NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.0% → 99.6%

Time: 9.7s

Alternatives: 8

Speedup: 1.0×

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: 79.0% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 73.2%

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

   2. *-commutative 73.2%

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

   3. times-frac 73.2%

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

   4. difference-of-squares 84.9%

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

   5. associate-/r* 86.0%

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

   6. metadata-eval 86.0%

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

   7. sub-neg 86.0%

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

   8. distribute-neg-frac 86.0%

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

   9. metadata-eval 86.0%

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

3. Simplified 86.0%

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

   1. clear-num 84.9%

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

   2. inv-pow 84.9%

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

5. Applied egg-rr 84.9%

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

   1. unpow-1 84.9%

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

7. Applied egg-rr 84.9%

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

   1. distribute-lft-in 78.2%

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

   2. associate-*l/ 78.2%

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

   3. metadata-eval 78.2%

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

   4. associate-/r/ 78.2%

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

   5. associate-*l/ 78.2%

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

   6. metadata-eval 78.2%

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

   7. associate-/r/ 78.2%

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

9. Applied egg-rr 78.2%

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

    1. distribute-lft-in 84.8%

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

    2. associate-*l/ 84.8%

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

    3. *-commutative 84.8%

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

    4. associate-/r* 99.3%

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

    5. +-commutative 99.3%

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

    6. distribute-lft-in 99.3%

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

    7. associate-*r/ 99.6%

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

    8. metadata-eval 99.6%

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

    9. associate-*r/ 99.6%

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

    10. metadata-eval 99.6%

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

    11. +-commutative 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{a} \cdot \left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right)\\ \mathbf{if}\;b \leq 3.35 \cdot 10^{-243}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(b \cdot a\right)}{0.5}}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{b \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 a) (* 0.5 (/ (/ PI (+ b a)) (- b a))))))
   (if (<= b 3.35e-243)
     (/ PI (/ (* a (* b a)) 0.5))
     (if (<= b 1.3e-175)
       t_0
       (if (<= b 1.25e-111)
         (* (/ PI a) (/ (/ 0.5 b) a))
         (if (<= b 4e+141)
           t_0
           (/ (* (/ PI b) (+ (/ 1.0 a) (/ -1.0 b))) (* b 2.0))))))))

C

double code(double a, double b) {
	double t_0 = (1.0 / a) * (0.5 * ((((double) M_PI) / (b + a)) / (b - a)));
	double tmp;
	if (b <= 3.35e-243) {
		tmp = ((double) M_PI) / ((a * (b * a)) / 0.5);
	} else if (b <= 1.3e-175) {
		tmp = t_0;
	} else if (b <= 1.25e-111) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else if (b <= 4e+141) {
		tmp = t_0;
	} else {
		tmp = ((((double) M_PI) / b) * ((1.0 / a) + (-1.0 / b))) / (b * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (1.0 / a) * (0.5 * ((Math.PI / (b + a)) / (b - a)));
	double tmp;
	if (b <= 3.35e-243) {
		tmp = Math.PI / ((a * (b * a)) / 0.5);
	} else if (b <= 1.3e-175) {
		tmp = t_0;
	} else if (b <= 1.25e-111) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else if (b <= 4e+141) {
		tmp = t_0;
	} else {
		tmp = ((Math.PI / b) * ((1.0 / a) + (-1.0 / b))) / (b * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (1.0 / a) * (0.5 * ((math.pi / (b + a)) / (b - a)))
	tmp = 0
	if b <= 3.35e-243:
		tmp = math.pi / ((a * (b * a)) / 0.5)
	elif b <= 1.3e-175:
		tmp = t_0
	elif b <= 1.25e-111:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	elif b <= 4e+141:
		tmp = t_0
	else:
		tmp = ((math.pi / b) * ((1.0 / a) + (-1.0 / b))) / (b * 2.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(1.0 / a) * Float64(0.5 * Float64(Float64(pi / Float64(b + a)) / Float64(b - a))))
	tmp = 0.0
	if (b <= 3.35e-243)
		tmp = Float64(pi / Float64(Float64(a * Float64(b * a)) / 0.5));
	elseif (b <= 1.3e-175)
		tmp = t_0;
	elseif (b <= 1.25e-111)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	elseif (b <= 4e+141)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(pi / b) * Float64(Float64(1.0 / a) + Float64(-1.0 / b))) / Float64(b * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (1.0 / a) * (0.5 * ((pi / (b + a)) / (b - a)));
	tmp = 0.0;
	if (b <= 3.35e-243)
		tmp = pi / ((a * (b * a)) / 0.5);
	elseif (b <= 1.3e-175)
		tmp = t_0;
	elseif (b <= 1.25e-111)
		tmp = (pi / a) * ((0.5 / b) / a);
	elseif (b <= 4e+141)
		tmp = t_0;
	else
		tmp = ((pi / b) * ((1.0 / a) + (-1.0 / b))) / (b * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(1.0 / a), $MachinePrecision] * N[(0.5 * N[(N[(Pi / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.35e-243], N[(Pi / N[(N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e-175], t$95$0, If[LessEqual[b, 1.25e-111], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+141], t$95$0, N[(N[(N[(Pi / b), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 3.35000000000000004e-243

