NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.1% → 99.7%

Time: 10.8s

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: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

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(Float64(0.5 * 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[(N[(0.5 * N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*r* 79.5%

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

   3. associate-*r/ 79.5%

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

   4. associate-*r* 79.5%

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

   5. *-rgt-identity 79.5%

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

   6. sub-neg 79.5%

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

   7. distribute-neg-frac 79.5%

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

   8. metadata-eval 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in a around 0 56.3%

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

   1. *-un-lft-identity 56.3%

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

   2. difference-of-squares 60.2%

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

   3. times-frac 65.7%

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

   4. div-inv 65.7%

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

   5. metadata-eval 65.7%

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

   6. associate-*l/ 65.7%

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

   7. *-un-lft-identity 65.7%

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

   8. *-commutative 65.7%

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

   9. *-un-lft-identity 65.7%

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

   10. times-frac 65.7%

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

   11. metadata-eval 65.7%

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

7. Applied egg-rr 65.7%

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

   1. associate-*l/ 65.7%

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

   2. *-lft-identity 65.7%

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

   3. associate-/l* 65.7%

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

   4. +-commutative 65.7%

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

9. Simplified 65.7%

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

10. Taylor expanded in a around 0 99.7%

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

Alternative 2: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\frac{\pi}{b}}{a + b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{\frac{\pi}{b}}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5e+67)
   (* 0.5 (/ (/ (/ PI b) (+ a b)) a))
   (* (/ -0.5 b) (/ (/ PI b) (- a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.5e+67) {
		tmp = 0.5 * (((((double) M_PI) / b) / (a + b)) / a);
	} else {
		tmp = (-0.5 / b) * ((((double) M_PI) / b) / -a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.5e+67) {
		tmp = 0.5 * (((Math.PI / b) / (a + b)) / a);
	} else {
		tmp = (-0.5 / b) * ((Math.PI / b) / -a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.5e+67:
		tmp = 0.5 * (((math.pi / b) / (a + b)) / a)
	else:
		tmp = (-0.5 / b) * ((math.pi / b) / -a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.5e+67)
		tmp = Float64(0.5 * Float64(Float64(Float64(pi / b) / Float64(a + b)) / a));
	else
		tmp = Float64(Float64(-0.5 / b) * Float64(Float64(pi / b) / Float64(-a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.5e+67)
		tmp = 0.5 * (((pi / b) / (a + b)) / a);
	else
		tmp = (-0.5 / b) * ((pi / b) / -a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.5e+67], N[(0.5 * N[(N[(N[(Pi / b), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4999999999999995e67

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*r* 80.5%

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

      3. associate-*r/ 80.5%

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

      4. associate-*r* 80.5%

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

      5. *-rgt-identity 80.5%

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

      6. sub-neg 80.5%

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

      7. distribute-neg-frac 80.5%

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

      8. metadata-eval 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in a around inf 59.4%

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

      1. difference-of-squares 65.9%

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

      2. times-frac 75.9%

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

   7. Applied egg-rr 75.9%

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

   8. Taylor expanded in b around 0 99.6%

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

      1. neg-mul-1 99.6%

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

   10. Simplified 99.6%

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

       1. frac-2neg 99.6%

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

       2. frac-times 89.6%

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

       3. add-sqr-sqrt 44.3%

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

       4. sqrt-unprod 54.3%

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

       5. sqr-neg 54.3%

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

       6. sqrt-unprod 16.1%

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

       7. add-sqr-sqrt 27.9%

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

       8. add-sqr-sqrt 11.8%

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

       9. sqrt-unprod 51.7%

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

       10. sqr-neg 51.7%

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

       11. sqrt-unprod 45.3%

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

       12. add-sqr-sqrt 89.6%

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

   12. Applied egg-rr 89.6%

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

       1. distribute-rgt-neg-in 89.6%

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

       2. distribute-frac-neg 89.6%

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

       3. *-commutative 89.6%

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

       4. times-frac 97.4%

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

       5. associate-*l/ 97.4%

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

       6. associate-/l* 97.5%

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

       7. distribute-lft-neg-in 97.5%

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

       8. metadata-eval 97.5%

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

       9. +-commutative 97.5%

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

   14. Simplified 97.5%

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

   if 6.4999999999999995e67 < b

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-*r* 74.5%

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

      3. associate-*r/ 74.4%

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

      4. associate-*r* 74.4%

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

      5. *-rgt-identity 74.4%

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

      6. sub-neg 74.4%

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

      7. distribute-neg-frac 74.4%

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

      8. metadata-eval 74.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in a around inf 42.0%

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

      1. difference-of-squares 46.9%

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

      2. times-frac 46.9%

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

   7. Applied egg-rr 46.9%

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

   8. Taylor expanded in b around 0 99.7%

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

      1. neg-mul-1 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in b around inf 99.7%

