NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 12.1s

Alternatives: 5

Speedup: 1.8×

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: 78.4% 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, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{-0.5}{b + a} \cdot \frac{\frac{\pi}{-a}}{b} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ -0.5 (+ b a)) (/ (/ PI (- a)) b)))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return (-0.5 / (b + a)) * ((math.pi / -a) / b)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(-0.5 / Float64(b + a)) * Float64(Float64(pi / Float64(-a)) / b))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (-0.5 / (b + a)) * ((pi / -a) / b);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(-0.5 / N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi / (-a)), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-*r* 76.3%

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

   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-*r* 76.3%

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

   5. *-rgt-identity 76.3%

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

   6. sub-neg 76.3%

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

   7. distribute-neg-frac 76.3%

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

   8. metadata-eval 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in a around inf 55.1%

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

   1. difference-of-squares 64.9%

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

7. Applied egg-rr 64.9%

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

   1. times-frac 70.1%

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

9. Applied egg-rr 70.1%

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

10. Taylor expanded in b around 0 99.7%

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

    1. mul-1-neg 99.7%

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

    2. associate-/r* 99.6%

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

    3. distribute-neg-frac 99.6%

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

    4. distribute-neg-frac2 99.6%

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

12. Simplified 99.6%

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

Alternative 2: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -9.4 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \pi}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -9.4e-82)
   (/ (/ (* -0.5 PI) (* b a)) (- b a))
   (/ (/ (* PI 0.5) (* b a)) (- b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -9.4e-82) {
		tmp = ((-0.5 * ((double) M_PI)) / (b * a)) / (b - a);
	} else {
		tmp = ((((double) M_PI) * 0.5) / (b * a)) / (b - a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -9.4e-82) {
		tmp = ((-0.5 * Math.PI) / (b * a)) / (b - a);
	} else {
		tmp = ((Math.PI * 0.5) / (b * a)) / (b - a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -9.4e-82:
		tmp = ((-0.5 * math.pi) / (b * a)) / (b - a)
	else:
		tmp = ((math.pi * 0.5) / (b * a)) / (b - a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -9.4e-82)
		tmp = Float64(Float64(Float64(-0.5 * pi) / Float64(b * a)) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b * a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9.4e-82)
		tmp = ((-0.5 * pi) / (b * a)) / (b - a);
	else
		tmp = ((pi * 0.5) / (b * a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -9.4e-82], N[(N[(N[(-0.5 * Pi), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.4000000000000001e-82

   1. Initial program 82.7%

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

      1. un-div-inv 82.7%

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

      2. difference-of-squares 92.4%

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

      3. associate-/r* 92.4%

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

      4. div-inv 92.4%

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

      5. metadata-eval 92.4%

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

   4. Applied egg-rr 92.4%

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

      1. associate-*l/ 99.5%

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

      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 0 87.9%

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

      1. associate-*r/ 87.9%

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

      2. *-commutative 87.9%

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

   9. Simplified 87.9%

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

   if -9.4000000000000001e-82 < a

   1. Initial program 73.7%

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

      1. un-div-inv 73.8%

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

      2. difference-of-squares 83.5%

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

      3. associate-/r* 85.0%

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

      4. div-inv 85.0%

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

      5. metadata-eval 85.0%

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

   4. Applied egg-rr 85.0%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 70.4%

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

      1. associate-*r/ 70.4%

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

   9. Simplified 70.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.4 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot \pi}{b \cdot a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{-0.5}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.2e-81)
   (/ (* (/ PI a) (/ -0.5 b)) (- b a))
   (/ (/ (* PI 0.5) (* b a)) (- b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3.2e-81) {
		tmp = ((((double) M_PI) / a) * (-0.5 / b)) / (b - a);
	} else {
		tmp = ((((double) M_PI) * 0.5) / (b * a)) / (b - a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3.2e-81) {
		tmp = ((Math.PI / a) * (-0.5 / b)) / (b - a);
	} else {
		tmp = ((Math.PI * 0.5) / (b * a)) / (b - a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3.2e-81:
		tmp = ((math.pi / a) * (-0.5 / b)) / (b - a)
	else:
		tmp = ((math.pi * 0.5) / (b * a)) / (b - a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3.2e-81)
		tmp = Float64(Float64(Float64(pi / a) * Float64(-0.5 / b)) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi * 0.5) / Float64(b * a)) / Float64(b - a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.2e-81)
		tmp = ((pi / a) * (-0.5 / b)) / (b - a);
	else
		tmp = ((pi * 0.5) / (b * a)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3.2e-81], N[(N[(N[(Pi / a), $MachinePrecision] * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi * 0.5), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.2e-81