   1. Initial program 71.9%

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

      1. times-frac 71.9%

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

      2. *-commutative 71.9%

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

      3. times-frac 71.9%

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

      4. difference-of-squares 86.0%

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

      5. associate-/r* 87.4%

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

      6. metadata-eval 87.4%

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

      7. sub-neg 87.4%

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

      8. distribute-neg-frac 87.4%

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

      9. metadata-eval 87.4%

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

   3. Simplified 87.4%

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

      1. add-cube-cbrt 87.1%

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

      2. pow3 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in b around 0 59.5%

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

      1. associate-*r/ 59.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. *-commutative 59.5%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

      3. unpow2 59.5%

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

      4. associate-/l* 59.5%

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

      5. associate-*l* 68.2%

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

   8. Simplified 68.2%

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

   if 3.35000000000000004e-243 < b < 1.3e-175 or 1.2500000000000001e-111 < b < 4.00000000000000007e141

   1. Initial program 87.4%

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

      1. times-frac 87.4%

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

      2. *-commutative 87.4%

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

      3. times-frac 87.4%

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

      4. difference-of-squares 92.2%

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

      5. associate-/r* 92.3%

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

      6. metadata-eval 92.3%

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

      7. sub-neg 92.3%

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

      8. distribute-neg-frac 92.3%

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

      9. metadata-eval 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in a around 0 77.9%

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

   if 1.3e-175 < b < 1.2500000000000001e-111

   1. Initial program 86.7%

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

      1. times-frac 86.6%

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

      2. *-commutative 86.6%

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

      3. times-frac 86.6%

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

      4. difference-of-squares 86.6%

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

      5. associate-/r* 86.7%

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

      6. metadata-eval 86.7%

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

      7. sub-neg 86.7%

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

      8. distribute-neg-frac 86.7%

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

      9. metadata-eval 86.7%

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

   3. Simplified 86.7%

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

      1. add-cube-cbrt 85.7%

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

      2. pow2 85.7%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right) \]

   5. Applied egg-rr 85.7%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)} \]

   6. Taylor expanded in b around 0 79.3%

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

      1. associate-*r/ 79.3%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. *-commutative 79.3%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

      3. times-frac 79.4%

         \[\leadsto \color{blue}{\frac{\pi}{{a}^{2}} \cdot \frac{0.5}{b}} \]

      4. unpow2 79.4%

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

   8. Simplified 79.4%

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

      1. associate-*l/ 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. times-frac 92.6%

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

   12. Simplified 92.6%

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

   if 4.00000000000000007e141 < b

   1. Initial program 42.9%

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

   2. Taylor expanded in b around inf 62.9%

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

      1. unpow2 62.9%

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

      2. associate-/r* 65.7%

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

   4. Simplified 65.7%

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

      1. frac-times 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. associate-*r/ 65.7%

