r/ 75.6%

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

       2. *-rgt-identity 75.6%

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

       3. unpow-1 75.6%

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

   12. Simplified 75.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b - a}}{a + b}}{\frac{a}{0.5}}\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 9: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.5e-28)
   (* (/ (/ PI (* a b)) (- b a)) (- 0.5))
   (* 0.5 (* PI (/ (/ 1.0 (* a b)) b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-28) {
		tmp = ((((double) M_PI) / (a * b)) / (b - a)) * -0.5;
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-28) {
		tmp = ((Math.PI / (a * b)) / (b - a)) * -0.5;
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.5e-28:
		tmp = ((math.pi / (a * b)) / (b - a)) * -0.5
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.5e-28)
		tmp = Float64(Float64(Float64(pi / Float64(a * b)) / Float64(b - a)) * Float64(-0.5));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-28)
		tmp = ((pi / (a * b)) / (b - a)) * -0.5;
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.5e-28], N[(N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision] * (-0.5)), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.4999999999999998e-28

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-/r/ 76.4%

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

      3. associate-*l/ 76.5%

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

      4. *-commutative 76.5%

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

      5. associate-/r/ 76.5%

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

      6. times-frac 76.5%

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

   3. Simplified 76.6%

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

      1. clear-num 76.6%

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

      2. inv-pow 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. unpow-1 76.6%

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

      2. fma-def 76.5%

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

      3. +-commutative 76.5%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-*r/ 76.6%

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

      7. mul-1-neg 76.6%

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

      8. unsub-neg 76.6%

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

   7. Simplified 76.6%

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

      1. div-sub 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. div-sub 76.6%

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

       2. difference-of-squares 88.5%

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

       3. *-commutative 88.5%

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

       4. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 88.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 61.6%

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

       3. associate-/r/ 61.6%

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

       4. +-commutative 61.6%

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

   13. Applied egg-rr 61.6%

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

       1. expm1-def 88.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

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

       3. associate-*l/ 99.8%

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

       4. *-lft-identity 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in a around inf 91.8%

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

       1. associate-*r/ 91.8%

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

       2. neg-mul-1 91.8%

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

   18. Simplified 91.8%

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

   if -4.4999999999999998e-28 < a

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-/r/ 80.9%

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

      3. associate-*l/ 81.0%

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

      4. *-commutative 81.0%

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

      5. associate-/r/ 81.0%

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

      6. times-frac 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in b around inf 68.5%

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

      1. unpow2 68.5%

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

   6. Simplified 68.5%

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

      1. div-inv 68.4%

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

   8. Applied egg-rr 68.4%

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

      1. inv-pow 68.4%

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

      2. associate-*r* 76.7%

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

      3. unpow-prod-down 76.4%

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

      4. inv-pow 76.4%

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

   10. Applied egg-rr 76.4%

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

       1. associate-*r/ 76.4%

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

       2. *-rgt-identity 76.4%

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

       3. unpow-1 76.4%

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

   12. Simplified 76.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{\pi}{a \cdot b}}{b - a} \cdot \left(-0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 10: 75.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.55e-30)
   (* 0.5 (/ PI (* a (* a b))))
   (* 0.5 (* PI (/ (/ 1.0 (* a b)) b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-30) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-30) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.55e-30:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.55e-30)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.55e-30)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.55e-30], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.54999999999999995e-30

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-/r/ 76.4%

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

      3. associate-*l/ 76.5%

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

      4. *-commutative 76.5%

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

      5. associate-/r/ 76.5%

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

      6. times-frac 76.5%

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

   3. Simplified 76.6%

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

      1. clear-num 76.6%

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

      2. inv-pow 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. unpow-1 76.6%

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

      2. fma-def 76.5%

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

      3. +-commutative 76.5%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-*r/ 76.6%

