NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 9.2s

Alternatives: 6

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: 78.8% 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

\[\begin{array}{l} \\ \frac{\frac{\pi}{2}}{b - a} \cdot \frac{\frac{1}{a} + \frac{-1}{b}}{b + a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-*r/ 75.6%

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

   2. *-rgt-identity 75.6%

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

   3. associate-*l/ 75.6%

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

   4. difference-of-squares 84.5%

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

   5. *-commutative 84.5%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 62.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-309} \lor \neg \left(b \leq 3.8 \cdot 10^{-213}\right):\\ \;\;\;\;\frac{-1}{b} \cdot \left(-0.5 \cdot \frac{\pi}{b \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b \cdot a} \cdot \frac{\pi}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -8e-309) (not (<= b 3.8e-213)))
   (* (/ -1.0 b) (* -0.5 (/ PI (* b a))))
   (* (/ -0.5 (* b a)) (/ PI b))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -8e-309) || !(b <= 3.8e-213)) {
		tmp = (-1.0 / b) * (-0.5 * (((double) M_PI) / (b * a)));
	} else {
		tmp = (-0.5 / (b * a)) * (((double) M_PI) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -8e-309) || !(b <= 3.8e-213)) {
		tmp = (-1.0 / b) * (-0.5 * (Math.PI / (b * a)));
	} else {
		tmp = (-0.5 / (b * a)) * (Math.PI / b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -8e-309) or not (b <= 3.8e-213):
		tmp = (-1.0 / b) * (-0.5 * (math.pi / (b * a)))
	else:
		tmp = (-0.5 / (b * a)) * (math.pi / b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -8e-309) || !(b <= 3.8e-213))
		tmp = Float64(Float64(-1.0 / b) * Float64(-0.5 * Float64(pi / Float64(b * a))));
	else
		tmp = Float64(Float64(-0.5 / Float64(b * a)) * Float64(pi / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -8e-309) || ~((b <= 3.8e-213)))
		tmp = (-1.0 / b) * (-0.5 * (pi / (b * a)));
	else
		tmp = (-0.5 / (b * a)) * (pi / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -8e-309], N[Not[LessEqual[b, 3.8e-213]], $MachinePrecision]], N[(N[(-1.0 / b), $MachinePrecision] * N[(-0.5 * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(Pi / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.0000000000000003e-309 or 3.8e-213 < b

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

      2. associate-*r* 76.2%

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

      3. associate-*r/ 76.3%

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

      4. associate-/l* 76.3%

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

      5. /-rgt-identity 76.3%

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

      6. associate-/l* 76.2%

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

      7. difference-of-squares 84.7%

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

      8. associate-/l* 84.7%

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

      9. associate-/l* 99.6%

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

      10. associate-*r/ 85.6%

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

      11. sub-neg 85.6%

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

      12. distribute-neg-frac 85.6%

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

      13. metadata-eval 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in a around inf 59.7%

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

   5. Taylor expanded in b around 0 93.5%

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

      1. associate-*r/ 93.5%

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

   7. Simplified 93.5%

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

   8. Taylor expanded in a around 0 65.1%

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

   if -8.0000000000000003e-309 < b < 3.8e-213

   1. Initial program 66.7%

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

      1. associate-*r/ 66.8%

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

      2. *-rgt-identity 66.8%

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

      3. associate-*l/ 66.7%

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

      4. difference-of-squares 82.5%

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

      5. *-commutative 82.5%

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

      6. times-frac 99.8%

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

      7. sub-neg 99.8%

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

      8. distribute-neg-frac 99.8%

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

      9. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 94.5%

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

   5. Taylor expanded in b around inf 27.5%

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

      1. associate-*r/ 27.5%

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

      2. associate-*r/ 27.5%

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

      3. *-commutative 27.5%

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

   7. Applied egg-rr 27.5%

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

      1. associate-*l/ 27.5%

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

      2. *-commutative 27.5%

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

      3. associate-*l* 27.5%

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

      4. metadata-eval 27.5%

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

      5. *-commutative 27.5%

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

   9. Simplified 27.5%

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

       1. associate-/l/ 27.5%

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

       2. *-commutative 27.5%

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

       3. times-frac 27.5%

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

   11. Applied egg-rr 27.5%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-309} \lor \neg \left(b \leq 3.8 \cdot 10^{-213}\right):\\ \;\;\;\;\frac{-1}{b} \cdot \left(-0.5 \cdot \frac{\pi}{b \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b \cdot a} \cdot \frac{\pi}{b}\\ \end{array} \]