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

      2. *-rgt-identity 65.7%

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

   8. Simplified 65.7%

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

      1. associate-*l/ 99.5%

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

      2. *-commutative 99.5%

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

   10. Applied egg-rr 99.5%

       \[\leadsto \color{blue}{\frac{\frac{\pi}{b} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.35 \cdot 10^{-243}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(b \cdot a\right)}{0.5}}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-111}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.5 \cdot \frac{\frac{\pi}{b + a}}{b - a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{b \cdot 2}\\ \end{array} \]

Alternative 3: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.3 \cdot 10^{-243}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(b \cdot a\right)}{0.5}}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\left(\frac{-0.5}{b} + \frac{0.5}{a}\right) \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{b \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.3e-243)
   (/ PI (/ (* a (* b a)) 0.5))
   (if (<= b 2e+131)
     (* (+ (/ -0.5 b) (/ 0.5 a)) (/ PI (- (* b b) (* a a))))
     (/ (* (/ PI b) (+ (/ 1.0 a) (/ -1.0 b))) (* b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.3e-243) {
		tmp = ((double) M_PI) / ((a * (b * a)) / 0.5);
	} else if (b <= 2e+131) {
		tmp = ((-0.5 / b) + (0.5 / a)) * (((double) M_PI) / ((b * b) - (a * a)));
	} else {
		tmp = ((((double) M_PI) / b) * ((1.0 / a) + (-1.0 / b))) / (b * 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.3e-243) {
		tmp = Math.PI / ((a * (b * a)) / 0.5);
	} else if (b <= 2e+131) {
		tmp = ((-0.5 / b) + (0.5 / a)) * (Math.PI / ((b * b) - (a * a)));
	} else {
		tmp = ((Math.PI / b) * ((1.0 / a) + (-1.0 / b))) / (b * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.3e-243:
		tmp = math.pi / ((a * (b * a)) / 0.5)
	elif b <= 2e+131:
		tmp = ((-0.5 / b) + (0.5 / a)) * (math.pi / ((b * b) - (a * a)))
	else:
		tmp = ((math.pi / b) * ((1.0 / a) + (-1.0 / b))) / (b * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.3e-243)
		tmp = Float64(pi / Float64(Float64(a * Float64(b * a)) / 0.5));
	elseif (b <= 2e+131)
		tmp = Float64(Float64(Float64(-0.5 / b) + Float64(0.5 / a)) * Float64(pi / Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(pi / b) * Float64(Float64(1.0 / a) + Float64(-1.0 / b))) / Float64(b * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.3e-243)
		tmp = pi / ((a * (b * a)) / 0.5);
	elseif (b <= 2e+131)
		tmp = ((-0.5 / b) + (0.5 / a)) * (pi / ((b * b) - (a * a)));
	else
		tmp = ((pi / b) * ((1.0 / a) + (-1.0 / b))) / (b * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.3e-243], N[(Pi / N[(N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+131], N[(N[(N[(-0.5 / b), $MachinePrecision] + N[(0.5 / a), $MachinePrecision]), $MachinePrecision] * N[(Pi / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.30000000000000013e-243

   1. Initial program 71.9%

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

      1. times-frac 71.9%

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

      2. *-commutative 71.9%

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

      3. times-frac 71.9%

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

      4. difference-of-squares 86.0%

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

      5. associate-/r* 87.4%

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

      6. metadata-eval 87.4%

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

      7. sub-neg 87.4%

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

      8. distribute-neg-frac 87.4%

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

      9. metadata-eval 87.4%

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

   3. Simplified 87.4%

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

      1. add-cube-cbrt 87.1%

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

      2. pow3 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in b around 0 59.5%

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

      1. associate-*r/ 59.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. *-commutative 59.5%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