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

      7. mul-1-neg 76.6%

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

      8. unsub-neg 76.6%

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

   7. Simplified 76.6%

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

      1. div-sub 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. div-sub 76.6%

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

       2. difference-of-squares 88.5%

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

       3. *-commutative 88.5%

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

       4. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

       1. expm1-log1p-u 88.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 61.6%

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

       3. associate-/r/ 61.6%

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

       4. +-commutative 61.6%

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

   13. Applied egg-rr 61.6%

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

       1. expm1-def 88.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

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

       3. associate-*l/ 99.8%

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

       4. *-lft-identity 99.8%

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

   15. Simplified 99.8%

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

   16. Taylor expanded in a around inf 73.0%

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

       1. unpow2 73.0%

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

       2. associate-*l* 84.2%

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

   18. Simplified 84.2%

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

   if -1.54999999999999995e-30 < a

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-/r/ 80.9%

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

      3. associate-*l/ 81.0%

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

      4. *-commutative 81.0%

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

      5. associate-/r/ 81.0%

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

      6. times-frac 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in b around inf 68.5%

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

      1. unpow2 68.5%

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

   6. Simplified 68.5%

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

      1. div-inv 68.4%

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

   8. Applied egg-rr 68.4%

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

      1. inv-pow 68.4%

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

      2. associate-*r* 76.7%

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

      3. unpow-prod-down 76.4%

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

      4. inv-pow 76.4%

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

   10. Applied egg-rr 76.4%

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

       1. associate-*r/ 76.4%

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

       2. *-rgt-identity 76.4%

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

       3. unpow-1 76.4%

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

   12. Simplified 76.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 11: 75.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{1}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.15e-29)
   (* 0.5 (/ 1.0 (* a (/ a (/ PI b)))))
   (* 0.5 (* PI (/ (/ 1.0 (* a b)) b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.15e-29) {
		tmp = 0.5 * (1.0 / (a * (a / (((double) M_PI) / b))));
	} else {
		tmp = 0.5 * (((double) M_PI) * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.15e-29) {
		tmp = 0.5 * (1.0 / (a * (a / (Math.PI / b))));
	} else {
		tmp = 0.5 * (Math.PI * ((1.0 / (a * b)) / b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.15e-29:
		tmp = 0.5 * (1.0 / (a * (a / (math.pi / b))))
	else:
		tmp = 0.5 * (math.pi * ((1.0 / (a * b)) / b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.15e-29)
		tmp = Float64(0.5 * Float64(1.0 / Float64(a * Float64(a / Float64(pi / b)))));
	else
		tmp = Float64(0.5 * Float64(pi * Float64(Float64(1.0 / Float64(a * b)) / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.15e-29)
		tmp = 0.5 * (1.0 / (a * (a / (pi / b))));
	else
		tmp = 0.5 * (pi * ((1.0 / (a * b)) / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.15e-29], N[(0.5 * N[(1.0 / N[(a * N[(a / N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi * N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.14999999999999996e-29

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-/r/ 76.4%

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

      3. associate-*l/ 76.5%

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

      4. *-commutative 76.5%

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

      5. associate-/r/ 76.5%

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

      6. times-frac 76.5%

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

   3. Simplified 76.6%

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

      1. clear-num 76.6%

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

      2. inv-pow 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. unpow-1 76.6%

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

      2. fma-def 76.5%

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

      3. +-commutative 76.5%

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

      4. associate-*r/ 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-*r/ 76.6%

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

      7. mul-1-neg 76.6%

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

      8. unsub-neg 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in b around 0 73.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{{a}^{2} \cdot b}{\pi}}} \cdot 0.5 \]
   9. Step-by-step derivation

      1. unpow2 73.1%

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

      2. associate-/l* 73.1%

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

      3. *-lft-identity 73.1%

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

      4. times-frac 84.3%

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

      5. /-rgt-identity 84.3%

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

   10. Simplified 84.3%

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

   if -1.14999999999999996e-29 < a

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-/r/ 80.9%

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

      3. associate-*l/ 81.0%

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

      4. *-commutative 81.0%

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

      5. associate-/r/ 81.0%

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

      6. times-frac 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in b around inf 68.5%

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

      1. unpow2 68.5%

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

   6. Simplified 68.5%

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

      1. div-inv 68.4%

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

   8. Applied egg-rr 68.4%

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

      1. inv-pow 68.4%

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

      2. associate-*r* 76.7%

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

      3. unpow-prod-down 76.4%

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

      4. inv-pow 76.4%

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

   10. Applied egg-rr 76.4%

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

       1. associate-*r/ 76.4%

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

       2. *-rgt-identity 76.4%

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

       3. unpow-1 76.4%

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

   12. Simplified 76.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{1}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\pi \cdot \frac{\frac{1}{a \cdot b}}{b}\right)\\ \end{array} \]

