NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.7%

Time: 7.8s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 78.6%

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

   1. associate-*r/ 78.6%

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

   2. *-rgt-identity 78.6%

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

   3. sub-neg 78.6%

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

   4. distribute-neg-frac 78.6%

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

   5. metadata-eval 78.6%

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

3. Simplified 78.6%

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

   1. frac-add 78.5%

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

   2. *-un-lft-identity 78.5%

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

5. Applied egg-rr 78.5%

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

   1. *-commutative 78.5%

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

   2. neg-mul-1 78.5%

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

   3. sub-neg 78.5%

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

7. Simplified 78.5%

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

   1. associate-*r/ 78.6%

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

   2. div-inv 78.6%

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

   3. metadata-eval 78.6%

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

   4. *-commutative 78.6%

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

9. Applied egg-rr 78.6%

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

    1. associate-*l/ 78.6%

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

11. Applied egg-rr 78.6%

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

    1. difference-of-squares 86.8%

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

    2. times-frac 99.7%

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

    3. *-commutative 99.7%

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

    4. +-commutative 99.7%

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

    5. *-inverses 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 85.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.58 \lor \neg \left(b \leq 6.2 \cdot 10^{-74}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -0.58) (not (<= b 6.2e-74)))
   (* 0.5 (/ PI (* b (* a b))))
   (* 0.5 (/ PI (* a (* a b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -0.58) || !(b <= 6.2e-74)) {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (a * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -0.58) || !(b <= 6.2e-74)) {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	} else {
		tmp = 0.5 * (Math.PI / (a * (a * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -0.58) or not (b <= 6.2e-74):
		tmp = 0.5 * (math.pi / (b * (a * b)))
	else:
		tmp = 0.5 * (math.pi / (a * (a * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -0.58) || !(b <= 6.2e-74))
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(a * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -0.58) || ~((b <= 6.2e-74)))
		tmp = 0.5 * (pi / (b * (a * b)));
	else
		tmp = 0.5 * (pi / (a * (a * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -0.58], N[Not[LessEqual[b, 6.2e-74]], $MachinePrecision]], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.57999999999999996 or 6.2000000000000003e-74 < b

   1. Initial program 82.1%

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

      1. associate-*r/ 82.1%

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

      2. *-rgt-identity 82.1%

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

      3. sub-neg 82.1%

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

      4. distribute-neg-frac 82.1%

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

      5. metadata-eval 82.1%

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

   3. Simplified 82.1%

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

      1. frac-add 82.0%

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

      2. *-un-lft-identity 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. *-commutative 82.0%

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

      2. neg-mul-1 82.0%

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

      3. sub-neg 82.0%

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

   7. Simplified 82.0%

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

      1. associate-*r/ 82.1%

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

      2. div-inv 82.1%

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

      3. metadata-eval 82.1%

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

      4. *-commutative 82.1%

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

   9. Applied egg-rr 82.1%

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

   10. Taylor expanded in b around inf 81.6%

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

       1. unpow2 81.6%

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

       2. associate-*r* 89.1%

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

   12. Simplified 89.1%

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

   if -0.57999999999999996 < b < 6.2000000000000003e-74

   1. Initial program 73.8%

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

      1. times-frac 73.9%

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

      2. *-commutative 73.9%

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

      3. times-frac 73.9%

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

      4. difference-of-squares 81.3%

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

      5. associate-/r* 83.5%

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

      6. metadata-eval 83.5%

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

      7. sub-neg 83.5%

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

      8. distribute-neg-frac 83.5%

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

      9. metadata-eval 83.5%

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

   3. Simplified 83.5%

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

      1. add-cube-cbrt 82.9%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)} \]

      2. pow3 82.9%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{3}} \]

   5. Applied egg-rr 82.9%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{3}} \]

   6. Taylor expanded in b around 0 72.4%

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

      1. unpow2 72.4%

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

      2. associate-*l* 90.5%

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

   8. Simplified 90.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.58 \lor \neg \left(b \leq 6.2 \cdot 10^{-74}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(a \cdot b\right)}\\ \end{array} \]

