NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.2% → 99.1%

Time: 10.9s

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: 79.2% 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: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. *-commutative 77.7%

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

   2. associate-/r/ 77.6%

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

   3. associate-*l/ 77.7%

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

   4. *-commutative 77.7%

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

   5. associate-/r/ 77.7%

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

   6. times-frac 77.7%

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

3. Simplified 77.7%

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

   1. clear-num 77.6%

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

   2. inv-pow 77.6%

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

5. Applied egg-rr 77.6%

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

   1. unpow-1 77.6%

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

   2. fma-def 77.6%

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

   3. +-commutative 77.6%

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

   4. associate-*r/ 77.6%

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

   5. *-commutative 77.6%

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

   6. associate-*r/ 77.6%

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

   7. mul-1-neg 77.6%

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

   8. unsub-neg 77.6%

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

7. Simplified 77.6%

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

   1. div-sub 69.4%

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

9. Applied egg-rr 69.4%

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

    1. div-sub 77.6%

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

    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* 99.3%

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

11. Simplified 99.3%

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

    1. inv-pow 99.3%

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

    2. associate-/r/ 99.3%

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

13. Applied egg-rr 99.3%

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

14. Final simplification 99.3%

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

Alternative 2: 99.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. *-commutative 77.7%

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

   2. associate-/r/ 77.6%

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

   3. associate-*l/ 77.7%

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

   4. *-commutative 77.7%

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

   5. associate-/r/ 77.7%

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

   6. times-frac 77.7%

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

3. Simplified 77.7%

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

   1. clear-num 77.6%

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

   2. inv-pow 77.6%

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

5. Applied egg-rr 77.6%

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

   1. unpow-1 77.6%

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

   2. fma-def 77.6%

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

   3. +-commutative 77.6%

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

   4. associate-*r/ 77.6%

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

   5. *-commutative 77.6%

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

   6. associate-*r/ 77.6%

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

   7. mul-1-neg 77.6%

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

   8. unsub-neg 77.6%

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

7. Simplified 77.6%

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

   1. div-sub 69.4%

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

9. Applied egg-rr 69.4%

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

    1. div-sub 77.6%

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

    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* 99.3%

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

11. Simplified 99.3%

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

    1. inv-pow 99.3%

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

    2. associate-/r/ 99.3%

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

13. Applied egg-rr 99.3%

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

14. Taylor expanded in b around 0 99.3%

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

    1. associate-/l* 99.3%

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

16. Simplified 99.3%

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

17. Final simplification 99.3%

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

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{b - a}{\frac{-\pi}{b \cdot a}}}\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{-233}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+151)
   (* 0.5 (/ 1.0 (/ (- b a) (/ (- PI) (* b a)))))
   (if (<= a -3.5e-233)
     (* (* 0.5 (/ (/ PI (+ b a)) (- b a))) (+ (/ 1.0 a) (/ -1.0 b)))
     (* (/ (/ PI a) b) (/ 0.5 b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+151) {
		tmp = 0.5 * (1.0 / ((b - a) / (-((double) M_PI) / (b * a))));
	} else if (a <= -3.5e-233) {
		tmp = (0.5 * ((((double) M_PI) / (b + a)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	} else {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+151) {
		tmp = 0.5 * (1.0 / ((b - a) / (-Math.PI / (b * a))));
	} else if (a <= -3.5e-233) {
		tmp = (0.5 * ((Math.PI / (b + a)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	} else {
		tmp = ((Math.PI / a) / b) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.5e+151:
		tmp = 0.5 * (1.0 / ((b - a) / (-math.pi / (b * a))))
	elif a <= -3.5e-233:
		tmp = (0.5 * ((math.pi / (b + a)) / (b - a))) * ((1.0 / a) + (-1.0 / b))
	else:
		tmp = ((math.pi / a) / b) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+151)
		tmp = Float64(0.5 * Float64(1.0 / Float64(Float64(b - a) / Float64(Float64(-pi) / Float64(b * a)))));
	elseif (a <= -3.5e-233)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / Float64(b + a)) / Float64(b - a))) * Float64(Float64(1.0 / a) + Float64(-1.0 / b)));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.5e+151)
		tmp = 0.5 * (1.0 / ((b - a) / (-pi / (b * a))));
	elseif (a <= -3.5e-233)
		tmp = (0.5 * ((pi / (b + a)) / (b - a))) * ((1.0 / a) + (-1.0 / b));
	else
		tmp = ((pi / a) / b) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+151], N[(0.5 * N[(1.0 / N[(N[(b - a), $MachinePrecision] / N[((-Pi) / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -3.5e-233], N[(N[(0.5 * N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e151

   1. Initial program 30.4%

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

      1. *-commutative 30.4%

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

      2. associate-/r/ 30.4%

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

      3. associate-*l/ 30.4%

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

      4. *-commutative 30.4%

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

      5. associate-/r/ 30.4%

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

      6. times-frac 30.4%

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

   3. Simplified 30.4%

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

      1. clear-num 30.4%

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

      2. inv-pow 30.4%

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

   5. Applied egg-rr 30.4%

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

      1. unpow-1 30.4%

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

      2. fma-def 30.4%

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

      3. +-commutative 30.4%

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

      4. associate-*r/ 30.4%

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

      5. *-commutative 30.4%

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

      6. associate-*r/ 30.4%

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

      7. mul-1-neg 30.4%

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

      8. unsub-neg 30.4%

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

   7. Simplified 30.4%

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

      1. div-sub 24.4%

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

   9. Applied egg-rr 24.4%

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

       1. div-sub 30.4%

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

       2. difference-of-squares 60.7%

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

       3. *-commutative 60.7%

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

       4. associate-/l* 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in a around inf 99.8%