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

Alternative 3: 73.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{--0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{\frac{\pi}{b}}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.8e-77)
   (* (/ (/ PI b) a) (/ (- -0.5) a))
   (* (/ -0.5 b) (/ (/ PI b) (- a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.8e-77) {
		tmp = ((((double) M_PI) / b) / a) * (-(-0.5) / a);
	} else {
		tmp = (-0.5 / b) * ((((double) M_PI) / b) / -a);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.8e-77) {
		tmp = ((Math.PI / b) / a) * (-(-0.5) / a);
	} else {
		tmp = (-0.5 / b) * ((Math.PI / b) / -a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.8e-77:
		tmp = ((math.pi / b) / a) * (-(-0.5) / a)
	else:
		tmp = (-0.5 / b) * ((math.pi / b) / -a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.8e-77)
		tmp = Float64(Float64(Float64(pi / b) / a) * Float64(Float64(-(-0.5)) / a));
	else
		tmp = Float64(Float64(-0.5 / b) * Float64(Float64(pi / b) / Float64(-a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.8e-77)
		tmp = ((pi / b) / a) * (-(-0.5) / a);
	else
		tmp = (-0.5 / b) * ((pi / b) / -a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.8e-77], N[(N[(N[(Pi / b), $MachinePrecision] / a), $MachinePrecision] * N[((--0.5) / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.79999999999999966e-77

   1. Initial program 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*r* 77.5%

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

      3. associate-*r/ 77.5%

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

      4. associate-*r* 77.5%

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

      5. *-rgt-identity 77.5%

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

      6. sub-neg 77.5%

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

      7. distribute-neg-frac 77.5%

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

      8. metadata-eval 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in a around inf 60.5%

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

      1. difference-of-squares 68.0%

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

      2. times-frac 79.5%

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

   7. Applied egg-rr 79.5%

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

   8. Taylor expanded in b around 0 99.6%

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

      1. neg-mul-1 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in b around 0 75.7%

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

   if 6.79999999999999966e-77 < b

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*r* 84.8%

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

      3. associate-*r/ 84.7%

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

      4. associate-*r* 84.7%

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

      5. *-rgt-identity 84.7%

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

      6. sub-neg 84.7%

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

      7. distribute-neg-frac 84.7%

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

      8. metadata-eval 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in a around inf 46.5%

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

      1. difference-of-squares 49.3%

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

      2. times-frac 49.4%

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

   7. Applied egg-rr 49.4%

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

   8. Taylor expanded in b around 0 99.7%

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

      1. neg-mul-1 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in b around inf 79.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\pi}{b}}{a} \cdot \frac{--0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{\frac{\pi}{b}}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*r* 79.5%

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

   3. associate-*r/ 79.5%

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

   4. associate-*r* 79.5%

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

   5. *-rgt-identity 79.5%

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

   6. sub-neg 79.5%

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

   7. distribute-neg-frac 79.5%

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

   8. metadata-eval 79.5%

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

3. Simplified 79.5%

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

   1. div-inv 79.5%

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

   2. neg-mul-1 79.5%

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

   3. sub-neg 79.5%

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

   4. frac-sub 79.5%

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

   5. *-un-lft-identity 79.5%

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

   6. *-rgt-identity 79.5%

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

6. Applied egg-rr 79.5%

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

   1. div-inv 79.5%

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

   2. div-inv 79.5%

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

   3. metadata-eval 79.5%

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

   4. associate-*l* 79.5%

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

   5. difference-of-squares 85.7%

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

   6. times-frac 99.6%

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

   7. metadata-eval 99.6%

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

   8. div-inv 99.6%

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

   9. frac-times 99.7%

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

   10. *-un-lft-identity 99.7%

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

   11. *-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

   1. +-commutative 99.7%

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

   2. associate-/l/ 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in b around inf 61.5%

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

    1. *-un-lft-identity 61.5%

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

    2. sub-neg 61.5%

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

    3. add-sqr-sqrt 30.0%

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

    4. sqrt-unprod 76.2%

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

    5. sqr-neg 76.2%

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

    6. sqrt-unprod 50.9%

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

    7. add-sqr-sqrt 99.6%

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

    8. *-commutative 99.6%

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

    9. *-commutative 99.6%

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

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

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

Alternative 5: 62.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*r* 79.5%

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

   3. associate-*r/ 79.5%

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

   4. associate-*r* 79.5%

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

   5. *-rgt-identity 79.5%

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

   6. sub-neg 79.5%

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

   7. distribute-neg-frac 79.5%

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

   8. metadata-eval 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in a around inf 56.6%

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

   1. difference-of-squares 62.9%

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

   2. times-frac 71.3%

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

7. Applied egg-rr 71.3%

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

8. Taylor expanded in b around 0 99.6%

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

   1. neg-mul-1 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in b around 0 65.4%

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

12. Final simplification 65.4%

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

Alternative 6: 62.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*r* 79.5%

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

   3. associate-*r/ 79.5%

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

   4. associate-*r* 79.5%

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

   5. *-rgt-identity 79.5%

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

   6. sub-neg 79.5%

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

   7. distribute-neg-frac 79.5%

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

   8. metadata-eval 79.5%

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

3. Simplified 79.5%

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

5. Taylor expanded in a around inf 56.6%

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

   1. difference-of-squares 62.9%

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

   2. times-frac 71.3%

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

7. Applied egg-rr 71.3%

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

8. Taylor expanded in b around 0 68.1%

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

9. Taylor expanded in b around 0 65.4%

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

    1. associate-*r/ 65.4%

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

    2. mul-1-neg 65.4%

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

11. Simplified 65.4%

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

12. Final simplification 65.4%

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

Reproduce

herbie shell --seed 2024135 
(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))))