NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 10.4s

Alternatives: 6

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-*l* 75.0%

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

   2. *-rgt-identity 75.0%

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

   3. associate-/l* 75.0%

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

   4. metadata-eval 75.0%

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

   5. associate-*l/ 75.1%

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

   6. *-lft-identity 75.1%

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

   7. sub-neg 75.1%

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

   8. distribute-neg-frac 75.1%

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

   9. metadata-eval 75.1%

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

3. Simplified 75.1%

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

   1. *-un-lft-identity 75.1%

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

   2. clear-num 74.8%

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

   3. frac-add 74.7%

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

   4. associate-/r/ 74.8%

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

   5. *-un-lft-identity 74.8%

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

   6. *-commutative 74.8%

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

   7. neg-mul-1 74.8%

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

   8. sub-neg 74.8%

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

   9. flip-+ 98.2%

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

   10. +-commutative 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-lft-identity 98.2%

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

   2. associate-/r* 99.6%

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

8. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

10. Applied egg-rr 99.6%

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

    1. un-div-inv 99.7%

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

    2. associate-/r* 98.3%

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

    3. times-frac 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 74.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{a \cdot b}\\ \mathbf{if}\;b \leq 4.6 \cdot 10^{-55}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* a b))))
   (if (<= b 4.6e-55) (* t_0 (/ PI a)) (* t_0 (/ PI b)))))

C

double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (b <= 4.6e-55) {
		tmp = t_0 * (((double) M_PI) / a);
	} else {
		tmp = t_0 * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 / (a * b);
	double tmp;
	if (b <= 4.6e-55) {
		tmp = t_0 * (Math.PI / a);
	} else {
		tmp = t_0 * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 / (a * b)
	tmp = 0
	if b <= 4.6e-55:
		tmp = t_0 * (math.pi / a)
	else:
		tmp = t_0 * (math.pi / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 / Float64(a * b))
	tmp = 0.0
	if (b <= 4.6e-55)
		tmp = Float64(t_0 * Float64(pi / a));
	else
		tmp = Float64(t_0 * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 / (a * b);
	tmp = 0.0;
	if (b <= 4.6e-55)
		tmp = t_0 * (pi / a);
	else
		tmp = t_0 * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.6e-55], N[(t$95$0 * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.60000000000000023e-55

   1. Initial program 74.6%

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

      1. associate-*l* 74.5%

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

      2. *-rgt-identity 74.5%

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

      3. associate-/l* 74.5%

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

      4. metadata-eval 74.5%

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

      5. associate-*l/ 74.6%

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

      6. *-lft-identity 74.6%

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

      7. sub-neg 74.6%

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

      8. distribute-neg-frac 74.6%

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

      9. metadata-eval 74.6%

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

   3. Simplified 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. clear-num 74.5%

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

      3. frac-add 74.5%

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

      4. associate-/r/ 74.5%

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

      5. *-un-lft-identity 74.5%

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

      6. *-commutative 74.5%

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

      7. neg-mul-1 74.5%

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

      8. sub-neg 74.5%

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

      9. flip-+ 98.4%

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

      10. +-commutative 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. un-div-inv 99.7%

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

       2. associate-/r* 98.4%

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

       3. times-frac 99.6%

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

   12. Applied egg-rr 99.6%

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

   13. Taylor expanded in a around inf 71.7%

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

   if 4.60000000000000023e-55 < b

   1. Initial program 76.0%

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

      1. associate-*l* 76.0%

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

      2. *-rgt-identity 76.0%

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

      3. associate-/l* 76.0%

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

      4. metadata-eval 76.0%

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

      5. associate-*l/ 76.1%

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

      6. *-lft-identity 76.1%

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

      7. sub-neg 76.1%

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

      8. distribute-neg-frac 76.1%

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

      9. metadata-eval 76.1%

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

   3. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. clear-num 75.3%

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

      3. frac-add 75.2%

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

      4. associate-/r/ 75.2%

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

      5. *-un-lft-identity 75.2%

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

      6. *-commutative 75.2%

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

      7. neg-mul-1 75.2%

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

      8. sub-neg 75.2%

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

      9. flip-+ 97.9%

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

      10. +-commutative 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. *-lft-identity 97.9%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. un-div-inv 99.7%