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

       1. mul-1-neg 99.8%

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

   14. Simplified 99.8%

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

   if -1.5e151 < a < -3.49999999999999991e-233

   1. Initial program 88.1%

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

      1. times-frac 88.2%

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

      2. *-commutative 88.2%

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

      3. times-frac 88.2%

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

      4. difference-of-squares 94.2%

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

      5. associate-/r* 94.3%

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

      6. metadata-eval 94.3%

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

      7. sub-neg 94.3%

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

      8. distribute-neg-frac 94.3%

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

      9. metadata-eval 94.3%

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

   3. Simplified 94.3%

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

   if -3.49999999999999991e-233 < a

   1. Initial program 82.6%

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

      1. times-frac 82.6%

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

      2. *-commutative 82.6%

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

      3. times-frac 82.6%

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

      4. difference-of-squares 91.9%

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

      5. associate-/r* 92.6%

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

      6. metadata-eval 92.6%

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

      7. sub-neg 92.6%

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

      8. distribute-neg-frac 92.6%

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

      9. metadata-eval 92.6%

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

   3. Simplified 92.6%

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

      1. frac-add 92.7%

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

      2. *-un-lft-identity 92.7%

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

   5. Applied egg-rr 92.7%

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

      1. *-commutative 92.7%

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

      2. neg-mul-1 92.7%

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

      3. sub-neg 92.7%

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

   7. Simplified 92.7%

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

   8. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. unpow2 68.9%

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

      3. *-commutative 68.9%

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

      4. times-frac 69.0%

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

   10. Simplified 69.0%

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

       1. associate-*r/ 69.1%

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

   12. Applied egg-rr 69.1%

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

       1. times-frac 72.9%

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

   14. Applied egg-rr 72.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{b - a}{\frac{-\pi}{b \cdot a}}}\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{-233}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 4: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b}\\ \mathbf{if}\;a \leq -6.1 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{b - a}{\frac{-\pi}{b \cdot a}}}\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-113}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{b}\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{-205}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI a) b)))
   (if (<= a -6.1e-37)
     (* 0.5 (/ 1.0 (/ (- b a) (/ (- PI) (* b a)))))
     (if (<= a -2.4e-113)
       (* t_0 (/ 0.5 b))
       (if (<= a -5.6e-205)
         (* (* 0.5 (/ (/ PI (+ b a)) (- b a))) (/ -1.0 b))
         (* 0.5 (/ t_0 b)))))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / a) / b;
	double tmp;
	if (a <= -6.1e-37) {
		tmp = 0.5 * (1.0 / ((b - a) / (-((double) M_PI) / (b * a))));
	} else if (a <= -2.4e-113) {
		tmp = t_0 * (0.5 / b);
	} else if (a <= -5.6e-205) {
		tmp = (0.5 * ((((double) M_PI) / (b + a)) / (b - a))) * (-1.0 / b);
	} else {
		tmp = 0.5 * (t_0 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / a) / b;
	double tmp;
	if (a <= -6.1e-37) {
		tmp = 0.5 * (1.0 / ((b - a) / (-Math.PI / (b * a))));
	} else if (a <= -2.4e-113) {
		tmp = t_0 * (0.5 / b);
	} else if (a <= -5.6e-205) {
		tmp = (0.5 * ((Math.PI / (b + a)) / (b - a))) * (-1.0 / b);
	} else {
		tmp = 0.5 * (t_0 / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / a) / b
	tmp = 0
	if a <= -6.1e-37:
		tmp = 0.5 * (1.0 / ((b - a) / (-math.pi / (b * a))))
	elif a <= -2.4e-113:
		tmp = t_0 * (0.5 / b)
	elif a <= -5.6e-205:
		tmp = (0.5 * ((math.pi / (b + a)) / (b - a))) * (-1.0 / b)
	else:
		tmp = 0.5 * (t_0 / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / a) / b)
	tmp = 0.0
	if (a <= -6.1e-37)
		tmp = Float64(0.5 * Float64(1.0 / Float64(Float64(b - a) / Float64(Float64(-pi) / Float64(b * a)))));
	elseif (a <= -2.4e-113)
		tmp = Float64(t_0 * Float64(0.5 / b));
	elseif (a <= -5.6e-205)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / Float64(b + a)) / Float64(b - a))) * Float64(-1.0 / b));
	else
		tmp = Float64(0.5 * Float64(t_0 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / a) / b;
	tmp = 0.0;
	if (a <= -6.1e-37)
		tmp = 0.5 * (1.0 / ((b - a) / (-pi / (b * a))));
	elseif (a <= -2.4e-113)
		tmp = t_0 * (0.5 / b);
	elseif (a <= -5.6e-205)
		tmp = (0.5 * ((pi / (b + a)) / (b - a))) * (-1.0 / b);
	else
		tmp = 0.5 * (t_0 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[a, -6.1e-37], N[(0.5 * N[(1.0 / N[(N[(b - a), $MachinePrecision] / N[((-Pi) / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.4e-113], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -5.6e-205], N[(N[(0.5 * N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.1000000000000003e-37

   1. Initial program 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-/r/ 65.5%

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

      3. associate-*l/ 65.6%

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

      4. *-commutative 65.6%

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

      5. associate-/r/ 65.6%

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

      6. times-frac 65.6%

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

   3. Simplified 65.6%

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

      1. clear-num 65.5%

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

      2. inv-pow 65.5%

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

   5. Applied egg-rr 65.5%

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

      1. unpow-1 65.5%

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

      2. fma-def 65.5%

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

      3. +-commutative 65.5%

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

      4. associate-*r/ 65.5%

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

      5. *-commutative 65.5%

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

      6. associate-*r/ 65.5%

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

      7. mul-1-neg 65.5%

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

      8. unsub-neg 65.5%

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

   7. Simplified 65.5%

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

      1. div-sub 56.5%

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

   9. Applied egg-rr 56.5%

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

       1. div-sub 65.5%

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

       2. difference-of-squares 80.4%

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

       3. *-commutative 80.4%

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

       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. Taylor expanded in a around inf 92.8%

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

       1. mul-1-neg 92.8%

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

   14. Simplified 92.8%

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

   if -6.1000000000000003e-37 < a < -2.40000000000000012e-113

   1. Initial program 90.9%

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

      1. times-frac 90.9%

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

      2. *-commutative 90.9%

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

      3. times-frac 90.9%

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

      4. difference-of-squares 90.9%

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

      5. associate-/r* 91.1%

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

      6. metadata-eval 91.1%

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

      7. sub-neg 91.1%

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

      8. distribute-neg-frac 91.1%

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

      9. metadata-eval 91.1%

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

   3. Simplified 91.1%

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

      1. frac-add 90.9%

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

      2. *-un-lft-identity 90.9%

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

   5. Applied egg-rr 90.9%

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

      1. *-commutative 90.9%

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

      2. neg-mul-1 90.9%

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

      3. sub-neg 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in b around inf 74.2%

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

      1. associate-*r/ 74.2%

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

      2. unpow2 74.2%

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

      3. *-commutative 74.2%

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

      4. times-frac 74.3%

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

   10. Simplified 74.3%

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

       1. associate-*r/ 74.5%

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

   12. Applied egg-rr 74.5%

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

       1. times-frac 83.0%

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

   14. Applied egg-rr 83.0%

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

   if -2.40000000000000012e-113 < a < -5.59999999999999983e-205

   1. Initial program 82.8%

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

      1. times-frac 82.8%

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

      2. *-commutative 82.8%

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

      3. times-frac 82.8%

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

      4. difference-of-squares 99.5%

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

      5. associate-/r* 99.8%

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

      6. metadata-eval 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-neg-frac 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 51.4%

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

   if -5.59999999999999983e-205 < a

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/r/ 80.7%

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

      3. associate-*l/ 80.6%

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

      4. *-commutative 80.6%

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

      5. associate-/r/ 80.6%

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

      6. times-frac 80.6%

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

   3. Simplified 80.8%

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

      1. clear-num 80.5%

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

      2. inv-pow 80.5%

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

   5. Applied egg-rr 80.5%

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

      1. unpow-1 80.5%

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

      2. fma-def 80.5%

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

      3. +-commutative 80.5%

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

      4. associate-*r/ 80.5%

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

      5. *-commutative 80.5%

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

      6. associate-*r/ 80.5%

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

      7. mul-1-neg 80.5%

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

      8. unsub-neg 80.5%

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

   7. Simplified 80.5%

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

      1. div-sub 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 80.5%

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

       2. difference-of-squares 90.4%

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

       3. *-commutative 90.4%

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

       4. associate-/l* 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in b around inf 67.9%

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

       1. associate-/r* 68.1%

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

       2. unpow2 68.1%

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

       3. associate-/r* 73.6%

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

   14. Simplified 73.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.1 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{b - a}{\frac{-\pi}{b \cdot a}}}\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{-205}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 10^{+104}:\\ \;\;\;\;\left(\frac{0.5}{a} + \frac{-0.5}{b}\right) \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.32e-202)
   (* 0.5 (/ PI (* a (* b a))))
   (if (<= b 1e+104)
     (* (+ (/ 0.5 a) (/ -0.5 b)) (/ PI (- (* b b) (* a a))))
     (* (/ (/ PI a) b) (/ 0.5 b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.32e-202) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else if (b <= 1e+104) {
		tmp = ((0.5 / a) + (-0.5 / b)) * (((double) M_PI) / ((b * b) - (a * a)));
	} else {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.32e-202) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else if (b <= 1e+104) {
		tmp = ((0.5 / a) + (-0.5 / b)) * (Math.PI / ((b * b) - (a * a)));
	} else {
		tmp = ((Math.PI / a) / b) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.32e-202:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	elif b <= 1e+104:
		tmp = ((0.5 / a) + (-0.5 / b)) * (math.pi / ((b * b) - (a * a)))
	else:
		tmp = ((math.pi / a) / b) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.32e-202)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	elseif (b <= 1e+104)
		tmp = Float64(Float64(Float64(0.5 / a) + Float64(-0.5 / b)) * Float64(pi / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.32e-202)
		tmp = 0.5 * (pi / (a * (b * a)));
	elseif (b <= 1e+104)
		tmp = ((0.5 / a) + (-0.5 / b)) * (pi / ((b * b) - (a * a)));
	else
		tmp = ((pi / a) / b) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.32e-202], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+104], N[(N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.32000000000000009e-202

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-/r/ 76.7%

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

      3. associate-*l/ 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-/r/ 76.7%

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

      6. times-frac 76.7%

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

   3. Simplified 76.7%

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

      1. clear-num 76.5%

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

      2. inv-pow 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. unpow-1 76.5%

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

      2. fma-def 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/ 76.5%

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

      5. *-commutative 76.5%

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

      6. associate-*r/ 76.5%

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

      7. mul-1-neg 76.5%

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

      8. unsub-neg 76.5%

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

   7. Simplified 76.5%

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

      1. div-sub 70.2%

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

   9. Applied egg-rr 70.2%

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

       1. div-sub 76.5%

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

       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* 99.4%

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

   11. Simplified 99.4%

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

       1. inv-pow 99.4%

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

       2. associate-/r/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in b around 0 56.0%

