NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.6%

Time: 9.6s

Alternatives: 8

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: 78.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. associate-*l* 75.4%

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

   2. *-rgt-identity 75.4%

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

   3. associate-/l* 75.4%

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

   4. metadata-eval 75.4%

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

   5. associate-*l/ 75.5%

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

   6. *-lft-identity 75.5%

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

   7. sub-neg 75.5%

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

   8. distribute-neg-frac 75.5%

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

   9. metadata-eval 75.5%

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

3. Simplified 75.5%

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

   1. metadata-eval 75.5%

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

   2. div-inv 75.5%

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

   3. clear-num 75.5%

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

   4. clear-num 75.0%

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

   5. frac-times 74.9%

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

   6. metadata-eval 74.9%

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

   7. frac-add 74.9%

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

   8. associate-/r/ 74.9%

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

   9. *-un-lft-identity 74.9%

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

   10. *-commutative 74.9%

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

   11. neg-mul-1 74.9%

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

   12. sub-neg 74.9%

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

   13. flip-+ 98.8%

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

   14. +-commutative 98.8%

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

6. Applied egg-rr 98.8%

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

   1. associate-/r* 98.9%

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

   2. associate-/r/ 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-commutative 98.9%

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

   5. times-frac 99.6%

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

8. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. clear-num 99.6%

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

   3. frac-times 98.9%

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

   4. metadata-eval 98.9%

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

10. Applied egg-rr 98.9%

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

    1. associate-/r* 99.7%

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

    2. *-commutative 99.7%

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

    3. +-commutative 99.7%

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

12. Simplified 99.7%

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

Alternative 2: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot \frac{b}{\pi}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.25e-35)
   (/ (/ 0.5 (* a (/ b PI))) a)
   (* (/ 0.5 (* b a)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.25e-35) {
		tmp = (0.5 / (a * (b / ((double) M_PI)))) / a;
	} else {
		tmp = (0.5 / (b * a)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.25e-35) {
		tmp = (0.5 / (a * (b / Math.PI))) / a;
	} else {
		tmp = (0.5 / (b * a)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.25e-35:
		tmp = (0.5 / (a * (b / math.pi))) / a
	else:
		tmp = (0.5 / (b * a)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.25e-35)
		tmp = Float64(Float64(0.5 / Float64(a * Float64(b / pi))) / a);
	else
		tmp = Float64(Float64(0.5 / Float64(b * a)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.25e-35)
		tmp = (0.5 / (a * (b / pi))) / a;
	else
		tmp = (0.5 / (b * a)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.25e-35], N[(N[(0.5 / N[(a * N[(b / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.25000000000000005e-35

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*r* 78.5%

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

      3. associate-*r/ 78.5%

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

      4. associate-*r* 78.5%

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

      5. *-rgt-identity 78.5%

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

      6. sub-neg 78.5%

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

      7. distribute-neg-frac 78.5%

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

      8. metadata-eval 78.5%

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

   3. Simplified 78.5%

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

      1. associate-*l/ 78.6%

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

      2. div-inv 78.6%

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

      3. metadata-eval 78.6%

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

      4. *-commutative 78.6%

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

      5. associate-*r* 78.6%

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

      6. *-commutative 78.6%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in a around inf 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-*r/ 69.5%

