NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.6%

Time: 14.5s

Alternatives: 4

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\pi \cdot 0.5}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (* (/ (* PI 0.5) (+ b a)) (- (/ 1.0 a) (/ 1.0 b))) (- b a)))

C

double code(double a, double b) {
	return (((((double) M_PI) * 0.5) / (b + a)) * ((1.0 / a) - (1.0 / b))) / (b - a);
}

Java

public static double code(double a, double b) {
	return (((Math.PI * 0.5) / (b + a)) * ((1.0 / a) - (1.0 / b))) / (b - a);
}

Python

def code(a, b):
	return (((math.pi * 0.5) / (b + a)) * ((1.0 / a) - (1.0 / b))) / (b - a)

Julia

function code(a, b)
	return Float64(Float64(Float64(Float64(pi * 0.5) / Float64(b + a)) * Float64(Float64(1.0 / a) - Float64(1.0 / b))) / Float64(b - a))
end

MATLAB

function tmp = code(a, b)
	tmp = (((pi * 0.5) / (b + a)) * ((1.0 / a) - (1.0 / b))) / (b - a);
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

   1. un-div-inv 77.1%

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

   2. difference-of-squares 86.5%

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

   3. associate-/r* 86.5%

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

   4. div-inv 86.5%

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

   5. metadata-eval 86.5%

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

4. Applied egg-rr 86.5%

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

   1. associate-*l/ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{\frac{\pi \cdot 0.5}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a} \]
8. Add Preprocessing

Alternative 2: 66.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{b \cdot a}\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\pi}{-a} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 0.5 (* b a))))
   (if (<= a -1.2e+109) (* (/ PI (- a)) t_0) (* (/ PI b) t_0))))

C

double code(double a, double b) {
	double t_0 = 0.5 / (b * a);
	double tmp;
	if (a <= -1.2e+109) {
		tmp = (((double) M_PI) / -a) * t_0;
	} else {
		tmp = (((double) M_PI) / b) * t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = 0.5 / (b * a);
	double tmp;
	if (a <= -1.2e+109) {
		tmp = (Math.PI / -a) * t_0;
	} else {
		tmp = (Math.PI / b) * t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 / (b * a)
	tmp = 0
	if a <= -1.2e+109:
		tmp = (math.pi / -a) * t_0
	else:
		tmp = (math.pi / b) * t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 / Float64(b * a))
	tmp = 0.0
	if (a <= -1.2e+109)
		tmp = Float64(Float64(pi / Float64(-a)) * t_0);
	else
		tmp = Float64(Float64(pi / b) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 / (b * a);
	tmp = 0.0;
	if (a <= -1.2e+109)
		tmp = (pi / -a) * t_0;
	else
		tmp = (pi / b) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 / N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.2e+109], N[(N[(Pi / (-a)), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(Pi / b), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.19999999999999994e109