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

       1. unpow2 56.0%

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

       2. associate-*l* 64.2%

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

   16. Simplified 64.2%

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

   if 1.32000000000000009e-202 < b < 1e104

   1. Initial program 89.8%

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

      1. times-frac 89.9%

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

      2. *-commutative 89.9%

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

      3. times-frac 89.9%

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

      4. difference-of-squares 89.9%

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

      5. associate-/r* 90.3%

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

      6. metadata-eval 90.3%

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

      7. sub-neg 90.3%

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

      8. distribute-neg-frac 90.3%

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

      9. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

      1. distribute-lft-in 84.8%

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

      2. associate-/l/ 84.7%

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

      3. associate-/l/ 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. distribute-lft-out 89.9%

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

      2. associate-*r* 89.9%

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

      3. associate-*l/ 89.9%

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

      4. *-commutative 89.9%

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

      5. difference-of-squares 90.0%

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

      6. associate-*l/ 89.9%

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

      7. distribute-lft-in 89.9%

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

      8. associate-*r/ 89.9%

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

      9. metadata-eval 89.9%

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

      10. associate-*r/ 89.9%

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

      11. metadata-eval 89.9%

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

   7. Simplified 89.9%

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

   if 1e104 < b

   1. Initial program 65.1%

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

      1. times-frac 65.1%

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

      2. *-commutative 65.1%

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

      3. times-frac 65.1%

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

      4. difference-of-squares 84.7%

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

      5. associate-/r* 84.7%

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

      6. metadata-eval 84.7%

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

      7. sub-neg 84.7%

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

      8. distribute-neg-frac 84.7%

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

      9. metadata-eval 84.7%

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

   3. Simplified 84.7%

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

      1. frac-add 84.7%

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

      2. *-un-lft-identity 84.7%

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

   5. Applied egg-rr 84.7%

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

      1. *-commutative 84.7%

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

      2. neg-mul-1 84.7%

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

      3. sub-neg 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in b around inf 84.7%

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

      1. associate-*r/ 84.7%

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

      2. unpow2 84.7%

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

      3. *-commutative 84.7%

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

      4. times-frac 84.7%

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

   10. Simplified 84.7%

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

       1. associate-*r/ 84.8%

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

   12. Applied egg-rr 84.8%

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

       1. times-frac 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-202}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 10^{+104}:\\ \;\;\;\;\left(\frac{0.5}{a} + \frac{-0.5}{b}\right) \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-205}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\pi \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.5e-205)
   (* 0.5 (/ PI (* a (* b a))))
   (if (<= b 5e+109)
     (/ (* PI (+ (/ 0.5 a) (/ -0.5 b))) (- (* b b) (* a a)))
     (* (/ (/ PI a) b) (/ 0.5 b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-205) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else if (b <= 5e+109) {
		tmp = (((double) M_PI) * ((0.5 / a) + (-0.5 / b))) / ((b * b) - (a * a));
	} else {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7.5e-205) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else if (b <= 5e+109) {
		tmp = (Math.PI * ((0.5 / a) + (-0.5 / b))) / ((b * b) - (a * a));
	} else {
		tmp = ((Math.PI / a) / b) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7.5e-205:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	elif b <= 5e+109:
		tmp = (math.pi * ((0.5 / a) + (-0.5 / b))) / ((b * b) - (a * a))
	else:
		tmp = ((math.pi / a) / b) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.5e-205)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	elseif (b <= 5e+109)
		tmp = Float64(Float64(pi * Float64(Float64(0.5 / a) + Float64(-0.5 / b))) / Float64(Float64(b * b) - Float64(a * a)));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7.5e-205)
		tmp = 0.5 * (pi / (a * (b * a)));
	elseif (b <= 5e+109)
		tmp = (pi * ((0.5 / a) + (-0.5 / b))) / ((b * b) - (a * a));
	else
		tmp = ((pi / a) / b) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.5e-205], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+109], N[(N[(Pi * N[(N[(0.5 / a), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.4999999999999996e-205

   1. Initial program 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-/r/ 76.7%

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

      3. associate-*l/ 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-/r/ 76.7%

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

      6. times-frac 76.7%

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

   3. Simplified 76.7%

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

      1. clear-num 76.5%

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

      2. inv-pow 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. unpow-1 76.5%

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

      2. fma-def 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/ 76.5%

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

      5. *-commutative 76.5%

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

      6. associate-*r/ 76.5%

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

      7. mul-1-neg 76.5%

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

      8. unsub-neg 76.5%

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

   7. Simplified 76.5%

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

      1. div-sub 70.2%

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

   9. Applied egg-rr 70.2%

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

       1. div-sub 76.5%

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

       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* 99.4%

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

   11. Simplified 99.4%

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

       1. inv-pow 99.4%

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

       2. associate-/r/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in b around 0 56.0%

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

       1. unpow2 56.0%

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

       2. associate-*l* 64.2%

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

   16. Simplified 64.2%

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

   if 7.4999999999999996e-205 < b < 5.0000000000000001e109

   1. Initial program 89.8%

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

      1. times-frac 89.9%

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

      2. *-commutative 89.9%

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

      3. times-frac 89.9%

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

      4. difference-of-squares 89.9%

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

      5. associate-/r* 90.3%

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

      6. metadata-eval 90.3%

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

      7. sub-neg 90.3%

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

      8. distribute-neg-frac 90.3%

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

      9. metadata-eval 90.3%

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

   3. Simplified 90.3%

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

      1. distribute-lft-in 84.8%

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

      2. associate-/l/ 84.7%

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

      3. associate-/l/ 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. distribute-lft-out 89.9%

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

      2. associate-*r* 89.9%

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

      3. associate-*l/ 89.9%

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

      4. *-commutative 89.9%

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

      5. difference-of-squares 90.0%

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

      6. associate-*l/ 89.9%

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

      7. distribute-lft-in 89.9%

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

      8. associate-*r/ 89.9%

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

      9. metadata-eval 89.9%

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

      10. associate-*r/ 89.9%

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

      11. metadata-eval 89.9%

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

   7. Simplified 89.9%

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

      1. associate-*l/ 90.0%

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

   9. Applied egg-rr 90.0%

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

   if 5.0000000000000001e109 < b

   1. Initial program 65.1%

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

      1. times-frac 65.1%

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

      2. *-commutative 65.1%

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

      3. times-frac 65.1%

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

      4. difference-of-squares 84.7%

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

      5. associate-/r* 84.7%

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

      6. metadata-eval 84.7%

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

      7. sub-neg 84.7%

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

      8. distribute-neg-frac 84.7%

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

      9. metadata-eval 84.7%

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

   3. Simplified 84.7%

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

      1. frac-add 84.7%

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

      2. *-un-lft-identity 84.7%

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

   5. Applied egg-rr 84.7%

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

      1. *-commutative 84.7%

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

      2. neg-mul-1 84.7%

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

      3. sub-neg 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in b around inf 84.7%