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

       2. associate-/r* 98.1%

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

       3. times-frac 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in a around 0 88.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25e-62)
   (* (/ 0.5 (* a b)) (/ PI a))
   (* (/ (/ PI b) a) (/ 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-62) {
		tmp = (0.5 / (a * b)) * (((double) M_PI) / a);
	} else {
		tmp = ((((double) M_PI) / b) / a) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.25e-62) {
		tmp = (0.5 / (a * b)) * (Math.PI / a);
	} else {
		tmp = ((Math.PI / b) / a) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.25e-62:
		tmp = (0.5 / (a * b)) * (math.pi / a)
	else:
		tmp = ((math.pi / b) / a) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.25e-62)
		tmp = Float64(Float64(0.5 / Float64(a * b)) * Float64(pi / a));
	else
		tmp = Float64(Float64(Float64(pi / b) / a) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.25e-62)
		tmp = (0.5 / (a * b)) * (pi / a);
	else
		tmp = ((pi / b) / a) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.25e-62], N[(N[(0.5 / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25e-62

   1. Initial program 74.4%

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

      1. associate-*l* 74.4%

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

      2. *-rgt-identity 74.4%

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

      3. associate-/l* 74.4%

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

      4. metadata-eval 74.4%

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

      5. associate-*l/ 74.4%

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

      6. *-lft-identity 74.4%

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

      7. sub-neg 74.4%

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

      8. distribute-neg-frac 74.4%

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

      9. metadata-eval 74.4%

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

   3. Simplified 74.4%

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

      1. *-un-lft-identity 74.4%

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

      2. clear-num 74.4%

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

      3. frac-add 74.3%

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

      4. associate-/r/ 74.4%

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

      5. *-un-lft-identity 74.4%

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

      6. *-commutative 74.4%

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

      7. neg-mul-1 74.4%

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

      8. sub-neg 74.4%

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

      9. flip-+ 98.3%

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

      10. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. *-lft-identity 98.3%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. un-div-inv 99.7%

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

       2. associate-/r* 98.4%

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

       3. times-frac 99.6%

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

   12. Applied egg-rr 99.6%

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

   13. Taylor expanded in a around inf 71.6%

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

   if 1.25e-62 < b

   1. Initial program 76.3%

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

      1. associate-*l* 76.3%

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

      2. *-rgt-identity 76.3%

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

      3. associate-/l* 76.3%

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

      4. metadata-eval 76.3%

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

      5. associate-*l/ 76.3%

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

      6. *-lft-identity 76.3%

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

      7. sub-neg 76.3%

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

      8. distribute-neg-frac 76.3%

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

      9. metadata-eval 76.3%

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

   3. Simplified 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. clear-num 75.5%

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

      3. frac-add 75.5%

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

      4. associate-/r/ 75.5%

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

      5. *-un-lft-identity 75.5%

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

      6. *-commutative 75.5%

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

      7. neg-mul-1 75.5%

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

      8. sub-neg 75.5%

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

      9. flip-+ 98.0%

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

      10. +-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-lft-identity 98.0%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in a around 0 87.2%

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

       1. *-commutative 87.2%

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

       2. times-frac 87.2%

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

   13. Applied egg-rr 87.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6e-54) (/ (* 0.5 (/ PI a)) (* a b)) (* (/ (/ PI b) a) (/ 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6e-54) {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	} else {
		tmp = ((((double) M_PI) / b) / a) * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.6e-54) {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	} else {
		tmp = ((Math.PI / b) / a) * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.6e-54:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	else:
		tmp = ((math.pi / b) / a) * (0.5 / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6e-54)
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	else
		tmp = Float64(Float64(Float64(pi / b) / a) * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.6e-54)
		tmp = (0.5 * (pi / a)) / (a * b);
	else
		tmp = ((pi / b) / a) * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6e-54], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.59999999999999999e-54

   1. Initial program 74.6%

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

      1. associate-*l* 74.5%

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

      2. *-rgt-identity 74.5%

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

      3. associate-/l* 74.5%

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

      4. metadata-eval 74.5%

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

      5. associate-*l/ 74.6%

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

      6. *-lft-identity 74.6%

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

      7. sub-neg 74.6%

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

      8. distribute-neg-frac 74.6%

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

      9. metadata-eval 74.6%

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

   3. Simplified 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. clear-num 74.5%

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

      3. frac-add 74.5%

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

      4. associate-/r/ 74.5%

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

      5. *-un-lft-identity 74.5%

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

      6. *-commutative 74.5%

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

      7. neg-mul-1 74.5%

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

      8. sub-neg 74.5%

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

      9. flip-+ 98.4%

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

      10. +-commutative 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. *-lft-identity 98.4%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in a around inf 71.7%