   1. Initial program 62.6%

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

      1. associate-*l* 62.5%

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

      2. *-rgt-identity 62.5%

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

      3. associate-/l* 62.5%

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

      4. metadata-eval 62.5%

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

      5. associate-*l/ 62.6%

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

      6. *-lft-identity 62.6%

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

      7. sub-neg 62.6%

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

      8. distribute-neg-frac 62.6%

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

      9. metadata-eval 62.6%

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

   3. Simplified 62.6%

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

      1. metadata-eval 62.6%

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

      2. div-inv 62.6%

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

      3. associate-*r/ 62.6%

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

      4. *-commutative 62.6%

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

      5. difference-of-squares 76.9%

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

      6. associate-/r* 99.8%

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

   6. Applied egg-rr 68.1%

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

   7. Taylor expanded in a around 0 68.1%

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

      1. associate-*r/ 68.1%

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

   9. Simplified 68.1%

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

       1. *-un-lft-identity 68.1%

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

       2. associate-/l/ 68.1%

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

       3. *-commutative 68.1%

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

       4. *-commutative 68.1%

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

   11. Applied egg-rr 68.1%

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

       1. *-lft-identity 68.1%

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

       2. times-frac 68.1%

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

       3. *-commutative 68.1%

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

   13. Simplified 68.1%

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

   14. Taylor expanded in b around 0 68.1%

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

       1. neg-mul-1 68.1%

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

       2. distribute-neg-frac2 68.1%

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

   16. Simplified 68.1%

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

   if -1.19999999999999994e109 < a

   1. Initial program 80.5%

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

      1. associate-*l* 80.5%

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

      2. *-rgt-identity 80.5%

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

      3. associate-/l* 80.5%

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

      4. metadata-eval 80.5%

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

      5. associate-*l/ 80.6%

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

      6. *-lft-identity 80.6%

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

      7. sub-neg 80.6%

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

      8. distribute-neg-frac 80.6%

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

      9. metadata-eval 80.6%

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

   3. Simplified 80.6%

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

      1. metadata-eval 80.6%

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

      2. div-inv 80.6%

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

      3. associate-*r/ 80.6%

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

      4. *-commutative 80.6%

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

      5. difference-of-squares 88.8%

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

      6. associate-/r* 99.6%

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

   6. Applied egg-rr 63.6%

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

   7. Taylor expanded in a around 0 63.6%

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

      1. associate-*r/ 63.6%

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

   9. Simplified 63.6%

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

       1. *-un-lft-identity 63.6%

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

       2. associate-/l/ 63.4%

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

       3. *-commutative 63.4%

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

       4. *-commutative 63.4%

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

   11. Applied egg-rr 63.4%

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

       1. *-lft-identity 63.4%

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

       2. times-frac 63.5%

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

       3. *-commutative 63.5%

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

   13. Simplified 63.5%

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

   14. Taylor expanded in b around inf 67.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\pi}{-a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{b} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. *-commutative 77.1%

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

   2. associate-*r* 77.1%

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

   3. associate-*r/ 77.1%

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

   4. associate-*r* 77.1%

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

   5. *-rgt-identity 77.1%

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

   6. sub-neg 77.1%

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

   7. distribute-neg-frac 77.1%

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

   8. metadata-eval 77.1%

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

3. Simplified 77.1%

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

5. Taylor expanded in a around 0 54.5%

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

   1. associate-*r/ 54.5%

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

7. Simplified 54.5%

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

   1. associate-/l* 54.5%

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

   2. difference-of-squares 60.8%

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

   3. times-frac 65.8%

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

9. Applied egg-rr 65.8%

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

10. Taylor expanded in a around 0 99.7%

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

    1. associate-/l/ 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 4: 62.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. associate-*l* 77.1%

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

   2. *-rgt-identity 77.1%

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

   3. associate-/l* 77.1%

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

   4. metadata-eval 77.1%

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

   5. associate-*l/ 77.1%

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

   6. *-lft-identity 77.1%

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

   7. sub-neg 77.1%

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

   8. distribute-neg-frac 77.1%

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

   9. metadata-eval 77.1%

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

3. Simplified 77.1%

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

   1. metadata-eval 77.1%

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

   2. div-inv 77.1%

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

   3. associate-*r/ 77.1%

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

   4. *-commutative 77.1%

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

   5. difference-of-squares 86.5%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 64.4%

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

7. Taylor expanded in a around 0 64.4%

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

   1. associate-*r/ 64.4%

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

9. Simplified 64.4%

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

    1. *-un-lft-identity 64.4%

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

    2. associate-/l/ 64.3%

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

    3. *-commutative 64.3%

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

    4. *-commutative 64.3%

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

11. Applied egg-rr 64.3%

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

    1. *-lft-identity 64.3%

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

    2. times-frac 64.4%

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

    3. *-commutative 64.4%

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

13. Simplified 64.4%

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

14. Taylor expanded in b around inf 63.4%

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

15. Final simplification 63.4%

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

Reproduce

herbie shell --seed 2024053 
(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))))