NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.7%

Time: 9.9s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. *-commutative 77.4%

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

   2. associate-*l/ 77.4%

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

   3. associate-*r/ 77.4%

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

   4. associate-/l* 77.4%

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

   5. sub-neg 77.4%

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

   6. distribute-neg-frac 77.4%

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

   7. metadata-eval 77.4%

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

   8. associate-*r/ 77.4%

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

   9. *-rgt-identity 77.4%

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

   10. difference-of-squares 85.2%

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

   11. associate-/r* 85.2%

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

3. Simplified 85.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
4. Step-by-step derivation

   1. div-inv 85.2%

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

   2. associate-/r/ 85.1%

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

5. Applied egg-rr 85.1%

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

   1. associate-*r/ 85.2%

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

   2. *-rgt-identity 85.2%

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

   3. associate-*l/ 85.3%

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

   4. +-commutative 85.3%

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

7. Simplified 85.3%

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

   1. associate-/r/ 99.6%

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

9. Applied egg-rr 99.6%

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

    1. associate-*l/ 99.6%

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

    2. *-commutative 99.6%

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

    3. times-frac 99.6%

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

    4. metadata-eval 99.6%

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

    5. distribute-neg-frac 99.6%

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

    6. sub-neg 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in a around 0 99.6%

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

    1. associate-*l/ 99.7%

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

    2. *-un-lft-identity 99.7%

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

    3. div-inv 99.7%

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

    4. metadata-eval 99.7%

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

14. Applied egg-rr 99.7%

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

    1. associate-/l* 99.7%

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

16. Simplified 99.7%

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

17. Final simplification 99.7%

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

Alternative 2: 73.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \frac{\frac{\pi}{b}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e-90)
   (/ (* PI (/ 0.5 a)) (* a b))
   (* (/ 1.0 (* a b)) (/ (/ PI b) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-90) {
		tmp = (((double) M_PI) * (0.5 / a)) / (a * b);
	} else {
		tmp = (1.0 / (a * b)) * ((((double) M_PI) / b) / 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-90) {
		tmp = (Math.PI * (0.5 / a)) / (a * b);
	} else {
		tmp = (1.0 / (a * b)) * ((Math.PI / b) / 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4e-90:
		tmp = (math.pi * (0.5 / a)) / (a * b)
	else:
		tmp = (1.0 / (a * b)) * ((math.pi / b) / 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e-90)
		tmp = Float64(Float64(pi * Float64(0.5 / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(1.0 / Float64(a * b)) * Float64(Float64(pi / b) / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4e-90)
		tmp = (pi * (0.5 / a)) / (a * b);
	else
		tmp = (1.0 / (a * b)) * ((pi / b) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e-90], N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4000000000000002e-90

   1. Initial program 81.0%

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

   2. Taylor expanded in b around 0 68.6%

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

      1. unpow2 68.6%

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

   4. Simplified 68.6%

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

      1. associate-*r/ 68.6%

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

      2. associate-*l* 78.7%

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

      3. frac-times 78.6%

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

      4. associate-*r/ 78.6%

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

   6. Applied egg-rr 78.6%

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

   if -2.4000000000000002e-90 < a

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l/ 75.7%

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

      3. associate-*r/ 75.7%

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

      4. associate-/l* 75.7%

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

      5. sub-neg 75.7%

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

      6. distribute-neg-frac 75.7%

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

      7. metadata-eval 75.7%

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

      8. associate-*r/ 75.8%

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

      9. *-rgt-identity 75.8%

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

      10. difference-of-squares 83.9%

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

      11. associate-/r* 83.9%

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

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
   4. Step-by-step derivation

      1. div-inv 83.8%

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

      2. associate-/r/ 83.7%

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

   5. Applied egg-rr 83.7%

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

      1. associate-*r/ 83.8%

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

      2. *-rgt-identity 83.8%

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

      3. associate-*l/ 83.9%

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

      4. +-commutative 83.9%

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

   7. Simplified 83.9%

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

      1. associate-/r/ 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.7%

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

       2. *-commutative 99.7%

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

       3. times-frac 99.7%

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

       4. metadata-eval 99.7%

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

       5. distribute-neg-frac 99.7%

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

       6. sub-neg 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in a around 0 99.6%