      3. unpow2 59.5%

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

      4. associate-/l* 59.5%

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

      5. associate-*l* 68.2%

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

   8. Simplified 68.2%

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

   if 3.30000000000000013e-243 < b < 1.9999999999999998e131

   1. Initial program 86.8%

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

      1. times-frac 86.8%

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

      2. *-commutative 86.8%

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

      3. times-frac 86.8%

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

      4. difference-of-squares 90.9%

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

      5. associate-/r* 90.9%

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

      6. metadata-eval 90.9%

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

      7. sub-neg 90.9%

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

      8. distribute-neg-frac 90.9%

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

      9. metadata-eval 90.9%

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

   3. Simplified 90.9%

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

      1. distribute-lft-in 82.8%

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

      2. associate-/l/ 82.7%

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

      3. associate-/l/ 82.8%

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

   5. Applied egg-rr 82.8%

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

      1. distribute-lft-out 90.9%

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

      2. associate-*r* 90.9%

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

      3. associate-*l/ 90.8%

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

      4. *-commutative 90.8%

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

      5. difference-of-squares 86.7%

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

      6. associate-*l/ 86.8%

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

      7. distribute-lft-in 86.8%

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

      8. associate-*r/ 86.8%

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

      9. metadata-eval 86.8%

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

      10. associate-*r/ 86.8%

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

      11. metadata-eval 86.8%

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

   7. Simplified 86.8%

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

   if 1.9999999999999998e131 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 66.2%

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

      1. unpow2 66.2%

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

      2. associate-/r* 68.6%

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

   4. Simplified 68.6%

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

      1. frac-times 68.8%

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

   6. Applied egg-rr 68.8%

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

      1. associate-*r/ 68.8%

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

      2. *-rgt-identity 68.8%

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

   8. Simplified 68.8%

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

      1. associate-*l/ 99.5%

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

      2. *-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.3 \cdot 10^{-243}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(b \cdot a\right)}{0.5}}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\left(\frac{-0.5}{b} + \frac{0.5}{a}\right) \cdot \frac{\pi}{b \cdot b - a \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{b} \cdot \left(\frac{1}{a} + \frac{-1}{b}\right)}{b \cdot 2}\\ \end{array} \]

Alternative 4: 69.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-31}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-31)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (* 0.5 (/ PI (* a (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-31) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = 0.5 * (((double) M_PI) / (a * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-31) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = 0.5 * (Math.PI / (a * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.3e-31:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = 0.5 * (math.pi / (a * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.3e-31)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(0.5 * Float64(pi / Float64(a * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.3e-31)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = 0.5 * (pi / (a * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.3e-31], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(Pi / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2999999999999998e-31

   1. Initial program 68.7%

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

      1. times-frac 68.8%

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

      2. *-commutative 68.8%

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

      3. times-frac 68.8%

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

      4. difference-of-squares 82.0%

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

      5. associate-/r* 83.0%

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

      6. metadata-eval 83.0%

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

      7. sub-neg 83.0%

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

      8. distribute-neg-frac 83.0%

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

      9. metadata-eval 83.0%

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

   3. Simplified 83.0%

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

      1. add-cube-cbrt 82.6%

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

      2. pow2 82.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right) \]

   5. Applied egg-rr 82.6%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)} \]

   6. Taylor expanded in b around 0 69.7%

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

      1. associate-*r/ 69.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. *-commutative 69.7%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

      3. times-frac 68.6%

         \[\leadsto \color{blue}{\frac{\pi}{{a}^{2}} \cdot \frac{0.5}{b}} \]

      4. unpow2 68.6%

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

   8. Simplified 68.6%

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

      1. associate-*l/ 67.3%

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

   10. Applied egg-rr 67.3%

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

       1. times-frac 84.8%

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

   12. Simplified 84.8%

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

   if -2.2999999999999998e-31 < a

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

      2. associate-/r/ 74.7%

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

      3. associate-*l/ 74.7%

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

      4. *-commutative 74.7%

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

      5. associate-/r/ 74.7%

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

      6. times-frac 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in b around inf 60.3%