Alternative 3: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a} \cdot \frac{-1}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(-0.5 \cdot \frac{\pi}{b \cdot a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.3e-64)
   (* (/ (* PI -0.5) a) (/ -1.0 (* b a)))
   (* (/ -1.0 b) (* -0.5 (/ PI (* b a))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.3e-64) {
		tmp = ((((double) M_PI) * -0.5) / a) * (-1.0 / (b * a));
	} else {
		tmp = (-1.0 / b) * (-0.5 * (((double) M_PI) / (b * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.3e-64) {
		tmp = ((Math.PI * -0.5) / a) * (-1.0 / (b * a));
	} else {
		tmp = (-1.0 / b) * (-0.5 * (Math.PI / (b * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.3e-64:
		tmp = ((math.pi * -0.5) / a) * (-1.0 / (b * a))
	else:
		tmp = (-1.0 / b) * (-0.5 * (math.pi / (b * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.3e-64)
		tmp = Float64(Float64(Float64(pi * -0.5) / a) * Float64(-1.0 / Float64(b * a)));
	else
		tmp = Float64(Float64(-1.0 / b) * Float64(-0.5 * Float64(pi / Float64(b * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.3e-64)
		tmp = ((pi * -0.5) / a) * (-1.0 / (b * a));
	else
		tmp = (-1.0 / b) * (-0.5 * (pi / (b * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.3e-64], N[(N[(N[(Pi * -0.5), $MachinePrecision] / a), $MachinePrecision] * N[(-1.0 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / b), $MachinePrecision] * N[(-0.5 * N[(Pi / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.3000000000000001e-64

   1. Initial program 75.8%

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

      1. associate-*r/ 75.8%

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

      2. *-rgt-identity 75.8%

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

      3. associate-*l/ 75.8%

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

      4. difference-of-squares 85.6%

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

      5. *-commutative 85.6%

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

      6. times-frac 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 70.0%

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

   5. Taylor expanded in b around 0 74.6%

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

      1. associate-*r/ 90.4%

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

   7. Simplified 74.6%

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

   if 2.3000000000000001e-64 < b

   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%

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

      2. associate-*r* 74.9%

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

      3. associate-*r/ 75.0%

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

      4. associate-/l* 75.0%

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

      5. /-rgt-identity 75.0%

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

      6. associate-/l* 74.9%

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

      7. difference-of-squares 81.7%

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

      8. associate-/l* 81.7%

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

      9. associate-/l* 99.7%

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

      10. associate-*r/ 82.2%

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

      11. sub-neg 82.2%

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

      12. distribute-neg-frac 82.2%

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

      13. metadata-eval 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in a around inf 46.7%

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

   5. Taylor expanded in b around 0 98.5%

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

      1. associate-*r/ 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in a around 0 87.7%

      \[\leadsto \frac{-1}{b} \cdot \color{blue}{\left(-0.5 \cdot \frac{\pi}{a \cdot b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\pi \cdot -0.5}{a} \cdot \frac{-1}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b} \cdot \left(-0.5 \cdot \frac{\pi}{b \cdot a}\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. *-commutative 75.5%