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

   13. Taylor expanded in a around 0 76.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot b} \cdot \frac{\frac{\pi}{b}}{2}\\ \end{array} \]

Alternative 3: 68.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e-90)
   (* 0.5 (/ PI (* a (* a b))))
   (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-90) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-90) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4e-90:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e-90)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4e-90)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e-90], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4000000000000002e-90

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-*l/ 81.0%

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

      3. associate-*r/ 81.0%

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

      4. associate-/l* 80.8%

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

      5. sub-neg 80.8%

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

      6. distribute-neg-frac 80.8%

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

      7. metadata-eval 80.8%

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

      8. associate-*r/ 80.9%

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

      9. *-rgt-identity 80.9%

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

      10. difference-of-squares 88.1%

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

      11. associate-/r* 88.1%

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

   3. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
   4. Step-by-step derivation

      1. clear-num 88.1%

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

      2. inv-pow 88.1%

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

      3. associate-/r/ 88.0%

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

   5. Applied egg-rr 88.0%

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

      1. unpow-1 88.0%

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

      2. associate-/l* 98.3%

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

      3. associate-/r/ 98.2%

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

      4. +-commutative 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around inf 68.6%

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

      1. unpow2 68.6%

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

      2. associate-*l* 78.7%

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

   10. Simplified 78.7%

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

   if -2.4000000000000002e-90 < a

   1. Initial program 75.7%

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

   2. Taylor expanded in b around inf 64.3%

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

      1. unpow2 64.3%

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

   4. Simplified 64.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 4: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.2e-92)
   (* 0.5 (/ PI (* a (* a b))))
   (* 0.5 (/ PI (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.2e-92) {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.2e-92) {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.2e-92:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.2e-92)
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.2e-92)
		tmp = 0.5 * (pi / (a * (a * b)));
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.2e-92], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.19999999999999987e-92

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-*l/ 81.0%

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

      3. associate-*r/ 81.0%

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

      4. associate-/l* 80.8%

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

      5. sub-neg 80.8%

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

      6. distribute-neg-frac 80.8%

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

      7. metadata-eval 80.8%

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

      8. associate-*r/ 80.9%

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

      9. *-rgt-identity 80.9%

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

      10. difference-of-squares 88.1%

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

      11. associate-/r* 88.1%

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

   3. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
   4. Step-by-step derivation

      1. clear-num 88.1%

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

      2. inv-pow 88.1%

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

      3. associate-/r/ 88.0%

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

   5. Applied egg-rr 88.0%

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

      1. unpow-1 88.0%

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

      2. associate-/l* 98.3%

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

      3. associate-/r/ 98.2%

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

      4. +-commutative 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in a around inf 68.6%

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

      1. unpow2 68.6%

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

      2. associate-*l* 78.7%

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

   10. Simplified 78.7%

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

   if -2.19999999999999987e-92 < a

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l/ 75.7%

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

      3. associate-*r/ 75.7%

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

      4. associate-/l* 75.7%

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

      5. sub-neg 75.7%

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

      6. distribute-neg-frac 75.7%

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

      7. metadata-eval 75.7%

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

      8. associate-*r/ 75.8%

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

      9. *-rgt-identity 75.8%

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

      10. difference-of-squares 83.9%

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

      11. associate-/r* 83.9%

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

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
   4. Step-by-step derivation

      1. clear-num 83.8%

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

      2. inv-pow 83.8%

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

      3. associate-/r/ 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. unpow-1 83.8%

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

      2. associate-/l* 98.9%

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

      3. associate-/r/ 98.9%

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

      4. +-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in a around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-lft-identity 64.3%

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

      3. *-lft-identity 64.3%

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

      4. associate-*r* 76.2%

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

   10. Simplified 76.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 5: 73.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e-90)
   (* (/ 0.5 a) (/ PI (* a b)))
   (* 0.5 (/ PI (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-90) {
		tmp = (0.5 / a) * (((double) M_PI) / (a * b));
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-90) {
		tmp = (0.5 / a) * (Math.PI / (a * b));
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4e-90:
		tmp = (0.5 / a) * (math.pi / (a * b))
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e-90)
		tmp = Float64(Float64(0.5 / a) * Float64(pi / Float64(a * b)));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4e-90)
		tmp = (0.5 / a) * (pi / (a * b));
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e-90], N[(N[(0.5 / a), $MachinePrecision] * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4000000000000002e-90