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

      1. unpow2 60.3%

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

   6. Simplified 60.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-31}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\pi}{a \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 5: 69.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.1e-33)
   (* (/ PI a) (/ (/ 0.5 b) a))
   (* 0.5 (/ (/ PI a) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.1e-33) {
		tmp = (((double) M_PI) / a) * ((0.5 / b) / a);
	} else {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.1e-33) {
		tmp = (Math.PI / a) * ((0.5 / b) / a);
	} else {
		tmp = 0.5 * ((Math.PI / a) / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.1e-33:
		tmp = (math.pi / a) * ((0.5 / b) / a)
	else:
		tmp = 0.5 * ((math.pi / a) / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.1e-33)
		tmp = Float64(Float64(pi / a) * Float64(Float64(0.5 / b) / a));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.1e-33)
		tmp = (pi / a) * ((0.5 / b) / a);
	else
		tmp = 0.5 * ((pi / a) / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.1e-33], N[(N[(Pi / a), $MachinePrecision] * N[(N[(0.5 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1e-33

   1. Initial program 68.7%

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

      1. times-frac 68.8%

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

      2. *-commutative 68.8%

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

      3. times-frac 68.8%

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

      4. difference-of-squares 82.0%

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

      5. associate-/r* 83.0%

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

      6. metadata-eval 83.0%

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

      7. sub-neg 83.0%

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

      8. distribute-neg-frac 83.0%

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

      9. metadata-eval 83.0%

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

   3. Simplified 83.0%

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

      1. add-cube-cbrt 82.6%

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

      2. pow2 82.6%

         \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right) \]

   5. Applied egg-rr 82.6%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)} \]

   6. Taylor expanded in b around 0 69.7%

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

      1. associate-*r/ 69.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. *-commutative 69.7%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

      3. times-frac 68.6%

         \[\leadsto \color{blue}{\frac{\pi}{{a}^{2}} \cdot \frac{0.5}{b}} \]

      4. unpow2 68.6%

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

   8. Simplified 68.6%

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

      1. associate-*l/ 67.3%

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

   10. Applied egg-rr 67.3%

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

       1. times-frac 84.8%

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

   12. Simplified 84.8%

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

   if -4.1e-33 < a

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

      2. associate-/r/ 74.7%

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

      3. associate-*l/ 74.7%

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

      4. *-commutative 74.7%

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

      5. associate-/r/ 74.7%

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

      6. times-frac 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in b around inf 60.3%

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

      1. unpow2 60.3%

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

   6. Simplified 60.3%

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

      1. div-inv 60.2%

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

   8. Applied egg-rr 60.2%

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

   9. Taylor expanded in a around 0 60.3%

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

       1. associate-/r* 60.3%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{a}}{{b}^{2}}} \cdot 0.5 \]

       2. unpow2 60.3%

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

   11. Simplified 60.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{\pi}{a} \cdot \frac{\frac{0.5}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \]

Alternative 6: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(b \cdot a\right)}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.1e-25)
   (/ PI (/ (* a (* b a)) 0.5))
   (* 0.5 (/ (/ PI a) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.1e-25) {
		tmp = ((double) M_PI) / ((a * (b * a)) / 0.5);
	} else {
		tmp = 0.5 * ((((double) M_PI) / a) / (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.1e-25) {
		tmp = Math.PI / ((a * (b * a)) / 0.5);
	} else {
		tmp = 0.5 * ((Math.PI / a) / (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.1e-25:
		tmp = math.pi / ((a * (b * a)) / 0.5)
	else:
		tmp = 0.5 * ((math.pi / a) / (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.1e-25)
		tmp = Float64(pi / Float64(Float64(a * Float64(b * a)) / 0.5));
	else
		tmp = Float64(0.5 * Float64(Float64(pi / a) / Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.1e-25)
		tmp = pi / ((a * (b * a)) / 0.5);
	else
		tmp = 0.5 * ((pi / a) / (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.1e-25], N[(Pi / N[(N[(a * N[(b * a), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(Pi / a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.09999999999999987e-25