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

      3. frac-times 70.2%

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

      4. div-inv 70.2%

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

      5. div-inv 70.2%

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

      6. associate-/l/ 70.2%

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

      7. associate-*r/ 70.2%

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

   9. Applied egg-rr 70.2%

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

       1. *-commutative 70.2%

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

       2. div-inv 70.2%

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

       3. associate-*l* 70.2%

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

       4. times-frac 70.3%

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

       5. *-un-lft-identity 70.3%

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

       6. clear-num 70.3%

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

       7. *-commutative 70.3%

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

       8. associate-*r/ 70.3%

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

       9. div-inv 70.3%

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

   11. Applied egg-rr 70.3%

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

   if 2.25000000000000005e-35 < b

   1. Initial program 68.1%

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

      1. associate-*l* 68.1%

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

      2. *-rgt-identity 68.1%

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

      3. associate-/l* 68.1%

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

      4. metadata-eval 68.1%

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

      5. associate-*l/ 68.1%

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

      6. *-lft-identity 68.1%

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

      7. sub-neg 68.1%

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

      8. distribute-neg-frac 68.1%

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

      9. metadata-eval 68.1%

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

   3. Simplified 68.1%

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

      1. metadata-eval 68.1%

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

      2. div-inv 68.1%

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

      3. clear-num 68.1%

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

      4. clear-num 67.4%

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

      5. frac-times 67.4%

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

      6. metadata-eval 67.4%

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

      7. frac-add 67.4%

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

      8. associate-/r/ 67.5%

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

      9. *-un-lft-identity 67.5%

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

      10. *-commutative 67.5%

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

      11. neg-mul-1 67.5%

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

      12. sub-neg 67.5%

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

      13. flip-+ 98.9%

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

      14. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. associate-/r* 98.9%

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

      2. associate-/r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-commutative 98.9%

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

      5. times-frac 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in a around 0 86.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{0.5}{a \cdot \frac{b}{\pi}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{b \cdot a}\\ \mathbf{if}\;b \leq 4 \cdot 10^{-36}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* b a))))
   (if (<= b 4e-36) (* t_0 (/ PI a)) (* t_0 (/ PI b)))))

C

double code(double a, double b) {
	double t_0 = 0.5 / (b * a);
	double tmp;
	if (b <= 4e-36) {
		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 / (b * a);
	double tmp;
	if (b <= 4e-36) {
		tmp = t_0 * (Math.PI / a);
	} else {
		tmp = t_0 * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 / (b * a)
	tmp = 0
	if b <= 4e-36:
		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(b * a))
	tmp = 0.0
	if (b <= 4e-36)
		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 / (b * a);
	tmp = 0.0;
	if (b <= 4e-36)
		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[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4e-36], 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 < 3.9999999999999998e-36

   1. Initial program 78.4%

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

      1. associate-*l* 78.5%

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

      2. *-rgt-identity 78.5%

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

      3. associate-/l* 78.5%

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

      4. metadata-eval 78.5%

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

      5. associate-*l/ 78.6%

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

      6. *-lft-identity 78.6%

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

      7. sub-neg 78.6%

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

      8. distribute-neg-frac 78.6%

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

      9. metadata-eval 78.6%

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

   3. Simplified 78.6%

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

      1. metadata-eval 78.6%

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

      2. div-inv 78.6%

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

      3. clear-num 78.6%

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

      4. clear-num 78.2%

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

      5. frac-times 78.1%

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

      6. metadata-eval 78.1%

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

      7. frac-add 78.1%

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

      8. associate-/r/ 78.1%

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

      9. *-un-lft-identity 78.1%

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

      10. *-commutative 78.1%

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

      11. neg-mul-1 78.1%

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

      12. sub-neg 78.1%

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

      13. flip-+ 98.8%

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

      14. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-/r* 98.9%

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

      2. associate-/r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-commutative 98.9%

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

      5. times-frac 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around inf 70.2%

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

   if 3.9999999999999998e-36 < b

   1. Initial program 68.1%

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

      1. associate-*l* 68.1%

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

      2. *-rgt-identity 68.1%

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

      3. associate-/l* 68.1%

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

      4. metadata-eval 68.1%

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

      5. associate-*l/ 68.1%

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

      6. *-lft-identity 68.1%

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

      7. sub-neg 68.1%

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

      8. distribute-neg-frac 68.1%

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

      9. metadata-eval 68.1%

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

   3. Simplified 68.1%

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

      1. metadata-eval 68.1%

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

      2. div-inv 68.1%

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

      3. clear-num 68.1%

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

      4. clear-num 67.4%

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

      5. frac-times 67.4%

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

      6. metadata-eval 67.4%

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

      7. frac-add 67.4%

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

      8. associate-/r/ 67.5%

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

      9. *-un-lft-identity 67.5%

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

      10. *-commutative 67.5%

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

      11. neg-mul-1 67.5%

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

      12. sub-neg 67.5%

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

      13. flip-+ 98.9%

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

      14. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. associate-/r* 98.9%

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

      2. associate-/r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-commutative 98.9%