Alternative 3: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -145 \lor \neg \left(b \leq 3.8 \cdot 10^{-74}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -145.0) (not (<= b 3.8e-74)))
   (* 0.5 (/ PI (* b (* a b))))
   (/ (* 0.5 (/ PI a)) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -145.0) || !(b <= 3.8e-74)) {
		tmp = 0.5 * (((double) M_PI) / (b * (a * b)));
	} else {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -145.0) || !(b <= 3.8e-74)) {
		tmp = 0.5 * (Math.PI / (b * (a * b)));
	} else {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -145.0) or not (b <= 3.8e-74):
		tmp = 0.5 * (math.pi / (b * (a * b)))
	else:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -145.0) || !(b <= 3.8e-74))
		tmp = Float64(0.5 * Float64(pi / Float64(b * Float64(a * b))));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -145.0) || ~((b <= 3.8e-74)))
		tmp = 0.5 * (pi / (b * (a * b)));
	else
		tmp = (0.5 * (pi / a)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -145.0], N[Not[LessEqual[b, 3.8e-74]], $MachinePrecision]], N[(0.5 * N[(Pi / N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -145 or 3.7999999999999996e-74 < b

   1. Initial program 82.1%

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

      1. associate-*r/ 82.1%

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

      2. *-rgt-identity 82.1%

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

      3. sub-neg 82.1%

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

      4. distribute-neg-frac 82.1%

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

      5. metadata-eval 82.1%

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

   3. Simplified 82.1%

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

      1. frac-add 82.0%

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

      2. *-un-lft-identity 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. *-commutative 82.0%

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

      2. neg-mul-1 82.0%

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

      3. sub-neg 82.0%

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

   7. Simplified 82.0%

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

      1. associate-*r/ 82.1%

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

      2. div-inv 82.1%

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

      3. metadata-eval 82.1%

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

      4. *-commutative 82.1%

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

   9. Applied egg-rr 82.1%

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

   10. Taylor expanded in b around inf 81.6%

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

       1. unpow2 81.6%

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

       2. associate-*r* 89.1%

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

   12. Simplified 89.1%

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

   if -145 < b < 3.7999999999999996e-74

   1. Initial program 73.8%

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

      1. associate-*r/ 73.9%

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

      2. *-rgt-identity 73.9%

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

      3. sub-neg 73.9%

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

      4. distribute-neg-frac 73.9%

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

      5. metadata-eval 73.9%

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

   3. Simplified 73.9%

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

      1. frac-add 73.8%

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

      2. *-un-lft-identity 73.8%

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

   5. Applied egg-rr 73.8%

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

      1. *-commutative 73.8%

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

      2. neg-mul-1 73.8%

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

      3. sub-neg 73.8%

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

   7. Simplified 73.8%

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

      1. associate-*r/ 73.7%

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

      2. div-inv 73.7%

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

      3. metadata-eval 73.7%

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

      4. *-commutative 73.7%

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

   9. Applied egg-rr 73.7%

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

   10. Taylor expanded in b around 0 90.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -145 \lor \neg \left(b \leq 3.8 \cdot 10^{-74}\right):\\ \;\;\;\;0.5 \cdot \frac{\pi}{b \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \end{array} \]