   1. Initial program 68.2%

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

      1. times-frac 68.3%

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

      2. *-commutative 68.3%

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

      3. times-frac 68.3%

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

      4. difference-of-squares 81.7%

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

      5. associate-/r* 82.7%

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

      6. metadata-eval 82.7%

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

      7. sub-neg 82.7%

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

      8. distribute-neg-frac 82.7%

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

      9. metadata-eval 82.7%

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

   3. Simplified 82.7%

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

      1. add-cube-cbrt 82.3%

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

      2. pow3 82.3%

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

   5. Applied egg-rr 82.3%

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

   6. Taylor expanded in b around 0 70.7%

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

      1. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

      2. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

      3. unpow2 70.7%

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

      4. associate-/l* 70.7%

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

      5. associate-*l* 87.2%

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

   8. Simplified 87.2%

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

   if -4.09999999999999987e-25 < a

   1. Initial program 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-/r/ 74.8%

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

      3. associate-*l/ 74.8%

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

      4. *-commutative 74.8%

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

      5. associate-/r/ 74.8%

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

      6. times-frac 74.8%

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

   3. Simplified 74.9%

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

   4. Taylor expanded in b around inf 60.5%

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

      1. unpow2 60.5%

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

   6. Simplified 60.5%

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

      1. div-inv 60.4%

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

   8. Applied egg-rr 60.4%

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

   9. Taylor expanded in a around 0 60.5%

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

       1. associate-/r* 60.5%

          \[\leadsto \color{blue}{\frac{\frac{\pi}{a}}{{b}^{2}}} \cdot 0.5 \]

       2. unpow2 60.5%

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

   11. Simplified 60.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{\pi}{\frac{a \cdot \left(b \cdot a\right)}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\pi}{a}}{b \cdot b}\\ \end{array} \]

Alternative 7: 63.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 73.2%

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

   2. *-commutative 73.2%

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

   3. times-frac 73.2%

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

   4. difference-of-squares 84.9%

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

   5. associate-/r* 86.0%

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

   6. metadata-eval 86.0%

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

   7. sub-neg 86.0%

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

   8. distribute-neg-frac 86.0%

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

   9. metadata-eval 86.0%

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

3. Simplified 86.0%

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

   1. add-cube-cbrt 85.6%

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

   2. pow2 85.6%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right) \]

5. Applied egg-rr 85.6%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)} \]

6. Taylor expanded in b around 0 54.0%

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

   1. associate-*r/ 54.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

   2. *-commutative 54.0%

      \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

   3. times-frac 53.7%

      \[\leadsto \color{blue}{\frac{\pi}{{a}^{2}} \cdot \frac{0.5}{b}} \]

   4. unpow2 53.7%

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

8. Simplified 53.7%

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

   1. associate-*l/ 53.4%

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

10. Applied egg-rr 53.4%

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

    1. times-frac 61.3%

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

12. Simplified 61.3%

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

13. Taylor expanded in b around 0 61.3%

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

14. Final simplification 61.3%

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

Alternative 8: 63.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. times-frac 73.2%

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

   2. *-commutative 73.2%

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

   3. times-frac 73.2%

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

   4. difference-of-squares 84.9%

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

   5. associate-/r* 86.0%

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

   6. metadata-eval 86.0%

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

   7. sub-neg 86.0%

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

   8. distribute-neg-frac 86.0%

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

   9. metadata-eval 86.0%

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

3. Simplified 86.0%

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

   1. add-cube-cbrt 85.6%

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

   2. pow2 85.6%

      \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right) \]

5. Applied egg-rr 85.6%

   \[\leadsto \left(\frac{\frac{\pi}{b + a}}{b - a} \cdot 0.5\right) \cdot \color{blue}{\left({\left(\sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{a} + \frac{-1}{b}}\right)} \]

6. Taylor expanded in b around 0 54.0%

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

   1. associate-*r/ 54.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \pi}{{a}^{2} \cdot b}} \]

   2. *-commutative 54.0%

      \[\leadsto \frac{\color{blue}{\pi \cdot 0.5}}{{a}^{2} \cdot b} \]

   3. times-frac 53.7%

      \[\leadsto \color{blue}{\frac{\pi}{{a}^{2}} \cdot \frac{0.5}{b}} \]

   4. unpow2 53.7%

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

8. Simplified 53.7%

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

   1. associate-*l/ 53.4%

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

10. Applied egg-rr 53.4%

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

    1. times-frac 61.3%

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

12. Simplified 61.3%

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

13. Final simplification 61.3%

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

Reproduce

herbie shell --seed 2023230 
(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))))