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-*l/ 81.0%

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

      3. associate-*r/ 81.0%

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

      4. associate-/l* 80.8%

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

      5. sub-neg 80.8%

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

      6. distribute-neg-frac 80.8%

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

      7. metadata-eval 80.8%

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

      8. associate-*r/ 80.9%

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

      9. *-rgt-identity 80.9%

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

      10. difference-of-squares 88.1%

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

      11. associate-/r* 88.1%

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

   3. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
   4. Step-by-step derivation

      1. div-inv 88.1%

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

      2. associate-/r/ 88.1%

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

   5. Applied egg-rr 88.1%

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

      1. associate-*r/ 88.1%

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

      2. *-rgt-identity 88.1%

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

      3. associate-*l/ 88.2%

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

      4. +-commutative 88.2%

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

   7. Simplified 88.2%

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

      1. associate-/r/ 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. associate-*l/ 99.6%

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

       2. *-commutative 99.6%

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

       3. times-frac 99.6%

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

       4. metadata-eval 99.6%

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

       5. distribute-neg-frac 99.6%

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

       6. sub-neg 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in a around inf 68.6%

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

       1. unpow2 68.6%

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

       2. associate-*r* 78.7%

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

       3. associate-*r/ 78.7%

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

       4. times-frac 78.6%

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

   14. Simplified 78.6%

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

   if -2.4000000000000002e-90 < a

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l/ 75.7%

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

      3. associate-*r/ 75.7%

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

      4. associate-/l* 75.7%

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

      5. sub-neg 75.7%

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

      6. distribute-neg-frac 75.7%

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

      7. metadata-eval 75.7%

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

      8. associate-*r/ 75.8%

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

      9. *-rgt-identity 75.8%

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

      10. difference-of-squares 83.9%

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

      11. associate-/r* 83.9%

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

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
   4. Step-by-step derivation

      1. clear-num 83.8%

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

      2. inv-pow 83.8%

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

      3. associate-/r/ 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. unpow-1 83.8%

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

      2. associate-/l* 98.9%

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

      3. associate-/r/ 98.9%

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

      4. +-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in a around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-lft-identity 64.3%

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

      3. *-lft-identity 64.3%

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

      4. associate-*r* 76.2%

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

   10. Simplified 76.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\pi}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 6: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e-90)
   (/ (* PI (/ 0.5 a)) (* a b))
   (* 0.5 (/ PI (* b (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-90) {
		tmp = (((double) M_PI) * (0.5 / a)) / (a * b);
	} else {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.4e-90) {
		tmp = (Math.PI * (0.5 / a)) / (a * b);
	} else {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4e-90:
		tmp = (math.pi * (0.5 / a)) / (a * b)
	else:
		tmp = 0.5 * (math.pi / (b * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e-90)
		tmp = Float64(Float64(pi * Float64(0.5 / a)) / Float64(a * b));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4e-90)
		tmp = (pi * (0.5 / a)) / (a * b);
	else
		tmp = 0.5 * (pi / (b * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e-90], N[(N[(Pi * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4000000000000002e-90

   1. Initial program 81.0%

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

   2. Taylor expanded in b around 0 68.6%

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

      1. unpow2 68.6%

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

   4. Simplified 68.6%

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

      1. associate-*r/ 68.6%

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

      2. associate-*l* 78.7%

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

      3. frac-times 78.6%

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

      4. associate-*r/ 78.6%

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

   6. Applied egg-rr 78.6%

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

   if -2.4000000000000002e-90 < a

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*l/ 75.7%

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

      3. associate-*r/ 75.7%

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

      4. associate-/l* 75.7%

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

      5. sub-neg 75.7%

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

      6. distribute-neg-frac 75.7%

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

      7. metadata-eval 75.7%

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

      8. associate-*r/ 75.8%

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

      9. *-rgt-identity 75.8%

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

      10. difference-of-squares 83.9%

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

      11. associate-/r* 83.9%

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

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
   4. Step-by-step derivation