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

      1. associate-*r/ 84.7%

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

      2. unpow2 84.7%

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

      3. *-commutative 84.7%

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

      4. times-frac 84.7%

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

   10. Simplified 84.7%

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

       1. associate-*r/ 84.8%

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

   12. Applied egg-rr 84.8%

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

       1. times-frac 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-205}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{\pi \cdot \left(\frac{0.5}{a} + \frac{-0.5}{b}\right)}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 7: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b}\\ t_1 := 0.5 \cdot \frac{1}{\frac{b - a}{\frac{-\pi}{b \cdot a}}}\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-113}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{b}\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI a) b))
        (t_1 (* 0.5 (/ 1.0 (/ (- b a) (/ (- PI) (* b a)))))))
   (if (<= a -1.45e-36)
     t_1
     (if (<= a -2.8e-113)
       (* t_0 (/ 0.5 b))
       (if (<= a -5.6e-205) t_1 (* 0.5 (/ t_0 b)))))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / a) / b;
	double t_1 = 0.5 * (1.0 / ((b - a) / (-((double) M_PI) / (b * a))));
	double tmp;
	if (a <= -1.45e-36) {
		tmp = t_1;
	} else if (a <= -2.8e-113) {
		tmp = t_0 * (0.5 / b);
	} else if (a <= -5.6e-205) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (t_0 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / a) / b;
	double t_1 = 0.5 * (1.0 / ((b - a) / (-Math.PI / (b * a))));
	double tmp;
	if (a <= -1.45e-36) {
		tmp = t_1;
	} else if (a <= -2.8e-113) {
		tmp = t_0 * (0.5 / b);
	} else if (a <= -5.6e-205) {
		tmp = t_1;
	} else {
		tmp = 0.5 * (t_0 / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / a) / b
	t_1 = 0.5 * (1.0 / ((b - a) / (-math.pi / (b * a))))
	tmp = 0
	if a <= -1.45e-36:
		tmp = t_1
	elif a <= -2.8e-113:
		tmp = t_0 * (0.5 / b)
	elif a <= -5.6e-205:
		tmp = t_1
	else:
		tmp = 0.5 * (t_0 / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / a) / b)
	t_1 = Float64(0.5 * Float64(1.0 / Float64(Float64(b - a) / Float64(Float64(-pi) / Float64(b * a)))))
	tmp = 0.0
	if (a <= -1.45e-36)
		tmp = t_1;
	elseif (a <= -2.8e-113)
		tmp = Float64(t_0 * Float64(0.5 / b));
	elseif (a <= -5.6e-205)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(t_0 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / a) / b;
	t_1 = 0.5 * (1.0 / ((b - a) / (-pi / (b * a))));
	tmp = 0.0;
	if (a <= -1.45e-36)
		tmp = t_1;
	elseif (a <= -2.8e-113)
		tmp = t_0 * (0.5 / b);
	elseif (a <= -5.6e-205)
		tmp = t_1;
	else
		tmp = 0.5 * (t_0 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(1.0 / N[(N[(b - a), $MachinePrecision] / N[((-Pi) / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.45e-36], t$95$1, If[LessEqual[a, -2.8e-113], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -5.6e-205], t$95$1, N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45000000000000006e-36 or -2.8e-113 < a < -5.59999999999999983e-205

   1. Initial program 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-/r/ 69.2%

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

      3. associate-*l/ 69.3%

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

      4. *-commutative 69.3%

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

      5. associate-/r/ 69.3%

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

      6. times-frac 69.3%

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

   3. Simplified 69.2%

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

      1. clear-num 69.2%

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

      2. inv-pow 69.2%

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

   5. Applied egg-rr 69.2%

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

      1. unpow-1 69.2%

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

      2. fma-def 69.2%

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

      3. +-commutative 69.2%

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

      4. associate-*r/ 69.2%

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

      5. *-commutative 69.2%

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

      6. associate-*r/ 69.2%

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

      7. mul-1-neg 69.2%

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

      8. unsub-neg 69.2%

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

   7. Simplified 69.2%

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

      1. div-sub 63.3%

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

   9. Applied egg-rr 63.3%

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

       1. div-sub 69.2%

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

       2. difference-of-squares 84.5%

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

       3. *-commutative 84.5%

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

       4. associate-/l* 99.6%

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

   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. Taylor expanded in a around inf 82.7%

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

       1. mul-1-neg 82.7%

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

   14. Simplified 82.7%

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

   if -1.45000000000000006e-36 < a < -2.8e-113

   1. Initial program 90.9%

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

      1. times-frac 90.9%

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

      2. *-commutative 90.9%

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

      3. times-frac 90.9%

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

      4. difference-of-squares 90.9%

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

      5. associate-/r* 91.1%

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

      6. metadata-eval 91.1%

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

      7. sub-neg 91.1%

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

      8. distribute-neg-frac 91.1%

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

      9. metadata-eval 91.1%

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

   3. Simplified 91.1%

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

      1. frac-add 90.9%

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

      2. *-un-lft-identity 90.9%

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

   5. Applied egg-rr 90.9%

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

      1. *-commutative 90.9%

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

      2. neg-mul-1 90.9%

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

      3. sub-neg 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in b around inf 74.2%

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

      1. associate-*r/ 74.2%

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

      2. unpow2 74.2%

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

      3. *-commutative 74.2%

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

      4. times-frac 74.3%

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

   10. Simplified 74.3%

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

       1. associate-*r/ 74.5%

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

   12. Applied egg-rr 74.5%

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

       1. times-frac 83.0%

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

   14. Applied egg-rr 83.0%

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

   if -5.59999999999999983e-205 < a

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/r/ 80.7%

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

      3. associate-*l/ 80.6%

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

      4. *-commutative 80.6%

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

      5. associate-/r/ 80.6%

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

      6. times-frac 80.6%

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

   3. Simplified 80.8%

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

      1. clear-num 80.5%

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

      2. inv-pow 80.5%

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

   5. Applied egg-rr 80.5%

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

      1. unpow-1 80.5%

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

      2. fma-def 80.5%

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

      3. +-commutative 80.5%

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

      4. associate-*r/ 80.5%

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

      5. *-commutative 80.5%

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

      6. associate-*r/ 80.5%

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

      7. mul-1-neg 80.5%

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

      8. unsub-neg 80.5%

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

   7. Simplified 80.5%

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

      1. div-sub 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. div-sub 80.5%

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

       2. difference-of-squares 90.4%

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

       3. *-commutative 90.4%

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

       4. associate-/l* 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in b around inf 67.9%

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

       1. associate-/r* 68.1%

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

       2. unpow2 68.1%

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

       3. associate-/r* 73.6%

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

   14. Simplified 73.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-36}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{b - a}{\frac{-\pi}{b \cdot a}}}\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{-205}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{b - a}{\frac{-\pi}{b \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 8: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-195}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.2e-195)
   (* 0.5 (/ PI (* a (* b a))))
   (if (<= b 1e+135)
     (* (* 0.5 (/ (/ PI (+ b a)) (- b a))) (/ 1.0 a))
     (* (/ (/ PI a) b) (/ 0.5 b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.2e-195) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else if (b <= 1e+135) {
		tmp = (0.5 * ((((double) M_PI) / (b + a)) / (b - a))) * (1.0 / a);
	} else {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.2e-195) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else if (b <= 1e+135) {
		tmp = (0.5 * ((Math.PI / (b + a)) / (b - a))) * (1.0 / a);
	} else {
		tmp = ((Math.PI / a) / b) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.2e-195:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	elif b <= 1e+135:
		tmp = (0.5 * ((math.pi / (b + a)) / (b - a))) * (1.0 / a)
	else:
		tmp = ((math.pi / a) / b) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.2e-195)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	elseif (b <= 1e+135)
		tmp = Float64(Float64(0.5 * Float64(Float64(pi / Float64(b + a)) / Float64(b - a))) * Float64(1.0 / a));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.2e-195)
		tmp = 0.5 * (pi / (a * (b * a)));
	elseif (b <= 1e+135)
		tmp = (0.5 * ((pi / (b + a)) / (b - a))) * (1.0 / a);
	else
		tmp = ((pi / a) / b) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.2e-195], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+135], N[(N[(0.5 * N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 5.2000000000000003e-195

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-/r/ 76.8%

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

      3. associate-*l/ 76.9%

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

      4. *-commutative 76.9%

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

      5. associate-/r/ 76.9%

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

      6. times-frac 76.9%

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

   3. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   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 76.6%

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

      3. +-commutative 76.6%

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

      4. associate-*r/ 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 70.4%

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

   9. Applied egg-rr 70.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   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 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* 99.4%

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

   11. Simplified 99.4%

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

       1. inv-pow 99.4%

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

       2. associate-/r/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in b around 0 56.3%

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

       1. unpow2 56.3%

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

       2. associate-*l* 64.4%

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

   16. Simplified 64.4%

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

   if 5.2000000000000003e-195 < b < 9.99999999999999962e134

   1. Initial program 90.3%

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

      1. times-frac 90.4%

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

      2. *-commutative 90.4%

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

      3. times-frac 90.4%

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

      4. difference-of-squares 90.4%

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

      5. associate-/r* 90.8%

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

      6. metadata-eval 90.8%

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

      7. sub-neg 90.8%

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

      8. distribute-neg-frac 90.8%

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

      9. metadata-eval 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in a around 0 67.7%

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

   if 9.99999999999999962e134 < b

   1. Initial program 61.5%

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

      1. times-frac 61.5%

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

      2. *-commutative 61.5%

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

      3. times-frac 61.5%

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

      4. difference-of-squares 83.1%

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

      5. associate-/r* 83.1%

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

      6. metadata-eval 83.1%

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

      7. sub-neg 83.1%

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

      8. distribute-neg-frac 83.1%

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

      9. metadata-eval 83.1%

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

   3. Simplified 83.1%

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

      1. frac-add 83.1%

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

      2. *-un-lft-identity 83.1%

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

   5. Applied egg-rr 83.1%

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

      1. *-commutative 83.1%

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

      2. neg-mul-1 83.1%

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

      3. sub-neg 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in b around inf 83.1%

