NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 22.6s

Alternatives: 6

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: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{\frac{1}{a}}{b}}{\frac{a + b}{0.5 \cdot \pi}} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ (/ 1.0 a) b) (/ (+ a b) (* 0.5 PI))))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return ((1.0 / a) / b) / ((a + b) / (0.5 * math.pi))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(1.0 / a) / b) / Float64(Float64(a + b) / Float64(0.5 * pi)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((1.0 / a) / b) / ((a + b) / (0.5 * pi));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(1.0 / a), $MachinePrecision] / b), $MachinePrecision] / N[(N[(a + b), $MachinePrecision] / N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. un-div-inv 76.6%

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

   2. difference-of-squares 86.0%

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

   3. associate-/r* 86.3%

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

   4. div-inv 86.3%

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

   5. metadata-eval 86.3%

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

4. Applied egg-rr 86.3%

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

   1. frac-sub 86.3%

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

   2. frac-times 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-rgt-identity 98.9%

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

   2. associate-/l* 98.8%

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

   3. associate-*r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. *-commutative 98.8%

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

8. Simplified 98.8%

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

   1. associate-*r/ 98.8%

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

   2. +-commutative 98.8%

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

   3. associate-/r* 99.6%

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

   4. pow1 99.6%

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

   5. pow1 99.6%

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

   6. pow-div 99.6%

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

   7. metadata-eval 99.6%

      \[\leadsto \frac{\pi \cdot 0.5}{b + a} \cdot \frac{{\left(b - a\right)}^{\color{blue}{0}}}{a \cdot b} \]

   8. metadata-eval 99.6%

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

   9. clear-num 99.6%

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

   10. frac-times 98.9%

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

   11. metadata-eval 98.9%

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

   12. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. *-commutative 98.9%

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

    2. associate-/r* 99.6%

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

    3. *-commutative 99.6%

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

    4. associate-/r* 99.7%

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

    5. +-commutative 99.7%

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

    6. *-commutative 99.7%

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

12. Simplified 99.7%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. un-div-inv 76.6%

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

   2. difference-of-squares 86.0%

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

   3. associate-/r* 86.3%

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

   4. div-inv 86.3%

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

   5. metadata-eval 86.3%

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

4. Applied egg-rr 86.3%

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

   1. frac-sub 86.3%

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

   2. frac-times 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-rgt-identity 98.9%

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

   2. associate-/l* 98.8%

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

   3. associate-*r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. *-commutative 98.8%

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

8. Simplified 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/r* 99.6%

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

   3. pow1 99.6%

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

   4. pow1 99.6%

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

   5. pow-div 99.6%

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

   6. metadata-eval 99.6%

      \[\leadsto \frac{{\left(b - a\right)}^{\color{blue}{0}}}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{a + b}\right) \]

   7. metadata-eval 99.6%

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

   8. associate-*r/ 99.6%

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

   9. +-commutative 99.6%

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

   10. frac-times 98.9%

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

   11. *-un-lft-identity 98.9%

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

   12. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. times-frac 99.6%

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

    2. *-commutative 99.6%

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

    3. +-commutative 99.6%

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

12. Applied egg-rr 99.6%

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

Alternative 3: 93.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. un-div-inv 76.6%

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

   2. difference-of-squares 86.0%

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

   3. associate-/r* 86.3%

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

   4. div-inv 86.3%

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

   5. metadata-eval 86.3%

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

4. Applied egg-rr 86.3%

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

   1. frac-sub 86.3%

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

   2. frac-times 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-rgt-identity 98.9%

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

   2. associate-/l* 98.8%

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

   3. associate-*r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. *-commutative 98.8%

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

8. Simplified 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/r* 99.6%

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

   3. pow1 99.6%

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

   4. pow1 99.6%

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

   5. pow-div 99.6%

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

   6. metadata-eval 99.6%

      \[\leadsto \frac{{\left(b - a\right)}^{\color{blue}{0}}}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{a + b}\right) \]

   7. metadata-eval 99.6%

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

   8. associate-*r/ 99.6%

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

   9. +-commutative 99.6%

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

   10. frac-times 98.9%

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

   11. *-un-lft-identity 98.9%

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

   12. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. associate-*l* 91.7%

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

    2. +-commutative 91.7%

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

    3. times-frac 92.4%

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

12. Applied egg-rr 92.4%

    \[\leadsto \color{blue}{\frac{\pi}{b} \cdot \frac{0.5}{a \cdot \left(a + b\right)}} \]
13. Add Preprocessing