      1. clear-num 83.8%

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

      2. inv-pow 83.8%

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

      3. associate-/r/ 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. unpow-1 83.8%

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

      2. associate-/l* 98.9%

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

      3. associate-/r/ 98.9%

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

      4. +-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in a around 0 64.3%

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

      1. unpow2 64.3%

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

      2. *-lft-identity 64.3%

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

      3. *-lft-identity 64.3%

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

      4. associate-*r* 76.2%

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

   10. Simplified 76.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 7: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. *-commutative 77.4%

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

   2. associate-*l/ 77.4%

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

   3. associate-*r/ 77.4%

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

   4. associate-/l* 77.4%

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

   5. sub-neg 77.4%

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

   6. distribute-neg-frac 77.4%

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

   7. metadata-eval 77.4%

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

   8. associate-*r/ 77.4%

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

   9. *-rgt-identity 77.4%

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

   10. difference-of-squares 85.2%

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

   11. associate-/r* 85.2%

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

3. Simplified 85.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
4. Step-by-step derivation

   1. div-inv 85.2%

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

   2. associate-/r/ 85.1%

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

5. Applied egg-rr 85.1%

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

   1. associate-*r/ 85.2%

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

   2. *-rgt-identity 85.2%

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

   3. associate-*l/ 85.3%

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

   4. +-commutative 85.3%

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

7. Simplified 85.3%

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

   1. associate-/r/ 99.6%

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

9. Applied egg-rr 99.6%

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

    1. associate-*l/ 99.6%

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

    2. *-commutative 99.6%

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

    3. times-frac 99.6%

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

    4. metadata-eval 99.6%

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

    5. distribute-neg-frac 99.6%

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

    6. sub-neg 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in a around 0 99.6%

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

    1. expm1-log1p-u 82.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a \cdot b} \cdot \frac{\frac{\pi}{a + b}}{2}\right)\right)} \]

    2. expm1-udef 50.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{a \cdot b} \cdot \frac{\frac{\pi}{a + b}}{2}\right)} - 1} \]

    3. frac-times 50.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \frac{\pi}{a + b}}{\left(a \cdot b\right) \cdot 2}}\right)} - 1 \]

    4. *-un-lft-identity 50.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{\pi}{a + b}}}{\left(a \cdot b\right) \cdot 2}\right)} - 1 \]

14. Applied egg-rr 50.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{a + b}}{\left(a \cdot b\right) \cdot 2}\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 82.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{a + b}}{\left(a \cdot b\right) \cdot 2}\right)\right)} \]

    2. expm1-log1p 99.7%

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

    3. associate-/l/ 98.8%

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

    4. associate-*l* 98.8%

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

16. Simplified 98.8%

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

17. Final simplification 98.8%

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

Alternative 8: 62.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. *-commutative 77.4%

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

   2. associate-*l/ 77.4%

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

   3. associate-*r/ 77.4%

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

   4. associate-/l* 77.4%

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

   5. sub-neg 77.4%

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

   6. distribute-neg-frac 77.4%

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

   7. metadata-eval 77.4%

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

   8. associate-*r/ 77.4%

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

   9. *-rgt-identity 77.4%

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

   10. difference-of-squares 85.2%

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

   11. associate-/r* 85.2%

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

3. Simplified 85.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{a} + \frac{-1}{b}}{\frac{2}{\frac{\frac{\pi}{b + a}}{b - a}}}} \]
4. Step-by-step derivation

   1. clear-num 85.2%

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

   2. inv-pow 85.2%

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

   3. associate-/r/ 85.2%

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

5. Applied egg-rr 85.2%

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

   1. unpow-1 85.2%

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

   2. associate-/l* 98.7%

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

   3. associate-/r/ 98.7%

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

   4. +-commutative 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in a around inf 54.3%

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

   1. unpow2 54.3%

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

   2. associate-*l* 59.8%

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

10. Simplified 59.8%

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

11. Final simplification 59.8%

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

Reproduce

herbie shell --seed 2023283 
(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))))