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

      1. associate-*r/ 83.1%

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

      2. unpow2 83.1%

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

      3. *-commutative 83.1%

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

      4. times-frac 83.1%

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

   10. Simplified 83.1%

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

       1. associate-*r/ 83.1%

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

   12. Applied egg-rr 83.1%

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

       1. times-frac 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-195}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 10^{+135}:\\ \;\;\;\;\left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 9: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;b \leq 4.5 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ PI (* a (* b a))))))
   (if (<= b 4.5e-152)
     t_0
     (if (<= b 1.2e-87)
       (* (/ -0.5 b) (/ PI (- (* b b) (* a a))))
       (if (<= b 1.05e-29) t_0 (* 0.5 (/ (/ (/ PI a) b) b)))))))

C

double code(double a, double b) {
	double t_0 = 0.5 * (((double) M_PI) / (a * (b * a)));
	double tmp;
	if (b <= 4.5e-152) {
		tmp = t_0;
	} else if (b <= 1.2e-87) {
		tmp = (-0.5 / b) * (((double) M_PI) / ((b * b) - (a * a)));
	} else if (b <= 1.05e-29) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (((((double) M_PI) / a) / b) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 * (Math.PI / (a * (b * a)));
	double tmp;
	if (b <= 4.5e-152) {
		tmp = t_0;
	} else if (b <= 1.2e-87) {
		tmp = (-0.5 / b) * (Math.PI / ((b * b) - (a * a)));
	} else if (b <= 1.05e-29) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (((Math.PI / a) / b) / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 * (math.pi / (a * (b * a)))
	tmp = 0
	if b <= 4.5e-152:
		tmp = t_0
	elif b <= 1.2e-87:
		tmp = (-0.5 / b) * (math.pi / ((b * b) - (a * a)))
	elif b <= 1.05e-29:
		tmp = t_0
	else:
		tmp = 0.5 * (((math.pi / a) / b) / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))))
	tmp = 0.0
	if (b <= 4.5e-152)
		tmp = t_0;
	elseif (b <= 1.2e-87)
		tmp = Float64(Float64(-0.5 / b) * Float64(pi / Float64(Float64(b * b) - Float64(a * a))));
	elseif (b <= 1.05e-29)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(pi / a) / b) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 * (pi / (a * (b * a)));
	tmp = 0.0;
	if (b <= 4.5e-152)
		tmp = t_0;
	elseif (b <= 1.2e-87)
		tmp = (-0.5 / b) * (pi / ((b * b) - (a * a)));
	elseif (b <= 1.05e-29)
		tmp = t_0;
	else
		tmp = 0.5 * (((pi / a) / b) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.5e-152], t$95$0, If[LessEqual[b, 1.2e-87], N[(N[(-0.5 / b), $MachinePrecision] * N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-29], t$95$0, N[(0.5 * N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.5000000000000004e-152 or 1.2e-87 < b < 1.04999999999999995e-29

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-/r/ 77.1%

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

      3. associate-*l/ 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-/r/ 77.1%

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

      6. times-frac 77.1%

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

   3. Simplified 77.2%

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

      1. clear-num 76.9%

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

      2. inv-pow 76.9%

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

   5. Applied egg-rr 76.9%

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

      1. unpow-1 76.9%

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

      2. fma-def 76.9%

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

      3. +-commutative 76.9%

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

      4. associate-*r/ 76.9%

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

      5. *-commutative 76.9%

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

      6. associate-*r/ 76.9%

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

      7. mul-1-neg 76.9%

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

      8. unsub-neg 76.9%

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

   7. Simplified 76.9%

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

      1. div-sub 71.2%

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

   9. Applied egg-rr 71.2%

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

       1. div-sub 76.9%

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

       2. difference-of-squares 88.4%

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

       3. *-commutative 88.4%

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

       4. associate-/l* 99.4%

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

   11. Simplified 99.4%

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

       1. inv-pow 99.4%

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

       2. associate-/r/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in b around 0 56.7%

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

       1. unpow2 56.7%

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

       2. associate-*l* 65.6%

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

   16. Simplified 65.6%

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

   if 4.5000000000000004e-152 < b < 1.2e-87

   1. Initial program 83.2%

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

      1. times-frac 83.3%

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

      2. *-commutative 83.3%

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

      3. times-frac 83.3%

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

      4. difference-of-squares 83.3%

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

      5. associate-/r* 84.7%

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

      6. metadata-eval 84.7%

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

      7. sub-neg 84.7%

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

      8. distribute-neg-frac 84.7%

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

      9. metadata-eval 84.7%

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

   3. Simplified 84.7%

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

      1. distribute-lft-in 72.1%

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

      2. associate-/l/ 72.1%

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

      3. associate-/l/ 70.9%

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

   5. Applied egg-rr 70.9%

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

      1. distribute-lft-out 83.3%

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

      2. associate-*r* 83.3%

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

      3. associate-*l/ 83.3%

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

      4. *-commutative 83.3%

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

      5. difference-of-squares 83.3%

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

      6. associate-*l/ 83.3%

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

      7. distribute-lft-in 83.3%

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

      8. associate-*r/ 83.3%

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

      9. metadata-eval 83.3%

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

      10. associate-*r/ 83.3%

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

      11. metadata-eval 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in a around inf 64.1%

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

   if 1.04999999999999995e-29 < b

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-/r/ 77.7%

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

      3. associate-*l/ 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-/r/ 77.8%

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

      6. times-frac 77.8%

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

   3. Simplified 77.9%

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

      1. clear-num 77.9%

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

      2. inv-pow 77.9%

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

   5. Applied egg-rr 77.9%

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

      1. unpow-1 77.9%

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

      2. fma-def 77.9%

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

      3. +-commutative 77.9%

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

      4. associate-*r/ 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*r/ 77.9%

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

      7. mul-1-neg 77.9%

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

      8. unsub-neg 77.9%

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

   7. Simplified 77.9%

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

      1. div-sub 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. div-sub 77.9%

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

       2. difference-of-squares 90.1%

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

       3. *-commutative 90.1%

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

       4. associate-/l* 98.7%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in b around inf 80.1%

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

       1. associate-/r* 80.1%

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

       2. unpow2 80.1%

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

       3. associate-/r* 89.7%

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

   14. Simplified 89.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-152}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 10: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\pi}{a}}{b}\\ \mathbf{if}\;b \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{a}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ PI a) b)))
   (if (<= b 1.02e-29) (* 0.5 (/ 1.0 (/ a t_0))) (* 0.5 (/ t_0 b)))))

C

double code(double a, double b) {
	double t_0 = (((double) M_PI) / a) / b;
	double tmp;
	if (b <= 1.02e-29) {
		tmp = 0.5 * (1.0 / (a / t_0));
	} else {
		tmp = 0.5 * (t_0 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.PI / a) / b;
	double tmp;
	if (b <= 1.02e-29) {
		tmp = 0.5 * (1.0 / (a / t_0));
	} else {
		tmp = 0.5 * (t_0 / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pi / a) / b
	tmp = 0
	if b <= 1.02e-29:
		tmp = 0.5 * (1.0 / (a / t_0))
	else:
		tmp = 0.5 * (t_0 / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(pi / a) / b)
	tmp = 0.0
	if (b <= 1.02e-29)
		tmp = Float64(0.5 * Float64(1.0 / Float64(a / t_0)));
	else
		tmp = Float64(0.5 * Float64(t_0 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (pi / a) / b;
	tmp = 0.0;
	if (b <= 1.02e-29)
		tmp = 0.5 * (1.0 / (a / t_0));
	else
		tmp = 0.5 * (t_0 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, 1.02e-29], N[(0.5 * N[(1.0 / N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.01999999999999994e-29