Alternative 12: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-155}:\\ \;\;\;\;0.5 \cdot \frac{1}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.5e-155)
   (* 0.5 (/ 1.0 (* a (/ a (/ PI b)))))
   (* 0.5 (/ (/ PI (* a b)) (- b a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.5e-155) {
		tmp = 0.5 * (1.0 / (a * (a / (((double) M_PI) / 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 <= 4.5e-155) {
		tmp = 0.5 * (1.0 / (a * (a / (Math.PI / b))));
	} else {
		tmp = 0.5 * ((Math.PI / (a * b)) / (b - a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.5e-155:
		tmp = 0.5 * (1.0 / (a * (a / (math.pi / b))))
	else:
		tmp = 0.5 * ((math.pi / (a * b)) / (b - a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.5e-155)
		tmp = Float64(0.5 * Float64(1.0 / Float64(a * Float64(a / Float64(pi / b)))));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / Float64(a * b)) / Float64(b - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.5e-155)
		tmp = 0.5 * (1.0 / (a * (a / (pi / b))));
	else
		tmp = 0.5 * ((pi / (a * b)) / (b - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.5e-155], N[(0.5 * N[(1.0 / N[(a * N[(a / N[(Pi / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5000000000000004e-155

   1. Initial program 78.5%

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

      1. *-commutative 78.5%

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

      2. associate-/r/ 78.5%

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

      3. associate-*l/ 78.6%

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

      4. *-commutative 78.6%

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

      5. associate-/r/ 78.6%

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

      6. times-frac 78.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 78.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 78.6%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 78.6%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 78.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 78.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 78.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 78.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 78.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 78.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 78.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 78.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   8. Taylor expanded in b around 0 59.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{{a}^{2} \cdot b}{\pi}}} \cdot 0.5 \]
   9. Step-by-step derivation

      1. unpow2 59.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(a \cdot a\right)} \cdot b}{\pi}} \cdot 0.5 \]

      2. associate-/l* 59.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot a}{\frac{\pi}{b}}}} \cdot 0.5 \]

      3. *-lft-identity 59.8%

         \[\leadsto \frac{1}{\frac{a \cdot a}{\color{blue}{1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      4. times-frac 65.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{a}{1} \cdot \frac{a}{\frac{\pi}{b}}}} \cdot 0.5 \]

      5. /-rgt-identity 65.2%

         \[\leadsto \frac{1}{\color{blue}{a} \cdot \frac{a}{\frac{\pi}{b}}} \cdot 0.5 \]

   10. Simplified 65.2%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{a}{\frac{\pi}{b}}}} \cdot 0.5 \]

   if 4.5000000000000004e-155 < b

   1. Initial program 81.3%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 81.3%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 81.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 81.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 81.4%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 81.4%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 81.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 81.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 81.4%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 81.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 81.3%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 81.3%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 81.4%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 81.4%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 81.4%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 81.4%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 81.4%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. div-sub 69.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   9. Applied egg-rr 69.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   10. Step-by-step derivation

       1. div-sub 81.4%

          \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

       2. difference-of-squares 89.3%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       3. *-commutative 89.3%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       4. associate-/l* 99.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]

   11. Simplified 99.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]
   12. Step-by-step derivation

       1. expm1-log1p-u 88.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 57.7%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)} - 1\right)} \cdot 0.5 \]

       3. associate-/r/ 57.7%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}\right)} - 1\right) \cdot 0.5 \]

       4. +-commutative 57.7%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{\color{blue}{a + b}}\right)} - 1\right) \cdot 0.5 \]

   13. Applied egg-rr 57.7%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)} - 1\right)} \cdot 0.5 \]
   14. Step-by-step derivation

       1. expm1-def 88.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

          \[\leadsto \color{blue}{\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)} \cdot 0.5 \]

       3. associate-*l/ 99.6%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{b - a}} \cdot 0.5 \]

       4. *-lft-identity 99.6%

          \[\leadsto \frac{\color{blue}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}}{b - a} \cdot 0.5 \]

   15. Simplified 99.6%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{b - a}} \cdot 0.5 \]

   16. Taylor expanded in a around 0 81.8%

       \[\leadsto \frac{\color{blue}{\frac{\pi}{a \cdot b}}}{b - a} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-155}:\\ \;\;\;\;0.5 \cdot \frac{1}{a \cdot \frac{a}{\frac{\pi}{b}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a \cdot b}}{b - a}\\ \end{array} \]