Alternative 4: 85.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.85 \lor \neg \left(b \leq 6.2 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -0.85) (not (<= b 6.2e-74)))
   (/ (* PI (/ 0.5 b)) (* a b))
   (/ (* 0.5 (/ PI a)) (* a b))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -0.85) || !(b <= 6.2e-74)) {
		tmp = (((double) M_PI) * (0.5 / b)) / (a * b);
	} else {
		tmp = (0.5 * (((double) M_PI) / a)) / (a * b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -0.85) || !(b <= 6.2e-74)) {
		tmp = (Math.PI * (0.5 / b)) / (a * b);
	} else {
		tmp = (0.5 * (Math.PI / a)) / (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -0.85) or not (b <= 6.2e-74):
		tmp = (math.pi * (0.5 / b)) / (a * b)
	else:
		tmp = (0.5 * (math.pi / a)) / (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -0.85) || !(b <= 6.2e-74))
		tmp = Float64(Float64(pi * Float64(0.5 / b)) / Float64(a * b));
	else
		tmp = Float64(Float64(0.5 * Float64(pi / a)) / Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -0.85) || ~((b <= 6.2e-74)))
		tmp = (pi * (0.5 / b)) / (a * b);
	else
		tmp = (0.5 * (pi / a)) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -0.85], N[Not[LessEqual[b, 6.2e-74]], $MachinePrecision]], N[(N[(Pi * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.849999999999999978 or 6.2000000000000003e-74 < b

   1. Initial program 82.1%

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

      1. associate-*r/ 82.1%

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

      2. *-rgt-identity 82.1%

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

      3. sub-neg 82.1%

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

      4. distribute-neg-frac 82.1%

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

      5. metadata-eval 82.1%

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

   3. Simplified 82.1%

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

      1. frac-add 82.0%

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

      2. *-un-lft-identity 82.0%

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

   5. Applied egg-rr 82.0%

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

      1. *-commutative 82.0%

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

      2. neg-mul-1 82.0%

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

      3. sub-neg 82.0%

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

   7. Simplified 82.0%

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

      1. associate-*r/ 82.1%

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

      2. div-inv 82.1%

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

      3. metadata-eval 82.1%

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

      4. *-commutative 82.1%

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

   9. Applied egg-rr 82.1%

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

   10. Taylor expanded in b around inf 90.7%

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

       1. associate-*r/ 90.7%

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

       2. *-commutative 90.7%

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

       3. associate-*r/ 90.6%

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

   12. Simplified 90.6%

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

   if -0.849999999999999978 < b < 6.2000000000000003e-74

   1. Initial program 73.8%

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

      1. associate-*r/ 73.9%

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

      2. *-rgt-identity 73.9%

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

      3. sub-neg 73.9%

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

      4. distribute-neg-frac 73.9%

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

      5. metadata-eval 73.9%

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

   3. Simplified 73.9%

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

      1. frac-add 73.8%

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

      2. *-un-lft-identity 73.8%

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

   5. Applied egg-rr 73.8%

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

      1. *-commutative 73.8%

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

      2. neg-mul-1 73.8%

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

      3. sub-neg 73.8%

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

   7. Simplified 73.8%

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

      1. associate-*r/ 73.7%

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

      2. div-inv 73.7%

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

      3. metadata-eval 73.7%

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

      4. *-commutative 73.7%

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

   9. Applied egg-rr 73.7%

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

   10. Taylor expanded in b around 0 90.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.85 \lor \neg \left(b \leq 6.2 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\pi \cdot \frac{0.5}{b}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\pi}{a}}{a \cdot b}\\ \end{array} \]

Alternative 5: 63.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. times-frac 78.6%

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

   2. *-commutative 78.6%

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

   3. times-frac 78.6%

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

   4. difference-of-squares 86.8%

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

   5. associate-/r* 87.8%

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

   6. metadata-eval 87.8%

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

   7. sub-neg 87.8%

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

   8. distribute-neg-frac 87.8%

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

   9. metadata-eval 87.8%

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

3. Simplified 87.8%

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

   1. add-cube-cbrt 87.3%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right) \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)} \]

   2. pow3 87.2%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{3}} \]

5. Applied egg-rr 87.2%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{3}} \]

6. Taylor expanded in b around 0 53.6%

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

   1. unpow2 53.6%

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

   2. associate-*l* 61.2%

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

8. Simplified 61.2%

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

9. Final simplification 61.2%

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

Reproduce

herbie shell --seed 2023195 
(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))))