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/r/ 77.6%

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

      3. associate-*l/ 77.6%

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

      4. *-commutative 77.6%

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

      5. associate-/r/ 77.6%

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

      6. times-frac 77.6%

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

   3. Simplified 77.7%

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

      1. clear-num 77.5%

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

      2. inv-pow 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. unpow-1 77.5%

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

      2. fma-def 77.5%

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

      3. +-commutative 77.5%

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

      4. associate-*r/ 77.5%

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

      5. *-commutative 77.5%

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

      6. associate-*r/ 77.5%

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

      7. mul-1-neg 77.5%

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

      8. unsub-neg 77.5%

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

   7. Simplified 77.5%

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

   8. Taylor expanded in b around 0 56.2%

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

      1. unpow2 56.2%

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

      2. *-commutative 56.2%

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

   10. Simplified 56.2%

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

   11. Taylor expanded in b around 0 56.2%

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

       1. associate-/l* 56.2%

          \[\leadsto \frac{1}{\color{blue}{\frac{{a}^{2}}{\frac{\pi}{b}}}} \cdot 0.5 \]

       2. unpow2 56.2%

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

       3. associate-/l* 65.8%

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

       4. associate-/r* 65.8%

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

       5. *-commutative 65.8%

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

       6. associate-/r* 65.8%

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

   13. Simplified 65.8%

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

   if 1.01999999999999994e-29 < b

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-/r/ 77.7%

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

      3. associate-*l/ 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-/r/ 77.8%

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

      6. times-frac 77.8%

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

   3. Simplified 77.9%

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

      1. clear-num 77.9%

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

      2. inv-pow 77.9%

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

   5. Applied egg-rr 77.9%

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

      1. unpow-1 77.9%

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

      2. fma-def 77.9%

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

      3. +-commutative 77.9%

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

      4. associate-*r/ 77.9%

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

      5. *-commutative 77.9%

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

      6. associate-*r/ 77.9%

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

      7. mul-1-neg 77.9%

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

      8. unsub-neg 77.9%

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

   7. Simplified 77.9%

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

      1. div-sub 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. div-sub 77.9%

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

       2. difference-of-squares 90.1%

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

       3. *-commutative 90.1%

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

       4. associate-/l* 98.7%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in b around inf 80.1%

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

       1. associate-/r* 80.1%

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

       2. unpow2 80.1%

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

       3. associate-/r* 89.7%

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

   14. Simplified 89.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{a}{\frac{\frac{\pi}{a}}{b}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 11: 67.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.02e-29)
   (* PI (/ 0.5 (* b (* a a))))
   (* (/ PI a) (/ 0.5 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.02e-29) {
		tmp = ((double) M_PI) * (0.5 / (b * (a * a)));
	} else {
		tmp = (((double) M_PI) / a) * (0.5 / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.02e-29) {
		tmp = Math.PI * (0.5 / (b * (a * a)));
	} else {
		tmp = (Math.PI / a) * (0.5 / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.02e-29:
		tmp = math.pi * (0.5 / (b * (a * a)))
	else:
		tmp = (math.pi / a) * (0.5 / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.02e-29)
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(a * a))));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.02e-29)
		tmp = pi * (0.5 / (b * (a * a)));
	else
		tmp = (pi / a) * (0.5 / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.02e-29], N[(Pi * N[(0.5 / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.01999999999999994e-29

   1. Initial program 77.6%

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

   2. Taylor expanded in b around 0 56.2%

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

      1. associate-*r/ 56.2%

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

      2. unpow2 56.2%

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

   4. Simplified 56.2%

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

   5. Taylor expanded in a around 0 56.2%

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

      1. associate-*r/ 56.2%

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

      2. *-commutative 56.2%

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

      3. *-rgt-identity 56.2%

         \[\leadsto \frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot 1}}{{a}^{2} \cdot b} \]

      4. *-commutative 56.2%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot 1}{\color{blue}{b \cdot {a}^{2}}} \]

      5. unpow2 56.2%

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

      6. associate-*r/ 56.2%

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

      7. associate-*l* 56.2%

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

      8. associate-*r/ 56.2%

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

      9. metadata-eval 56.2%

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

      10. unpow2 56.2%

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

      11. *-commutative 56.2%

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

      12. unpow2 56.2%

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

   7. Simplified 56.2%

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

   if 1.01999999999999994e-29 < b

   1. Initial program 77.8%

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

      1. times-frac 77.8%

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

      2. *-commutative 77.8%

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

      3. times-frac 77.8%

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

      4. difference-of-squares 90.1%

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

      5. associate-/r* 90.2%

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

      6. metadata-eval 90.2%

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

      7. sub-neg 90.2%

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

      8. distribute-neg-frac 90.2%

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

      9. metadata-eval 90.2%

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

   3. Simplified 90.2%

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

      1. frac-add 90.2%

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

      2. *-un-lft-identity 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. *-commutative 90.2%

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

      2. neg-mul-1 90.2%

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

      3. sub-neg 90.2%

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

   7. Simplified 90.2%

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

   8. Taylor expanded in b around inf 80.1%

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

      1. associate-*r/ 80.1%

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

      2. unpow2 80.1%

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

      3. *-commutative 80.1%

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

      4. times-frac 80.1%

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

   10. Simplified 80.1%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 12: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7e-30) (* 0.5 (/ PI (* a (* b a)))) (* (/ PI a) (/ 0.5 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7e-30) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else {
		tmp = (((double) M_PI) / a) * (0.5 / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7e-30) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else {
		tmp = (Math.PI / a) * (0.5 / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7e-30:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	else:
		tmp = (math.pi / a) * (0.5 / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7e-30)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	else
		tmp = Float64(Float64(pi / a) * Float64(0.5 / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7e-30)
		tmp = 0.5 * (pi / (a * (b * a)));
	else
		tmp = (pi / a) * (0.5 / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7e-30], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.0000000000000006e-30

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/r/ 77.6%

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

      3. associate-*l/ 77.6%

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

      4. *-commutative 77.6%

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

      5. associate-/r/ 77.6%

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

      6. times-frac 77.6%

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

   3. Simplified 77.7%

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

      1. clear-num 77.5%

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

      2. inv-pow 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. unpow-1 77.5%

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

      2. fma-def 77.5%

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

      3. +-commutative 77.5%

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

      4. associate-*r/ 77.5%

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

      5. *-commutative 77.5%

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

      6. associate-*r/ 77.5%

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

      7. mul-1-neg 77.5%

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

      8. unsub-neg 77.5%

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

   7. Simplified 77.5%

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

      1. div-sub 69.6%

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

   9. Applied egg-rr 69.6%

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

       1. div-sub 77.5%

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

       2. difference-of-squares 87.9%

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

       3. *-commutative 87.9%

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

       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. inv-pow 99.5%

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

       2. associate-/r/ 99.5%

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

   13. Applied egg-rr 99.5%

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

   14. Taylor expanded in b around 0 56.2%

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

       1. unpow2 56.2%

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

       2. associate-*l* 65.4%

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

   16. Simplified 65.4%

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

   if 7.0000000000000006e-30 < b

   1. Initial program 77.8%

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

      1. times-frac 77.8%

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

      2. *-commutative 77.8%

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

      3. times-frac 77.8%

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

      4. difference-of-squares 90.1%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 90.2%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 90.2%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. frac-add 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

      2. *-un-lft-identity 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

   5. Applied egg-rr 90.2%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
   6. Step-by-step derivation

      1. *-commutative 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

      2. neg-mul-1 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

      3. sub-neg 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

   7. Simplified 90.2%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]

   8. Taylor expanded in b around inf 80.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. unpow2 80.1%

         \[\leadsto \frac{0.5 \cdot \pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \]

      3. *-commutative 80.1%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot \left(b \cdot b\right)} \]

      4. times-frac 80.1%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]

   10. Simplified 80.1%

       \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}\\ \end{array} \]

Alternative 13: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.2e-30) (* 0.5 (/ PI (* a (* b a)))) (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.2e-30) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.2e-30) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.2e-30:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.2e-30)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	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 (b <= 6.2e-30)
		tmp = 0.5 * (pi / (a * (b * a)));
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.2e-30], N[(0.5 * N[(Pi / N[(a * N[(b * a), $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 b < 6.19999999999999982e-30

   1. Initial program 77.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 77.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 77.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 77.6%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 77.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 77.5%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 77.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. div-sub 69.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   9. Applied egg-rr 69.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   10. Step-by-step derivation

       1. div-sub 77.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

       2. difference-of-squares 87.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       3. *-commutative 87.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       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. inv-pow 99.5%

          \[\leadsto \color{blue}{{\left(\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}\right)}^{-1}} \cdot 0.5 \]