Alternative 4: 93.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. un-div-inv 76.6%

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

   2. difference-of-squares 86.0%

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

   3. associate-/r* 86.3%

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

   4. div-inv 86.3%

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

   5. metadata-eval 86.3%

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

4. Applied egg-rr 86.3%

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

   1. frac-sub 86.3%

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

   2. frac-times 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-rgt-identity 98.9%

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

   2. associate-/l* 98.8%

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

   3. associate-*r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. *-commutative 98.8%

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

8. Simplified 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/r* 99.6%

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

   3. pow1 99.6%

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

   4. pow1 99.6%

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

   5. pow-div 99.6%

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

   6. metadata-eval 99.6%

      \[\leadsto \frac{{\left(b - a\right)}^{\color{blue}{0}}}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{a + b}\right) \]

   7. metadata-eval 99.6%

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

   8. associate-*r/ 99.6%

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

   9. +-commutative 99.6%

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

   10. frac-times 98.9%

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

   11. *-un-lft-identity 98.9%

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

   12. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. associate-/l* 98.9%

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

    2. associate-*l* 91.6%

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

    3. +-commutative 91.6%

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

12. Simplified 91.6%

    \[\leadsto \color{blue}{\pi \cdot \frac{0.5}{b \cdot \left(a \cdot \left(a + b\right)\right)}} \]
13. Add Preprocessing

Alternative 5: 62.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. un-div-inv 76.6%

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

   2. difference-of-squares 86.0%

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

   3. associate-/r* 86.3%

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

   4. div-inv 86.3%

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

   5. metadata-eval 86.3%

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

4. Applied egg-rr 86.3%

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

   1. frac-sub 86.3%

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

   2. frac-times 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-rgt-identity 98.9%

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

   2. associate-/l* 98.8%

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

   3. associate-*r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. *-commutative 98.8%

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

8. Simplified 98.8%

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

   1. associate-*r/ 98.8%

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

   2. +-commutative 98.8%

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

   3. associate-/r* 99.6%

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

   4. pow1 99.6%

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

   5. pow1 99.6%

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

   6. pow-div 99.6%

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

   7. metadata-eval 99.6%

      \[\leadsto \frac{\pi \cdot 0.5}{b + a} \cdot \frac{{\left(b - a\right)}^{\color{blue}{0}}}{a \cdot b} \]

   8. metadata-eval 99.6%

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

   9. frac-2neg 99.6%

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

   10. frac-times 98.9%

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

   11. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. *-rgt-identity 98.9%

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

    2. distribute-rgt-neg-in 98.9%

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

    3. metadata-eval 98.9%

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

    4. *-commutative 98.9%

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

    5. associate-*l* 91.7%

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

    6. +-commutative 91.7%

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

12. Simplified 91.7%

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

13. Taylor expanded in a around 0 60.5%

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

    1. mul-1-neg 60.5%

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

    2. distribute-rgt-neg-in 60.5%

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

15. Simplified 60.5%

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

Alternative 6: 62.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. un-div-inv 76.6%

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

   2. difference-of-squares 86.0%

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

   3. associate-/r* 86.3%

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

   4. div-inv 86.3%

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

   5. metadata-eval 86.3%

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

4. Applied egg-rr 86.3%

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

   1. frac-sub 86.3%

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

   2. frac-times 98.9%

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

   3. *-un-lft-identity 98.9%

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

   4. *-commutative 98.9%

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

6. Applied egg-rr 98.9%

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

   1. *-rgt-identity 98.9%

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

   2. associate-/l* 98.8%

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

   3. associate-*r/ 98.8%

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

   4. +-commutative 98.8%

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

   5. *-commutative 98.8%

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

8. Simplified 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-/r* 99.6%

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

   3. pow1 99.6%

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

   4. pow1 99.6%

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

   5. pow-div 99.6%

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

   6. metadata-eval 99.6%

      \[\leadsto \frac{{\left(b - a\right)}^{\color{blue}{0}}}{a \cdot b} \cdot \left(\pi \cdot \frac{0.5}{a + b}\right) \]

   7. metadata-eval 99.6%

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

   8. associate-*r/ 99.6%

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

   9. +-commutative 99.6%

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

   10. frac-times 98.9%

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

   11. *-un-lft-identity 98.9%

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

   12. *-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. associate-/l* 98.9%

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

    2. associate-*l* 91.6%

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

    3. +-commutative 91.6%

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

12. Simplified 91.6%

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

13. Taylor expanded in a around 0 60.4%

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

Reproduce

herbie shell --seed 2024143 
(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))))