Alternative 13: 68.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.8e-25)
   (* 0.5 (/ PI (* a (* a b))))
   (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.8e-25) {
		tmp = 0.5 * (((double) M_PI) / (a * (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 <= -1.8e-25) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.8e-25:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.8e-25)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.8e-25)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.8e-25], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e-25

   1. Initial program 76.4%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 76.4%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 76.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 76.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 76.5%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 76.5%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 76.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 76.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 76.6%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 76.6%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 76.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 76.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 76.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. div-sub 69.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   9. Applied egg-rr 69.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   10. Step-by-step derivation

       1. div-sub 76.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

       2. difference-of-squares 88.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       3. *-commutative 88.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       4. associate-/l* 99.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]

   11. Simplified 99.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]
   12. Step-by-step derivation

       1. expm1-log1p-u 88.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 61.6%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)} - 1\right)} \cdot 0.5 \]

       3. associate-/r/ 61.6%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}\right)} - 1\right) \cdot 0.5 \]

       4. +-commutative 61.6%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{\color{blue}{a + b}}\right)} - 1\right) \cdot 0.5 \]

   13. Applied egg-rr 61.6%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)} - 1\right)} \cdot 0.5 \]
   14. Step-by-step derivation

       1. expm1-def 88.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

          \[\leadsto \color{blue}{\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)} \cdot 0.5 \]

       3. associate-*l/ 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{b - a}} \cdot 0.5 \]

       4. *-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}}{b - a} \cdot 0.5 \]

   15. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{b - a}} \cdot 0.5 \]

   16. Taylor expanded in a around inf 73.0%

       \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   17. Step-by-step derivation

       1. unpow2 73.0%

          \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

       2. associate-*l* 84.2%

          \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   18. Simplified 84.2%

       \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   if -1.8e-25 < a

   1. Initial program 81.0%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 81.0%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 80.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 81.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 81.0%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 81.0%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 81.0%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]

   4. Taylor expanded in b around inf 68.5%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 68.5%

         \[\leadsto \frac{\pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \cdot 0.5 \]

   6. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(b \cdot b\right)}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-25}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 14: 75.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\frac{b \cdot \left(a \cdot b\right)}{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.55e-30)
   (* 0.5 (/ PI (* a (* a b))))
   (/ PI (/ (* b (* a b)) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-30) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = ((double) M_PI) / ((b * (a * b)) / 0.5);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.55e-30) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = Math.PI / ((b * (a * b)) / 0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.55e-30:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = math.pi / ((b * (a * b)) / 0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.55e-30)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(pi / Float64(Float64(b * Float64(a * b)) / 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.55e-30)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = pi / ((b * (a * b)) / 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.55e-30], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi / N[(N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.54999999999999995e-30

   1. Initial program 76.4%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 76.4%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 76.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 76.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 76.5%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 76.5%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 76.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 76.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 76.6%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 76.6%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 76.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 76.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 76.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 76.6%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. div-sub 69.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   9. Applied egg-rr 69.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   10. Step-by-step derivation

       1. div-sub 76.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

       2. difference-of-squares 88.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       3. *-commutative 88.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       4. associate-/l* 99.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]

   11. Simplified 99.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]
   12. Step-by-step derivation

       1. expm1-log1p-u 88.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

       2. expm1-udef 61.6%

          \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)} - 1\right)} \cdot 0.5 \]

       3. associate-/r/ 61.6%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}\right)} - 1\right) \cdot 0.5 \]

       4. +-commutative 61.6%

          \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{\color{blue}{a + b}}\right)} - 1\right) \cdot 0.5 \]

   13. Applied egg-rr 61.6%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)} - 1\right)} \cdot 0.5 \]
   14. Step-by-step derivation

       1. expm1-def 88.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

       2. expm1-log1p 99.6%

          \[\leadsto \color{blue}{\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)} \cdot 0.5 \]

       3. associate-*l/ 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{b - a}} \cdot 0.5 \]

       4. *-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}}{b - a} \cdot 0.5 \]

   15. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{b - a}} \cdot 0.5 \]

   16. Taylor expanded in a around inf 73.0%

       \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   17. Step-by-step derivation