       2. associate-/r/ 99.5%

          \[\leadsto {\color{blue}{\left(\frac{b - a}{\frac{\pi}{a} - \frac{\pi}{b}} \cdot \left(b + a\right)\right)}}^{-1} \cdot 0.5 \]

   13. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{{\left(\frac{b - a}{\frac{\pi}{a} - \frac{\pi}{b}} \cdot \left(b + a\right)\right)}^{-1}} \cdot 0.5 \]

   14. Taylor expanded in b around 0 56.2%

       \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   15. Step-by-step derivation

       1. unpow2 56.2%

          \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

       2. associate-*l* 65.4%

          \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   16. Simplified 65.4%

       \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   if 6.19999999999999982e-30 < b

   1. Initial program 77.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 77.8%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 77.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 77.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 77.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 77.8%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 77.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]

   4. Taylor expanded in b around inf 80.1%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 80.1%

         \[\leadsto \frac{\pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \cdot 0.5 \]

   6. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(b \cdot b\right)}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 14: 75.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.4e-30) (* 0.5 (/ PI (* a (* b a)))) (* 0.5 (/ PI (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.4e-30) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.4e-30) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else {
		tmp = 0.5 * (Math.PI / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.4e-30:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	else:
		tmp = 0.5 * (math.pi / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.4e-30)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.4e-30)
		tmp = 0.5 * (pi / (a * (b * a)));
	else
		tmp = 0.5 * (pi / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.4e-30], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.39999999999999967e-30

   1. Initial program 77.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 77.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 77.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 77.6%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 77.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 77.5%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 77.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. div-sub 69.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   9. Applied egg-rr 69.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   10. Step-by-step derivation

       1. div-sub 77.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

       2. difference-of-squares 87.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       3. *-commutative 87.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       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. inv-pow 99.5%

          \[\leadsto \color{blue}{{\left(\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}\right)}^{-1}} \cdot 0.5 \]

       2. associate-/r/ 99.5%

          \[\leadsto {\color{blue}{\left(\frac{b - a}{\frac{\pi}{a} - \frac{\pi}{b}} \cdot \left(b + a\right)\right)}}^{-1} \cdot 0.5 \]

   13. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{{\left(\frac{b - a}{\frac{\pi}{a} - \frac{\pi}{b}} \cdot \left(b + a\right)\right)}^{-1}} \cdot 0.5 \]

   14. Taylor expanded in b around 0 56.2%

       \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   15. Step-by-step derivation

       1. unpow2 56.2%

          \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

       2. associate-*l* 65.4%

          \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   16. Simplified 65.4%

       \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   if 4.39999999999999967e-30 < b

   1. Initial program 77.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 77.8%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 77.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 77.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 77.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 77.8%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 77.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]

   4. Taylor expanded in b around inf 80.1%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   5. Step-by-step derivation

      1. unpow2 80.1%

         \[\leadsto \frac{\pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \cdot 0.5 \]

   6. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(b \cdot b\right)}} \cdot 0.5 \]

   7. Taylor expanded in a around 0 80.1%

      \[\leadsto \frac{\pi}{\color{blue}{a \cdot {b}^{2}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. unpow2 80.1%

         \[\leadsto \frac{\pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \cdot 0.5 \]

      2. *-commutative 80.1%

         \[\leadsto \frac{\pi}{\color{blue}{\left(b \cdot b\right) \cdot a}} \cdot 0.5 \]

      3. associate-*l* 88.6%

         \[\leadsto \frac{\pi}{\color{blue}{b \cdot \left(b \cdot a\right)}} \cdot 0.5 \]

   9. Simplified 88.6%

      \[\leadsto \frac{\pi}{\color{blue}{b \cdot \left(b \cdot a\right)}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-30}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]

Alternative 15: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e-29) (* 0.5 (/ PI (* a (* b a)))) (* (/ (/ PI a) b) (/ 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e-29) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else {
		tmp = ((((double) M_PI) / a) / b) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1e-29) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else {
		tmp = ((Math.PI / a) / b) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1e-29:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	else:
		tmp = ((math.pi / a) / b) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e-29)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	else
		tmp = Float64(Float64(Float64(pi / a) / b) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1e-29)
		tmp = 0.5 * (pi / (a * (b * a)));
	else
		tmp = ((pi / a) / b) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e-29], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999943e-30

   1. Initial program 77.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 77.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 77.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 77.6%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 77.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 77.5%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 77.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. div-sub 69.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   9. Applied egg-rr 69.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   10. Step-by-step derivation

       1. div-sub 77.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

       2. difference-of-squares 87.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       3. *-commutative 87.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       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. inv-pow 99.5%

          \[\leadsto \color{blue}{{\left(\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}\right)}^{-1}} \cdot 0.5 \]

       2. associate-/r/ 99.5%

          \[\leadsto {\color{blue}{\left(\frac{b - a}{\frac{\pi}{a} - \frac{\pi}{b}} \cdot \left(b + a\right)\right)}}^{-1} \cdot 0.5 \]

   13. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{{\left(\frac{b - a}{\frac{\pi}{a} - \frac{\pi}{b}} \cdot \left(b + a\right)\right)}^{-1}} \cdot 0.5 \]

   14. Taylor expanded in b around 0 56.2%

       \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   15. Step-by-step derivation

       1. unpow2 56.2%

          \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

       2. associate-*l* 65.4%

          \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   16. Simplified 65.4%

       \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   if 9.99999999999999943e-30 < b

   1. Initial program 77.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. times-frac 77.8%

         \[\leadsto \color{blue}{\frac{\pi \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. *-commutative 77.8%

         \[\leadsto \frac{\pi \cdot 1}{\color{blue}{\left(b \cdot b - a \cdot a\right) \cdot 2}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. times-frac 77.8%

         \[\leadsto \color{blue}{\left(\frac{\pi}{b \cdot b - a \cdot a} \cdot \frac{1}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. difference-of-squares 90.1%

         \[\leadsto \left(\frac{\pi}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. associate-/r* 90.2%

         \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{b + a}}{b - a}} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. metadata-eval 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot \color{blue}{0.5}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. sub-neg 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)} \]

      8. distribute-neg-frac 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \color{blue}{\frac{-1}{b}}\right) \]

      9. metadata-eval 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{\color{blue}{-1}}{b}\right) \]

   3. Simplified 90.2%

      \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)} \]
   4. Step-by-step derivation

      1. frac-add 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{1 \cdot b + a \cdot -1}{a \cdot b}} \]

      2. *-un-lft-identity 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b} + a \cdot -1}{a \cdot b} \]

   5. Applied egg-rr 90.2%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b + a \cdot -1}{a \cdot b}} \]
   6. Step-by-step derivation

      1. *-commutative 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{-1 \cdot a}}{a \cdot b} \]

      2. neg-mul-1 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{b + \color{blue}{\left(-a\right)}}{a \cdot b} \]

      3. sub-neg 90.2%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \frac{\color{blue}{b - a}}{a \cdot b} \]

   7. Simplified 90.2%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\frac{b - a}{a \cdot b}} \]

   8. Taylor expanded in b around inf 80.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{a \cdot {b}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 80.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{a \cdot {b}^{2}}} \]

      2. unpow2 80.1%

         \[\leadsto \frac{0.5 \cdot \pi}{a \cdot \color{blue}{\left(b \cdot b\right)}} \]

      3. *-commutative 80.1%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{a \cdot \left(b \cdot b\right)} \]

      4. times-frac 80.1%

         \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]

   10. Simplified 80.1%

       \[\leadsto \color{blue}{\frac{\pi}{a} \cdot \frac{0.5}{b \cdot b}} \]
   11. Step-by-step derivation

       1. associate-*r/ 80.1%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{a} \cdot 0.5}{b \cdot b}} \]

   12. Applied egg-rr 80.1%

       \[\leadsto \color{blue}{\frac{\frac{\pi}{a} \cdot 0.5}{b \cdot b}} \]
   13. Step-by-step derivation

       1. times-frac 89.7%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}} \]

   14. Applied egg-rr 89.7%

       \[\leadsto \color{blue}{\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a}}{b} \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 16: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.05e-29)
   (* 0.5 (/ PI (* a (* b a))))
   (* 0.5 (/ (/ (/ PI a) b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.05e-29) {
		tmp = 0.5 * (((double) M_PI) / (a * (b * a)));
	} else {
		tmp = 0.5 * (((((double) M_PI) / a) / b) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.05e-29) {
		tmp = 0.5 * (Math.PI / (a * (b * a)));
	} else {
		tmp = 0.5 * (((Math.PI / a) / b) / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.05e-29:
		tmp = 0.5 * (math.pi / (a * (b * a)))
	else:
		tmp = 0.5 * (((math.pi / a) / b) / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.05e-29)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * a))));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(pi / a) / b) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.05e-29)
		tmp = 0.5 * (pi / (a * (b * a)));
	else
		tmp = 0.5 * (((pi / a) / b) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.05e-29], N[(0.5 * N[(Pi / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04999999999999995e-29