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

      5. times-frac 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in a around 0 86.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-36}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-34}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e-34) (* (/ 0.5 (* b a)) (/ PI a)) (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e-34) {
		tmp = (0.5 / (b * a)) * (((double) M_PI) / a);
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1e-34) {
		tmp = (0.5 / (b * a)) * (Math.PI / a);
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1e-34:
		tmp = (0.5 / (b * a)) * (math.pi / a)
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e-34)
		tmp = Float64(Float64(0.5 / Float64(b * a)) * Float64(pi / a));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1e-34)
		tmp = (0.5 / (b * a)) * (pi / a);
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e-34], N[(N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(Pi / a), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999928e-35

   1. Initial program 78.4%

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

      1. associate-*l* 78.5%

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

      2. *-rgt-identity 78.5%

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

      3. associate-/l* 78.5%

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

      4. metadata-eval 78.5%

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

      5. associate-*l/ 78.6%

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

      6. *-lft-identity 78.6%

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

      7. sub-neg 78.6%

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

      8. distribute-neg-frac 78.6%

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

      9. metadata-eval 78.6%

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

   3. Simplified 78.6%

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

      1. metadata-eval 78.6%

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

      2. div-inv 78.6%

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

      3. clear-num 78.6%

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

      4. clear-num 78.2%

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

      5. frac-times 78.1%

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

      6. metadata-eval 78.1%

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

      7. frac-add 78.1%

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

      8. associate-/r/ 78.1%

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

      9. *-un-lft-identity 78.1%

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

      10. *-commutative 78.1%

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

      11. neg-mul-1 78.1%

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

      12. sub-neg 78.1%

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

      13. flip-+ 98.8%

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

      14. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-/r* 98.9%

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

      2. associate-/r/ 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-commutative 98.9%

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

      5. times-frac 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in a around inf 70.2%

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

   if 9.99999999999999928e-35 < b

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-*r* 68.0%

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

      3. associate-*r/ 68.1%

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

      4. associate-*r* 68.1%

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

      5. *-rgt-identity 68.1%

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

      6. sub-neg 68.1%

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

      7. distribute-neg-frac 68.1%

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

      8. metadata-eval 68.1%

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

   3. Simplified 68.1%

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

      1. associate-*l/ 68.1%

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

      2. div-inv 68.1%

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

      3. metadata-eval 68.1%

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

      4. *-commutative 68.1%

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

      5. associate-*r* 68.1%

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

      6. *-commutative 68.1%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in a around 0 87.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-34}:\\ \;\;\;\;\frac{0.5}{b \cdot a} \cdot \frac{\pi}{a}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-34}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-34) (* PI (/ 0.5 (* a (* b a)))) (* PI (/ 0.5 (* b (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-34) {
		tmp = ((double) M_PI) * (0.5 / (a * (b * a)));
	} else {
		tmp = ((double) M_PI) * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-34) {
		tmp = Math.PI * (0.5 / (a * (b * a)));
	} else {
		tmp = Math.PI * (0.5 / (b * (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.1e-34:
		tmp = math.pi * (0.5 / (a * (b * a)))
	else:
		tmp = math.pi * (0.5 / (b * (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e-34)
		tmp = Float64(pi * Float64(0.5 / Float64(a * Float64(b * a))));
	else
		tmp = Float64(pi * Float64(0.5 / Float64(b * Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.1e-34)
		tmp = pi * (0.5 / (a * (b * a)));
	else
		tmp = pi * (0.5 / (b * (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e-34], N[(Pi * N[(0.5 / N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * N[(0.5 / N[(b * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0999999999999999e-34