       1. unpow2 73.0%

          \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

       2. associate-*l* 84.2%

          \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   18. Simplified 84.2%

       \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   if -1.54999999999999995e-30 < a

   1. Initial program 81.0%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 81.0%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 81.0%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 81.0%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 89.3%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 89.6%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. frac-add 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

      2. *-un-lft-identity 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

   5. Applied egg-rr 89.6%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
   6. Step-by-step derivation

      1. *-commutative 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

      2. neg-mul-1 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

      3. sub-neg 89.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

   7. Simplified 89.6%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]

   8. Taylor expanded in b around inf 68.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 68.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. *-commutative 68.5%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot {b}^{2}} \]

      3. unpow2 68.5%

         \[\leadsto \frac{\pi \cdot 0.5}{a \cdot \color{blue}{\left(b \cdot b\right)}} \]

      4. associate-/l* 68.5%

         \[\leadsto \color{blue}{\frac{\pi}{\frac{a \cdot \left(b \cdot b\right)}{0.5}}} \]

      5. *-commutative 68.5%

         \[\leadsto \frac{\pi}{\frac{\color{blue}{\left(b \cdot b\right) \cdot a}}{0.5}} \]

      6. associate-*l* 76.7%

         \[\leadsto \frac{\pi}{\frac{\color{blue}{b \cdot \left(b \cdot a\right)}}{0.5}} \]

   10. Simplified 76.7%

       \[\leadsto \color{blue}{\frac{\pi}{\frac{b \cdot \left(b \cdot a\right)}{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\frac{b \cdot \left(a \cdot b\right)}{0.5}}\\ \end{array} \]

Alternative 15: 56.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* 0.5 (/ (/ PI b) (* a a))))

C

double code(double a, double b) {
	return 0.5 * ((((double) M_PI) / b) / (a * a));
}

Java

public static double code(double a, double b) {
	return 0.5 * ((Math.PI / b) / (a * a));
}

Python

def code(a, b):
	return 0.5 * ((math.pi / b) / (a * a))

Julia

function code(a, b)
	return Float64(0.5 * Float64(Float64(pi / b) / Float64(a * a)))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 * ((pi / b) / (a * a));
end

Wolfram

code[a_, b_] := N[(0.5 * N[(N[(Pi / b), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. times-frac 79.7%

      \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. *-commutative 79.7%

      \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. times-frac 79.7%

      \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. difference-of-squares 89.1%

      \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. associate-/r* 90.0%

      \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. metadata-eval 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. sub-neg 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

   8. distribute-neg-frac 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

   9. metadata-eval 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Simplified 90.0%

   \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
4. Step-by-step derivation

   1. frac-add 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

   2. *-un-lft-identity 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

5. Applied egg-rr 90.0%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
6. Step-by-step derivation

   1. *-commutative 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

   2. neg-mul-1 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

   3. sub-neg 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

7. Simplified 90.0%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]

8. Taylor expanded in b around 0 55.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
9. Step-by-step derivation

   1. unpow2 55.4%

      \[\leadsto 0.5 \cdot \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]

   2. *-commutative 55.4%

      \[\leadsto 0.5 \cdot \frac{\pi}{\color{blue}{b \cdot \left(a \cdot a\right)}} \]

   3. associate-/r* 55.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{\pi}{b}}{a \cdot a}} \]

10. Simplified 55.4%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a}} \]

11. Final simplification 55.4%

    \[\leadsto 0.5 \cdot \frac{\frac{\pi}{b}}{a \cdot a} \]

Alternative 16: 56.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\frac{\frac{\pi}{a}}{a}}{b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* 0.5 (/ (/ (/ PI a) a) b)))

C

double code(double a, double b) {
	return 0.5 * (((((double) M_PI) / a) / a) / b);
}

Java

public static double code(double a, double b) {
	return 0.5 * (((Math.PI / a) / a) / b);
}

Python

def code(a, b):
	return 0.5 * (((math.pi / a) / a) / b)

Julia

function code(a, b)
	return Float64(0.5 * Float64(Float64(Float64(pi / a) / a) / b))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 * (((pi / a) / a) / b);
end

Wolfram

code[a_, b_] := N[(0.5 * N[(N[(N[(Pi / a), $MachinePrecision] / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. times-frac 79.7%