   1. Initial program 77.6%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 77.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 77.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 77.6%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 77.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 77.5%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 77.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 77.5%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. div-sub 69.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   9. Applied egg-rr 69.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   10. Step-by-step derivation

       1. div-sub 77.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

       2. difference-of-squares 87.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       3. *-commutative 87.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       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. inv-pow 99.5%

          \[\leadsto \color{blue}{{\left(\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}\right)}^{-1}} \cdot 0.5 \]

       2. associate-/r/ 99.5%

          \[\leadsto {\color{blue}{\left(\frac{b - a}{\frac{\pi}{a} - \frac{\pi}{b}} \cdot \left(b + a\right)\right)}}^{-1} \cdot 0.5 \]

   13. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{{\left(\frac{b - a}{\frac{\pi}{a} - \frac{\pi}{b}} \cdot \left(b + a\right)\right)}^{-1}} \cdot 0.5 \]

   14. Taylor expanded in b around 0 56.2%

       \[\leadsto \color{blue}{\frac{\pi}{{a}^{2} \cdot b}} \cdot 0.5 \]
   15. Step-by-step derivation

       1. unpow2 56.2%

          \[\leadsto \frac{\pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \cdot 0.5 \]

       2. associate-*l* 65.4%

          \[\leadsto \frac{\pi}{\color{blue}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   16. Simplified 65.4%

       \[\leadsto \color{blue}{\frac{\pi}{a \cdot \left(a \cdot b\right)}} \cdot 0.5 \]

   if 1.04999999999999995e-29 < b

   1. Initial program 77.8%

      \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Step-by-step derivation

      1. *-commutative 77.8%

         \[\leadsto \color{blue}{\left(\frac{1}{b \cdot b - a \cdot a} \cdot \frac{\pi}{2}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. associate-/r/ 77.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. associate-*l/ 77.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}}} \]

      4. *-commutative 77.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{2}}} \]

      5. associate-/r/ 77.8%

         \[\leadsto \frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\pi} \cdot 2}} \]

      6. times-frac 77.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b \cdot b - a \cdot a}{\pi}} \cdot \frac{1}{2}} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b \cdot b - a \cdot a} \cdot 0.5} \]
   4. Step-by-step derivation

      1. clear-num 77.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. inv-pow 77.9%

         \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]

   5. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{{\left(\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}\right)}^{-1}} \cdot 0.5 \]
   6. Step-by-step derivation

      1. unpow-1 77.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}}} \cdot 0.5 \]

      2. fma-def 77.9%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\pi \cdot \frac{-1}{b} + \frac{\pi}{a}}}} \cdot 0.5 \]

      3. +-commutative 77.9%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} + \pi \cdot \frac{-1}{b}}}} \cdot 0.5 \]

      4. associate-*r/ 77.9%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\frac{\pi \cdot -1}{b}}}} \cdot 0.5 \]

      5. *-commutative 77.9%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \frac{\color{blue}{-1 \cdot \pi}}{b}}} \cdot 0.5 \]

      6. associate-*r/ 77.9%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{-1 \cdot \frac{\pi}{b}}}} \cdot 0.5 \]

      7. mul-1-neg 77.9%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} + \color{blue}{\left(-\frac{\pi}{b}\right)}}} \cdot 0.5 \]

      8. unsub-neg 77.9%

         \[\leadsto \frac{1}{\frac{b \cdot b - a \cdot a}{\color{blue}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   7. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   8. Step-by-step derivation

      1. div-sub 68.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

   9. Applied egg-rr 68.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b}{\frac{\pi}{a} - \frac{\pi}{b}} - \frac{a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]
   10. Step-by-step derivation

       1. div-sub 77.9%

          \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - a \cdot a}{\frac{\pi}{a} - \frac{\pi}{b}}}} \cdot 0.5 \]

       2. difference-of-squares 90.1%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       3. *-commutative 90.1%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}{\frac{\pi}{a} - \frac{\pi}{b}}} \cdot 0.5 \]

       4. associate-/l* 98.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]

   11. Simplified 98.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{b - a}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}}} \cdot 0.5 \]

   12. Taylor expanded in b around inf 80.1%

       \[\leadsto \color{blue}{\frac{\pi}{a \cdot {b}^{2}}} \cdot 0.5 \]
   13. Step-by-step derivation

       1. associate-/r* 80.1%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{a}}{{b}^{2}}} \cdot 0.5 \]

       2. unpow2 80.1%

          \[\leadsto \frac{\frac{\pi}{a}}{\color{blue}{b \cdot b}} \cdot 0.5 \]

       3. associate-/r* 89.7%

          \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a}}{b}}{b}} \cdot 0.5 \]

   14. Simplified 89.7%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{a}}{b}}{b}} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{a}}{b}}{b}\\ \end{array} \]

Alternative 17: 58.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* PI (/ 0.5 (* b (* a a)))))

C

double code(double a, double b) {
	return ((double) M_PI) * (0.5 / (b * (a * a)));
}

Java

public static double code(double a, double b) {
	return Math.PI * (0.5 / (b * (a * a)));
}

Python

def code(a, b):
	return math.pi * (0.5 / (b * (a * a)))

Julia

function code(a, b)
	return Float64(pi * Float64(0.5 / Float64(b * Float64(a * a))))
end

MATLAB

function tmp = code(a, b)
	tmp = pi * (0.5 / (b * (a * a)));
end

Wolfram

code[a_, b_] := N[(Pi * N[(0.5 / N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

2. Taylor expanded in b around 0 51.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
3. Step-by-step derivation

   1. associate-*r/ 51.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

   2. unpow2 51.8%

      \[\leadsto \frac{0.5 \cdot \pi}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]

4. Simplified 51.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{\left(a \cdot a\right) \cdot b}} \]

5. Taylor expanded in a around 0 51.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\pi}{{a}^{2} \cdot b}} \]
6. Step-by-step derivation

   1. associate-*r/ 51.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

   2. *-commutative 51.8%

      \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

   3. *-rgt-identity 51.8%

      \[\leadsto \frac{\color{blue}{\left(\pi \cdot 0.5\right) \cdot 1}}{{a}^{2} \cdot b} \]

   4. *-commutative 51.8%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot 1}{\color{blue}{b \cdot {a}^{2}}} \]

   5. unpow2 51.8%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot 1}{b \cdot \color{blue}{\left(a \cdot a\right)}} \]

   6. associate-*r/ 51.7%

      \[\leadsto \color{blue}{\left(\pi \cdot 0.5\right) \cdot \frac{1}{b \cdot \left(a \cdot a\right)}} \]

   7. associate-*l* 51.7%

      \[\leadsto \color{blue}{\pi \cdot \left(0.5 \cdot \frac{1}{b \cdot \left(a \cdot a\right)}\right)} \]

   8. associate-*r/ 51.7%

      \[\leadsto \pi \cdot \color{blue}{\frac{0.5 \cdot 1}{b \cdot \left(a \cdot a\right)}} \]

   9. metadata-eval 51.7%

      \[\leadsto \pi \cdot \frac{\color{blue}{0.5}}{b \cdot \left(a \cdot a\right)} \]

   10. unpow2 51.7%

       \[\leadsto \pi \cdot \frac{0.5}{b \cdot \color{blue}{{a}^{2}}} \]

   11. *-commutative 51.7%

       \[\leadsto \pi \cdot \frac{0.5}{\color{blue}{{a}^{2} \cdot b}} \]

   12. unpow2 51.7%

       \[\leadsto \pi \cdot \frac{0.5}{\color{blue}{\left(a \cdot a\right)} \cdot b} \]

7. Simplified 51.7%

   \[\leadsto \color{blue}{\pi \cdot \frac{0.5}{\left(a \cdot a\right) \cdot b}} \]

8. Final simplification 51.7%

   \[\leadsto \pi \cdot \frac{0.5}{b \cdot \left(a \cdot a\right)} \]

Reproduce

herbie shell --seed 2023224 
(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))))