NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.7%

Time: 10.2s

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: 78.1% 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: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. *-commutative 79.4%

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

   2. associate-*r* 79.1%

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

   3. associate-*r/ 79.1%

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

   4. associate-*r* 79.1%

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

   5. *-rgt-identity 79.1%

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

   6. sub-neg 79.1%

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

   7. distribute-neg-frac 79.1%

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

   8. metadata-eval 79.1%

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

3. Simplified 79.1%

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

5. Taylor expanded in a around 0 53.9%

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

   1. difference-of-squares 59.4%

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

   2. associate-*r/ 59.4%

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

   3. *-commutative 59.4%

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

   4. *-un-lft-identity 59.4%

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

   5. associate-*l/ 59.4%

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

   6. times-frac 66.4%

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

7. Applied egg-rr 66.4%

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

8. Taylor expanded in b around inf 93.3%

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

   1. associate-*l/ 99.6%

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

   2. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. frac-times 99.7%

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

   5. *-un-lft-identity 99.7%

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

   6. *-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. un-div-inv 79.4%

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

   2. difference-of-squares 86.4%

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

   3. associate-/r* 86.6%

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

   4. div-inv 86.6%

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

   5. metadata-eval 86.6%

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

4. Applied egg-rr 86.6%

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

   1. frac-sub 86.6%

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

   2. frac-times 98.8%

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

   3. *-un-lft-identity 98.8%

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

   4. *-commutative 98.8%

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

6. Applied egg-rr 98.8%

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

   1. *-rgt-identity 98.8%

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

   2. *-commutative 98.8%

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

   3. *-commutative 98.8%

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

   4. times-frac 99.7%

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

   5. associate-/l* 99.6%

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

   6. *-commutative 99.6%

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

8. Applied egg-rr 99.6%

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

   1. *-inverses 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. associate-/l* 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 93.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. *-commutative 79.4%

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

   2. associate-*r* 79.1%

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

   3. associate-*r/ 79.1%

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

   4. associate-*r* 79.1%

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

   5. *-rgt-identity 79.1%

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

   6. sub-neg 79.1%

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

   7. distribute-neg-frac 79.1%

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

   8. metadata-eval 79.1%

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

3. Simplified 79.1%

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

5. Taylor expanded in a around 0 53.9%

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

   1. difference-of-squares 59.4%

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

   2. associate-*r/ 59.4%

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

   3. *-commutative 59.4%

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

   4. *-un-lft-identity 59.4%

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

   5. associate-*l/ 59.4%

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

   6. times-frac 66.4%

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

7. Applied egg-rr 66.4%

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

8. Taylor expanded in b around inf 93.3%

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

   1. associate-/l/ 93.1%

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

   2. associate-*r/ 93.1%

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

   3. *-commutative 93.1%

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

   4. frac-times 92.4%

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

   5. *-un-lft-identity 92.4%

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

   6. *-commutative 92.4%

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

10. Applied egg-rr 92.4%

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

    1. associate-/l* 92.3%

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

    2. *-commutative 92.3%

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

    3. +-commutative 92.3%

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

12. Simplified 92.3%

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

13. Final simplification 92.3%

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

Alternative 4: 66.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

   1. *-commutative 79.4%

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

   2. associate-*r* 79.1%

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

   3. associate-*r/ 79.1%

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

   4. associate-*r* 79.1%

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

   5. *-rgt-identity 79.1%

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

   6. sub-neg 79.1%

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

   7. distribute-neg-frac 79.1%

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

   8. metadata-eval 79.1%

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

3. Simplified 79.1%

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

5. Taylor expanded in a around 0 53.9%

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

   1. difference-of-squares 59.4%

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

   2. *-commutative 59.4%

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

   3. times-frac 66.0%

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

7. Applied egg-rr 66.0%

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

8. Taylor expanded in a around 0 61.8%

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

   1. frac-times 61.1%

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

   2. *-commutative 61.1%

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

   3. *-commutative 61.1%

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

10. Applied egg-rr 61.1%

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

    1. *-commutative 61.1%

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

    2. metadata-eval 61.1%

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

    3. distribute-lft-neg-in 61.1%

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

    4. distribute-frac-neg 61.1%

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

    5. distribute-frac-neg2 61.1%

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

    6. associate-/l* 61.1%

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

    7. distribute-frac-neg2 61.1%

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

    8. associate-/l/ 61.8%

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

    9. distribute-rgt-neg-in 61.8%

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

    10. distribute-lft-neg-in 61.8%

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

    11. metadata-eval 61.8%

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

    12. associate-/r* 61.8%

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

12. Simplified 61.8%

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

Reproduce

herbie shell --seed 2024149 
(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))))