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

   2. associate-*r* 75.5%

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

   3. associate-*r/ 75.6%

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

   4. associate-/l* 75.6%

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

   5. /-rgt-identity 75.6%

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

   6. associate-/l* 75.5%

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

   7. difference-of-squares 84.5%

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

   8. associate-/l* 84.5%

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

   9. associate-/l* 99.6%

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

   10. associate-*r/ 85.4%

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

   11. sub-neg 85.4%

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

   12. distribute-neg-frac 85.4%

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

   13. metadata-eval 85.4%

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

3. Simplified 85.4%

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

4. Taylor expanded in a around inf 61.0%

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

5. Taylor expanded in b around 0 92.7%

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

   1. associate-*r/ 92.7%

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

7. Simplified 92.7%

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

   1. associate-*r/ 99.6%

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

   2. associate-/l* 99.6%

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

9. Applied egg-rr 99.6%

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

    1. expm1-log1p-u 79.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{-1}{b} \cdot \frac{-0.5}{\frac{a}{\pi}}}{b + a}\right)\right)} \]

    2. expm1-udef 50.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{-1}{b} \cdot \frac{-0.5}{\frac{a}{\pi}}}{b + a}\right)} - 1} \]

    3. frac-times 50.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{-1 \cdot -0.5}{b \cdot \frac{a}{\pi}}}}{b + a}\right)} - 1 \]

    4. metadata-eval 50.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0.5}}{b \cdot \frac{a}{\pi}}}{b + a}\right)} - 1 \]

    5. +-commutative 50.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{0.5}{b \cdot \frac{a}{\pi}}}{\color{blue}{a + b}}\right)} - 1 \]

11. Applied egg-rr 50.0%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{0.5}{b \cdot \frac{a}{\pi}}}{a + b}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 79.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.5}{b \cdot \frac{a}{\pi}}}{a + b}\right)\right)} \]

    2. expm1-log1p 99.6%

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

    3. associate-/r* 99.6%

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

    4. metadata-eval 99.6%

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

    5. associate-*l/ 99.6%

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

    6. associate-*r/ 99.6%

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

    7. associate-/l* 99.6%

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

    8. associate-*l/ 99.6%

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

    9. associate-*l/ 99.6%

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

    10. associate-*r* 99.6%

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

    11. associate-*r/ 99.6%

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

    12. metadata-eval 99.6%

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

    13. associate-*l/ 99.6%

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

    14. associate-*r/ 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 5: 30.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-*r/ 75.6%

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

   2. *-rgt-identity 75.6%

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

   3. associate-*l/ 75.6%

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

   4. difference-of-squares 84.5%

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

   5. *-commutative 84.5%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a around inf 63.3%

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

5. Taylor expanded in b around inf 26.4%

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

   1. associate-*r/ 26.4%

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

   2. associate-*r/ 26.4%

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

   3. *-commutative 26.4%

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

7. Applied egg-rr 26.4%

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

   1. associate-*l/ 26.4%

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

   2. *-commutative 26.4%

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

   3. associate-*l* 26.4%

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

   4. metadata-eval 26.4%

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

   5. *-commutative 26.4%

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

9. Simplified 26.4%

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

    1. associate-/l/ 26.4%

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

    2. *-commutative 26.4%

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

    3. times-frac 26.4%

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

11. Applied egg-rr 26.4%

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

12. Final simplification 26.4%

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

Alternative 6: 30.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-*r/ 75.6%

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

   2. *-rgt-identity 75.6%

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

   3. associate-*l/ 75.6%

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

   4. difference-of-squares 84.5%

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

   5. *-commutative 84.5%

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

   6. times-frac 99.7%

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

   7. sub-neg 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a around inf 63.3%

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

   1. *-commutative 63.3%

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

   2. clear-num 63.4%

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

   3. frac-times 63.2%

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

   4. metadata-eval 63.2%

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

   5. div-inv 63.2%

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

   6. metadata-eval 63.2%

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

6. Applied egg-rr 63.2%

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

   1. associate-*r/ 63.2%

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

   2. associate-/l* 63.2%

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

   3. neg-mul-1 63.2%

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

   4. *-commutative 63.2%

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

   5. distribute-rgt-neg-in 63.2%

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

   6. metadata-eval 63.2%

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

   7. associate-*r* 56.0%

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

   8. *-commutative 56.0%

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

8. Simplified 56.0%

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

9. Taylor expanded in a around 0 26.4%

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

10. Final simplification 26.4%

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

Reproduce

herbie shell --seed 2023315 
(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))))