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

   if 1.59999999999999999e-54 < b

   1. Initial program 76.0%

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

      1. associate-*l* 76.0%

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

      2. *-rgt-identity 76.0%

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

      3. associate-/l* 76.0%

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

      4. metadata-eval 76.0%

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

      5. associate-*l/ 76.1%

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

      6. *-lft-identity 76.1%

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

      7. sub-neg 76.1%

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

      8. distribute-neg-frac 76.1%

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

      9. metadata-eval 76.1%

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

   3. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. clear-num 75.3%

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

      3. frac-add 75.2%

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

      4. associate-/r/ 75.2%

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

      5. *-un-lft-identity 75.2%

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

      6. *-commutative 75.2%

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

      7. neg-mul-1 75.2%

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

      8. sub-neg 75.2%

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

      9. flip-+ 97.9%

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

      10. +-commutative 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. *-lft-identity 97.9%

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

      2. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in a around 0 88.1%

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

       1. *-commutative 88.1%

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

       2. times-frac 88.1%

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

   13. Applied egg-rr 88.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-*l* 75.0%

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

   2. *-rgt-identity 75.0%

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

   3. associate-/l* 75.0%

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

   4. metadata-eval 75.0%

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

   5. associate-*l/ 75.1%

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

   6. *-lft-identity 75.1%

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

   7. sub-neg 75.1%

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

   8. distribute-neg-frac 75.1%

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

   9. metadata-eval 75.1%

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

3. Simplified 75.1%

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

   1. metadata-eval 75.1%

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

   2. div-inv 75.1%

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

   3. clear-num 74.8%

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

   4. un-div-inv 74.9%

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

   5. div-inv 74.9%

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

   6. metadata-eval 74.9%

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

   7. frac-add 74.8%

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

   8. associate-/r/ 74.8%

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

   9. *-un-lft-identity 74.8%

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

   10. *-commutative 74.8%

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

   11. neg-mul-1 74.8%

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

   12. sub-neg 74.8%

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

   13. flip-+ 98.3%

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

   14. +-commutative 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in a around 0 86.4%

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

   1. *-commutative 86.4%

      \[\leadsto \frac{\pi \cdot 0.5}{a \cdot \left(\color{blue}{b \cdot a} + {b}^{2}\right)} \]

   2. unpow2 86.4%

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

   3. distribute-lft-out 91.5%

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

9. Simplified 91.5%

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

    1. *-commutative 91.5%

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

    2. associate-*r* 98.3%

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

    3. frac-times 99.6%

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

    4. *-commutative 99.6%

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

    5. associate-*l/ 99.6%

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

    6. div-inv 99.6%

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

    7. associate-*r* 99.6%

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

    8. div-inv 99.6%

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

    9. *-commutative 99.6%

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

    10. associate-*r/ 99.6%

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

    11. *-un-lft-identity 99.6%

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

    12. times-frac 99.6%

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

    13. metadata-eval 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 6: 62.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

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

   1. associate-*l* 75.0%

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

   2. *-rgt-identity 75.0%

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

   3. associate-/l* 75.0%

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

   4. metadata-eval 75.0%

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

   5. associate-*l/ 75.1%

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

   6. *-lft-identity 75.1%

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

   7. sub-neg 75.1%

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

   8. distribute-neg-frac 75.1%

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

   9. metadata-eval 75.1%

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

3. Simplified 75.1%

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

   1. *-un-lft-identity 75.1%

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

   2. clear-num 74.8%

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

   3. frac-add 74.7%

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

   4. associate-/r/ 74.8%

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

   5. *-un-lft-identity 74.8%

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

   6. *-commutative 74.8%

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

   7. neg-mul-1 74.8%

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

   8. sub-neg 74.8%

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

   9. flip-+ 98.2%

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

   10. +-commutative 98.2%

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

6. Applied egg-rr 98.2%

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

   1. *-lft-identity 98.2%

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

   2. associate-/r* 99.6%

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

8. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

10. Applied egg-rr 99.6%

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

    1. un-div-inv 99.7%

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

    2. associate-/r* 98.3%

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

    3. times-frac 99.6%

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

12. Applied egg-rr 99.6%

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

13. Taylor expanded in a around inf 59.0%

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

14. Final simplification 59.0%

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

Reproduce

herbie shell --seed 2024115 
(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))))