NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 11.7s

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: 99.6% accurate, 1.9× speedup

Math

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

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(Float64(pi / 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[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*r* 76.9%

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

   3. associate-*r/ 76.9%

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

   4. associate-*r* 76.9%

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

   5. *-rgt-identity 76.9%

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

   6. sub-neg 76.9%

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

   7. distribute-neg-frac 76.9%

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

   8. metadata-eval 76.9%

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

3. Simplified 76.9%

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

   1. *-commutative 76.9%

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

   2. difference-of-squares 88.2%

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

   3. times-frac 99.7%

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

   4. div-inv 99.7%

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

   5. metadata-eval 99.7%

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

   6. add-sqr-sqrt 48.6%

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

   7. sqrt-unprod 71.7%

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

   8. frac-times 71.7%

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

   9. metadata-eval 71.7%

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

   10. metadata-eval 71.7%

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

   11. frac-times 71.7%

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

   12. sqrt-unprod 33.0%

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

   13. add-sqr-sqrt 66.4%

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

6. Applied egg-rr 66.4%

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

   1. associate-*l/ 66.4%

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

   2. *-commutative 66.4%

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

   3. +-commutative 66.4%

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

   4. +-commutative 66.4%

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

8. Simplified 66.4%

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

9. Taylor expanded in b around inf 99.7%

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

    1. associate-/r* 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 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 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*r* 76.9%

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

   3. associate-*r/ 76.9%

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

   4. associate-*r* 76.9%

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

   5. *-rgt-identity 76.9%

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

   6. sub-neg 76.9%

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

   7. distribute-neg-frac 76.9%

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

   8. metadata-eval 76.9%

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

3. Simplified 76.9%

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

   1. *-commutative 76.9%

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

   2. difference-of-squares 88.2%

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

   3. times-frac 99.7%

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

   4. div-inv 99.7%

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

   5. metadata-eval 99.7%

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

   6. add-sqr-sqrt 48.6%

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

   7. sqrt-unprod 71.7%

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

   8. frac-times 71.7%

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

   9. metadata-eval 71.7%

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

   10. metadata-eval 71.7%

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

   11. frac-times 71.7%

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

   12. sqrt-unprod 33.0%

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

   13. add-sqr-sqrt 66.4%

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

6. Applied egg-rr 66.4%

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

   1. associate-*l/ 66.4%

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

   2. *-commutative 66.4%

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

   3. +-commutative 66.4%

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

   4. +-commutative 66.4%

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

8. Simplified 66.4%

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

9. Taylor expanded in b around inf 99.7%

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

    1. associate-/r* 99.7%

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

11. Simplified 99.7%

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

    1. associate-/r* 99.7%

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

    2. associate-/l* 99.7%

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

13. Applied egg-rr 99.7%

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

    1. associate-*r/ 99.7%

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

    2. *-commutative 99.7%

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

    3. associate-*r/ 99.6%

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

    4. associate-/r* 99.6%

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

15. Simplified 99.6%

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

16. Taylor expanded in a around 0 99.6%

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

17. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

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(Float64(pi / 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[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision] * N[(0.5 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*r* 76.9%

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

   3. associate-*r/ 76.9%

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

   4. associate-*r* 76.9%

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

   5. *-rgt-identity 76.9%

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

   6. sub-neg 76.9%

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

   7. distribute-neg-frac 76.9%

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

   8. metadata-eval 76.9%

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

3. Simplified 76.9%

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

   1. *-commutative 76.9%

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

   2. difference-of-squares 88.2%

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

   3. times-frac 99.7%

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

   4. div-inv 99.7%

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

   5. metadata-eval 99.7%

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

   6. add-sqr-sqrt 48.6%

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

   7. sqrt-unprod 71.7%

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

   8. frac-times 71.7%

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

   9. metadata-eval 71.7%

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

   10. metadata-eval 71.7%

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

   11. frac-times 71.7%

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

   12. sqrt-unprod 33.0%

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

   13. add-sqr-sqrt 66.4%

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

6. Applied egg-rr 66.4%

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

   1. associate-*l/ 66.4%

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

   2. *-commutative 66.4%

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

   3. +-commutative 66.4%

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

   4. +-commutative 66.4%

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

8. Simplified 66.4%

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

9. Taylor expanded in b around inf 99.7%

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

    1. associate-/r* 99.7%

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

11. Simplified 99.7%

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

    1. associate-/r* 99.7%

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

    2. associate-/l* 99.7%

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

13. Applied egg-rr 99.7%

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

    1. associate-*r/ 99.7%

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

    2. *-commutative 99.7%

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

    3. associate-*r/ 99.6%

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

    4. associate-/r* 99.6%

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

15. Simplified 99.6%

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

16. Final simplification 99.6%

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

Alternative 4: 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 76.9%

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

   1. *-commutative 76.9%

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

   2. associate-*r* 76.9%

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

   3. associate-*r/ 76.9%

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

   4. associate-*r* 76.9%

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

   5. *-rgt-identity 76.9%

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

   6. sub-neg 76.9%

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

   7. distribute-neg-frac 76.9%

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

   8. metadata-eval 76.9%

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

3. Simplified 76.9%

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

   1. *-commutative 76.9%

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

   2. difference-of-squares 88.2%

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

   3. times-frac 99.7%

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

   4. div-inv 99.7%

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

   5. metadata-eval 99.7%

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

   6. add-sqr-sqrt 48.6%

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

   7. sqrt-unprod 71.7%

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

   8. frac-times 71.7%

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

   9. metadata-eval 71.7%

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

   10. metadata-eval 71.7%

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

   11. frac-times 71.7%

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

   12. sqrt-unprod 33.0%

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

   13. add-sqr-sqrt 66.4%

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

6. Applied egg-rr 66.4%

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

   1. associate-*l/ 66.4%

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

   2. *-commutative 66.4%

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

   3. +-commutative 66.4%

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

   4. +-commutative 66.4%

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

8. Simplified 66.4%

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

9. Taylor expanded in b around inf 99.7%

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

10. Final simplification 99.7%

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

Reproduce

herbie shell --seed 2024080 
(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))))