   1. Initial program 82.7%

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

      1. un-div-inv 82.7%

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

      2. difference-of-squares 92.4%

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

      3. associate-/r* 92.4%

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

      4. div-inv 92.4%

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

      5. metadata-eval 92.4%

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

   4. Applied egg-rr 92.4%

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

      1. associate-*l/ 99.5%

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

      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 0 87.9%

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

      1. associate-*r/ 87.9%

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

      2. *-commutative 87.9%

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

      3. times-frac 87.8%

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

   9. Simplified 87.8%

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

   if -3.2e-81 < a

   1. Initial program 73.7%

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

      1. un-div-inv 73.8%

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

      2. difference-of-squares 83.5%

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

      3. associate-/r* 85.0%

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

      4. div-inv 85.0%

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

      5. metadata-eval 85.0%

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

   4. Applied egg-rr 85.0%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 70.4%

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

      1. associate-*r/ 70.4%

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

   9. Simplified 70.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{-0.5}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi \cdot 0.5}{b \cdot a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{-0.5}{b}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.15e-80)
   (/ (* (/ PI a) (/ -0.5 b)) (- b a))
   (/ (* (/ PI a) (/ 0.5 b)) (- b a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.15e-80) {
		tmp = ((((double) M_PI) / a) * (-0.5 / b)) / (b - a);
	} else {
		tmp = ((((double) M_PI) / a) * (0.5 / b)) / (b - a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.15e-80) {
		tmp = ((Math.PI / a) * (-0.5 / b)) / (b - a);
	} else {
		tmp = ((Math.PI / a) * (0.5 / b)) / (b - a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.15e-80:
		tmp = ((math.pi / a) * (-0.5 / b)) / (b - a)
	else:
		tmp = ((math.pi / a) * (0.5 / b)) / (b - a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.15e-80)
		tmp = Float64(Float64(Float64(pi / a) * Float64(-0.5 / b)) / Float64(b - a));
	else
		tmp = Float64(Float64(Float64(pi / a) * Float64(0.5 / b)) / Float64(b - a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.15e-80)
		tmp = ((pi / a) * (-0.5 / b)) / (b - a);
	else
		tmp = ((pi / a) * (0.5 / b)) / (b - a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.15e-80], N[(N[(N[(Pi / a), $MachinePrecision] * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / a), $MachinePrecision] * N[(0.5 / b), $MachinePrecision]), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1499999999999999e-80

   1. Initial program 82.7%

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

      1. un-div-inv 82.7%

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

      2. difference-of-squares 92.4%

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

      3. associate-/r* 92.4%

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

      4. div-inv 92.4%

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

      5. metadata-eval 92.4%

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

   4. Applied egg-rr 92.4%

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

      1. associate-*l/ 99.5%

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

      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 0 87.9%

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

      1. associate-*r/ 87.9%

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

      2. *-commutative 87.9%

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

      3. times-frac 87.8%

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

   9. Simplified 87.8%

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

   if -1.1499999999999999e-80 < a

   1. Initial program 73.7%

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

      1. un-div-inv 73.8%

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

      2. difference-of-squares 83.5%

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

      3. associate-/r* 85.0%

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

      4. div-inv 85.0%

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

      5. metadata-eval 85.0%

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

   4. Applied egg-rr 85.0%

      \[\leadsto \color{blue}{\frac{\frac{\pi \cdot 0.5}{b + a}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 70.4%

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

      1. associate-*r/ 70.4%

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

      2. *-commutative 70.4%

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

      3. times-frac 70.4%

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

   9. Simplified 70.4%

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

Alternative 5: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{-0.5}{a} \cdot \frac{\frac{\pi}{b}}{b - a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ -0.5 a) (/ (/ PI b) (- b a))))

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(-0.5 / a) * Float64(Float64(pi / b) / Float64(b - a)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (-0.5 / a) * ((pi / b) / (b - a));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(-0.5 / a), $MachinePrecision] * N[(N[(Pi / b), $MachinePrecision] / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. *-commutative 76.3%

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

   2. associate-*r* 76.3%

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

   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-*r* 76.3%

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

   5. *-rgt-identity 76.3%

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

   6. sub-neg 76.3%

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

   7. distribute-neg-frac 76.3%

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

   8. metadata-eval 76.3%

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

3. Simplified 76.3%

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

5. Taylor expanded in a around inf 55.1%

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

   1. difference-of-squares 64.9%

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

7. Applied egg-rr 64.9%

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

   1. times-frac 70.1%

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

9. Applied egg-rr 70.1%

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

10. Taylor expanded in b around 0 66.9%

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

Reproduce

herbie shell --seed 2024085 
(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))))