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*r* 78.5%

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

      3. associate-*r/ 78.5%

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

      4. associate-*r* 78.5%

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

      5. *-rgt-identity 78.5%

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

      6. sub-neg 78.5%

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

      7. distribute-neg-frac 78.5%

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

      8. metadata-eval 78.5%

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

   3. Simplified 78.5%

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

      1. associate-*l/ 78.6%

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

      2. div-inv 78.6%

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

      3. metadata-eval 78.6%

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

      4. *-commutative 78.6%

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

      5. associate-*r* 78.6%

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

      6. *-commutative 78.6%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in a around inf 69.5%

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

   if 1.0999999999999999e-34 < b

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-*r* 68.0%

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

      3. associate-*r/ 68.1%

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

      4. associate-*r* 68.1%

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

      5. *-rgt-identity 68.1%

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

      6. sub-neg 68.1%

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

      7. distribute-neg-frac 68.1%

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

      8. metadata-eval 68.1%

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

   3. Simplified 68.1%

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

      1. associate-*l/ 68.1%

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

      2. div-inv 68.1%

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

      3. metadata-eval 68.1%

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

      4. *-commutative 68.1%

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

      5. associate-*r* 68.1%

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

      6. *-commutative 68.1%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in a around 0 87.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-34}:\\ \;\;\;\;\pi \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. associate-*l* 75.4%

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

   2. *-rgt-identity 75.4%

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

   3. associate-/l* 75.4%

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

   4. metadata-eval 75.4%

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

   5. associate-*l/ 75.5%

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

   6. *-lft-identity 75.5%

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

   7. sub-neg 75.5%

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

   8. distribute-neg-frac 75.5%

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

   9. metadata-eval 75.5%

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

3. Simplified 75.5%

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

   1. metadata-eval 75.5%

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

   2. div-inv 75.5%

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

   3. clear-num 75.5%

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

   4. clear-num 75.0%

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

   5. frac-times 74.9%

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

   6. metadata-eval 74.9%

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

   7. frac-add 74.9%

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

   8. associate-/r/ 74.9%

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

   9. *-un-lft-identity 74.9%

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

   10. *-commutative 74.9%

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

   11. neg-mul-1 74.9%

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

   12. sub-neg 74.9%

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

   13. flip-+ 98.8%

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

   14. +-commutative 98.8%

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

6. Applied egg-rr 98.8%

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

   1. associate-/r* 98.9%

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

   2. associate-/r/ 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-commutative 98.9%

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

   5. times-frac 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 7: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-*r* 75.4%

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

   3. associate-*r/ 75.4%

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

   4. associate-*r* 75.4%

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

   5. *-rgt-identity 75.4%

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

   6. sub-neg 75.4%

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

   7. distribute-neg-frac 75.4%

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

   8. metadata-eval 75.4%

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

3. Simplified 75.4%

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

   1. associate-*l/ 75.5%

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

   2. div-inv 75.5%

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

   3. metadata-eval 75.5%

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

   4. *-commutative 75.5%

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

   5. associate-*r* 75.4%

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

   6. *-commutative 75.4%

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 8: 63.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. *-commutative 75.4%

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

   2. associate-*r* 75.4%

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

   3. associate-*r/ 75.4%

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

   4. associate-*r* 75.4%

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

   5. *-rgt-identity 75.4%

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

   6. sub-neg 75.4%

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

   7. distribute-neg-frac 75.4%

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

   8. metadata-eval 75.4%

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

3. Simplified 75.4%

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

   1. associate-*l/ 75.5%

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

   2. div-inv 75.5%

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

   3. metadata-eval 75.5%

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

   4. *-commutative 75.5%

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

   5. associate-*r* 75.4%

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

   6. *-commutative 75.4%

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in a around inf 61.1%

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

8. Final simplification 61.1%

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

Reproduce

herbie shell --seed 2024118 
(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))))