NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.1% → 99.6%

Time: 10.7s

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: 79.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. un-div-inv 74.3%

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

   2. difference-of-squares 84.5%

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

   3. associate-/r* 85.2%

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

   4. div-inv 85.2%

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

   5. metadata-eval 85.2%

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

4. Applied egg-rr 85.2%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

   1. div-inv 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 2: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. *-commutative 74.3%

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

   2. associate-*r* 74.3%

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

   3. associate-*r/ 74.4%

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

   4. associate-*r* 74.4%

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

   5. *-rgt-identity 74.4%

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

   6. sub-neg 74.4%

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

   7. distribute-neg-frac 74.4%

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

   8. metadata-eval 74.4%

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

3. Simplified 74.4%

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

   1. *-un-lft-identity 74.4%

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

   2. difference-of-squares 84.5%

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

   3. times-frac 99.6%

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

   4. add-sqr-sqrt 48.2%

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

   5. sqrt-unprod 73.1%

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

   6. frac-times 73.1%

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

   7. metadata-eval 73.1%

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

   8. metadata-eval 73.1%

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

   9. frac-times 73.1%

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

   10. sqrt-unprod 32.1%

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

   11. add-sqr-sqrt 62.9%

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

   12. div-inv 62.9%

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

   13. metadata-eval 62.9%

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

6. Applied egg-rr 62.9%

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

   1. associate-*l/ 62.9%

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

   2. associate-/l* 62.9%

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

   3. +-commutative 62.9%

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

   4. *-commutative 62.9%

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

   5. +-commutative 62.9%

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

8. Simplified 62.9%

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

9. Taylor expanded in b around inf 99.6%

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

    1. *-un-lft-identity 99.6%

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

    2. *-un-lft-identity 99.6%

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

    3. times-frac 99.6%

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

    4. metadata-eval 99.6%

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

    5. associate-/r* 99.7%

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

    6. +-commutative 99.7%

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

11. Applied egg-rr 99.7%

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

    1. associate-*r/ 99.7%

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

    2. +-commutative 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 3: 66.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. un-div-inv 74.3%

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

   2. difference-of-squares 84.5%

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

   3. associate-/r* 85.2%

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

   4. div-inv 85.2%

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

   5. metadata-eval 85.2%

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

4. Applied egg-rr 85.2%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in b around 0 66.4%

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

   1. associate-*r/ 66.4%

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

   2. *-commutative 66.4%

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

   3. *-commutative 66.4%

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

   4. times-frac 66.4%

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

9. Simplified 66.4%

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

    1. *-commutative 66.4%

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

    2. times-frac 66.4%

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

    3. associate-/l/ 66.1%

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

    4. *-commutative 66.1%

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

    5. times-frac 66.4%

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

    6. *-commutative 66.4%

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

11. Applied egg-rr 66.4%

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

Alternative 4: 66.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. un-div-inv 74.3%

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

   2. difference-of-squares 84.5%

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

   3. associate-/r* 85.2%

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

   4. div-inv 85.2%

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

   5. metadata-eval 85.2%

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

4. Applied egg-rr 85.2%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in b around 0 66.4%

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

   1. associate-*r/ 66.4%

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

   2. *-commutative 66.4%

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

   3. *-commutative 66.4%

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

   4. times-frac 66.4%

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

9. Simplified 66.4%

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

    1. *-commutative 66.4%

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

    2. times-frac 66.4%

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

    3. associate-/l/ 66.1%

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

    4. times-frac 66.4%

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

    5. associate-/r* 66.4%

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

11. Applied egg-rr 66.4%

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

12. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2024103 
(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))))