      \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. *-commutative 79.7%

      \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. times-frac 79.7%

      \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. difference-of-squares 89.1%

      \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. associate-/r* 90.0%

      \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. metadata-eval 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. sub-neg 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

   8. distribute-neg-frac 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

   9. metadata-eval 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

3. Simplified 90.0%

   \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
4. Step-by-step derivation

   1. frac-add 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

   2. *-un-lft-identity 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

5. Applied egg-rr 90.0%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
6. Step-by-step derivation

   1. *-commutative 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

   2. neg-mul-1 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

   3. sub-neg 90.0%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

7. Simplified 90.0%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]

8. Taylor expanded in b around 0 55.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
9. Step-by-step derivation

   1. unpow2 55.4%

      \[\leadsto 0.5 \cdot \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]

   2. associate-/r* 55.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{\pi}{a \cdot a}}{b}} \]

   3. associate-/r* 56.1%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{\frac{\pi}{a}}{a}}}{b} \]

10. Simplified 56.1%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{\frac{\pi}{a}}{a}}{b}} \]

11. Final simplification 56.1%

    \[\leadsto 0.5 \cdot \frac{\frac{\frac{\pi}{a}}{a}}{b} \]

Alternative 17: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* 0.5 (/ PI (* a (* a b)))))

C

double code(double a, double b) {
	return 0.5 * (((double) M_PI) / (a * (a * b)));
}

Java

public static double code(double a, double b) {
	return 0.5 * (Math.PI / (a * (a * b)));
}

Python

def code(a, b):
	return 0.5 * (math.pi / (a * (a * b)))

Julia

function code(a, b)
	return Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 * (pi / (a * (a * b)));
end

Wolfram

code[a_, b_] := N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Step-by-step derivation

   1. *-commutative 79.6%

      \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. associate-/r/ 79.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. associate-*l/ 79.7%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

   4. *-commutative 79.7%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

   5. associate-/r/ 79.7%

      \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

   6. times-frac 79.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

3. Simplified 79.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
4. Step-by-step derivation

   1. clear-num 79.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

   2. inv-pow 79.7%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

5. Applied egg-rr 79.7%

   \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
6. Step-by-step derivation

   1. unpow-1 79.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

   2. fma-def 79.7%

      \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

   3. +-commutative 79.7%

      \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

   4. associate-*r/ 79.7%

      \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

   5. *-commutative 79.7%

      \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

   6. associate-*r/ 79.7%

      \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

   7. mul-1-neg 79.7%

      \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

   8. unsub-neg 79.7%

      \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

7. Simplified 79.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
8. Step-by-step derivation

   1. div-sub 68.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

9. Applied egg-rr 68.0%

   \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
10. Step-by-step derivation

    1. div-sub 79.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

    2. difference-of-squares 89.0%

       \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

    3. *-commutative 89.0%

       \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

    4. associate-/l* 99.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]

11. Simplified 99.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]
12. Step-by-step derivation

    1. expm1-log1p-u 77.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)\right)} \cdot 0.5 \]

    2. expm1-udef 51.8%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}\right)} - 1\right)} \cdot 0.5 \]

    3. associate-/r/ 51.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}\right)} - 1\right) \cdot 0.5 \]

    4. +-commutative 51.8%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{\color{blue}{a + b}}\right)} - 1\right) \cdot 0.5 \]

13. Applied egg-rr 51.8%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)} - 1\right)} \cdot 0.5 \]
14. Step-by-step derivation

    1. expm1-def 77.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)\right)} \cdot 0.5 \]

    2. expm1-log1p 99.6%

       \[\leadsto \color{blue}{\left(\frac{1}{b - a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}\right)} \cdot 0.5 \]

    3. associate-*l/ 99.7%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{b - a}} \cdot 0.5 \]

    4. *-lft-identity 99.7%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}}{b - a} \cdot 0.5 \]

15. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{a + b}}{b - a}} \cdot 0.5 \]

16. Taylor expanded in a around inf 55.4%

    \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
17. Step-by-step derivation

    1. unpow2 55.4%

       \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

    2. associate-*l* 60.1%

       \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

18. Simplified 60.1%

    \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

19. Final simplification 60.1%

    \[\leadsto 0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)} \]

Reproduce

herbie shell --seed 2023208 
(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))))