NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.7%

Time: 10.3s

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: 78.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Add Preprocessing
3. Step-by-step derivation

   1. un-div-inv 75.6%

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

   2. difference-of-squares 83.0%

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

   3. associate-/r* 83.5%

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

   4. div-inv 83.5%

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

   5. metadata-eval 83.5%

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

4. Applied egg-rr 83.5%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

   6. distribute-neg-frac 99.6%

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

   7. metadata-eval 99.6%

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

   8. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in b around 0 99.6%

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

    1. associate-*l/ 99.6%

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

    2. associate-/r* 99.6%

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

11. Applied egg-rr 99.6%

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

    1. associate-*l* 99.6%

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

    2. associate-*r/ 99.6%

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

    3. associate-*r/ 99.6%

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

    4. *-rgt-identity 99.6%

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

    5. associate-/r* 99.7%

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

13. Simplified 99.7%

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

Alternative 2: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-120} \lor \neg \left(b \leq 4.6 \cdot 10^{-307}\right):\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{b}}{a - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -4.2e-120) (not (<= b 4.6e-307)))
   (* (/ 0.5 b) (/ (/ PI a) (- b a)))
   (/ (/ (/ PI b) b) (- a b))))

C

double code(double a, double b) {
	double tmp;
	if ((b <= -4.2e-120) || !(b <= 4.6e-307)) {
		tmp = (0.5 / b) * ((((double) M_PI) / a) / (b - a));
	} else {
		tmp = ((((double) M_PI) / b) / b) / (a - b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((b <= -4.2e-120) || !(b <= 4.6e-307)) {
		tmp = (0.5 / b) * ((Math.PI / a) / (b - a));
	} else {
		tmp = ((Math.PI / b) / b) / (a - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b <= -4.2e-120) or not (b <= 4.6e-307):
		tmp = (0.5 / b) * ((math.pi / a) / (b - a))
	else:
		tmp = ((math.pi / b) / b) / (a - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((b <= -4.2e-120) || !(b <= 4.6e-307))
		tmp = Float64(Float64(0.5 / b) * Float64(Float64(pi / a) / Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(pi / b) / b) / Float64(a - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b <= -4.2e-120) || ~((b <= 4.6e-307)))
		tmp = (0.5 / b) * ((pi / a) / (b - a));
	else
		tmp = ((pi / b) / b) / (a - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[b, -4.2e-120], N[Not[LessEqual[b, 4.6e-307]], $MachinePrecision]], N[(N[(0.5 / b), $MachinePrecision] * N[(N[(Pi / a), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / b), $MachinePrecision] / b), $MachinePrecision] / N[(a - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.2000000000000001e-120 or 4.5999999999999998e-307 < b

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. associate-*r* 75.7%

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

      3. associate-*r/ 75.7%

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

      4. associate-*r* 75.7%

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

      5. *-rgt-identity 75.7%

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

      6. sub-neg 75.7%

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

      7. distribute-neg-frac 75.7%

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

      8. metadata-eval 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in a around 0 60.2%

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

      1. difference-of-squares 64.6%

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

   7. Applied egg-rr 64.6%

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

      1. frac-times 73.1%

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

      2. +-commutative 73.1%

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

   9. Applied egg-rr 73.1%

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

   10. Taylor expanded in a around 0 71.4%

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

   if -4.2000000000000001e-120 < b < 4.5999999999999998e-307

   1. Initial program 74.2%

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

      1. un-div-inv 74.2%

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

      2. difference-of-squares 77.8%

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

      3. associate-/r* 79.9%

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

      4. div-inv 79.9%

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

      5. metadata-eval 79.9%

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

   4. Applied egg-rr 79.9%

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

      1. associate-*l/ 99.6%

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

      2. associate-/l* 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around inf 23.3%

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

      1. distribute-lft-out 23.3%

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

      2. mul-1-neg 23.3%

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

   9. Simplified 23.3%

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

   10. Taylor expanded in a around inf 12.6%

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

       1. mul-1-neg 12.6%

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

       2. distribute-frac-neg 12.6%

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

   12. Simplified 12.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-120} \lor \neg \left(b \leq 4.6 \cdot 10^{-307}\right):\\ \;\;\;\;\frac{0.5}{b} \cdot \frac{\frac{\pi}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\pi}{b}}{b}}{a - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(a + b), $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* 75.6%

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

   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-*r* 75.6%

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

   5. *-rgt-identity 75.6%

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

   6. sub-neg 75.6%

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

   7. distribute-neg-frac 75.6%

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

   8. metadata-eval 75.6%

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

3. Simplified 75.6%

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

5. Taylor expanded in a around 0 56.5%

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

   1. difference-of-squares 60.8%

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

7. Applied egg-rr 60.8%

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

   1. frac-times 68.3%

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

   2. +-commutative 68.3%

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

9. Applied egg-rr 68.3%

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

10. Taylor expanded in a around 0 99.6%

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

11. Final simplification 99.6%

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

Alternative 4: 34.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(Pi / b), $MachinePrecision] / b), $MachinePrecision] / N[(a - 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. Add Preprocessing
3. Step-by-step derivation

   1. un-div-inv 75.6%

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

   2. difference-of-squares 83.0%

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

   3. associate-/r* 83.5%

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

   4. div-inv 83.5%

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

   5. metadata-eval 83.5%

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

4. Applied egg-rr 83.5%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in b around inf 64.3%

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

   1. distribute-lft-out 64.3%

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

   2. mul-1-neg 64.3%

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

9. Simplified 64.3%

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

10. Taylor expanded in a around inf 30.3%

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

    1. mul-1-neg 30.3%

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

    2. distribute-frac-neg 30.3%

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

12. Simplified 30.3%

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

13. Final simplification 30.3%

    \[\leadsto \frac{\frac{\frac{\pi}{b}}{b}}{a - b} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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))))