NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.0% → 99.6%

Time: 14.0s

Alternatives: 5

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: 79.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. *-commutative 77.1%

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

   2. associate-*r* 77.1%

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

   3. associate-*r/ 77.1%

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

   4. associate-*r* 77.1%

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

   5. *-rgt-identity 77.1%

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

   6. sub-neg 77.1%

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

   7. distribute-neg-frac 77.1%

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

   8. metadata-eval 77.1%

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

3. Simplified 77.1%

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

   1. *-un-lft-identity 77.1%

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

   2. difference-of-squares 86.5%

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

   3. times-frac 99.5%

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

   4. add-sqr-sqrt 50.1%

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

   5. sqrt-unprod 71.3%

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

   6. frac-times 71.3%

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

   7. metadata-eval 71.3%

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

   8. metadata-eval 71.3%

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

   9. frac-times 71.3%

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

   10. sqrt-unprod 30.7%

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

   11. add-sqr-sqrt 64.4%

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

   12. div-inv 64.4%

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

   13. metadata-eval 64.4%

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

6. Applied egg-rr 64.4%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/r* 99.7%

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

   5. +-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

    1. *-commutative 99.7%

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

    2. associate-/r* 99.7%

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

    3. associate-*r/ 99.7%

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

    4. *-commutative 99.7%

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

    5. *-commutative 99.7%

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

    6. times-frac 99.6%

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

11. Simplified 99.6%

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

    1. clear-num 99.6%

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

    2. frac-times 99.7%

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

    3. metadata-eval 99.7%

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

13. Applied egg-rr 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 99.0% accurate, 1.9× speedup

Math

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

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 a)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.1%

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

   1. *-commutative 77.1%

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

   2. associate-*r* 77.1%

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

   3. associate-*r/ 77.1%

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

   4. associate-*r* 77.1%

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

   5. *-rgt-identity 77.1%

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

   6. sub-neg 77.1%

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

   7. distribute-neg-frac 77.1%

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

   8. metadata-eval 77.1%

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

3. Simplified 77.1%

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

   1. *-un-lft-identity 77.1%

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

   2. difference-of-squares 86.5%

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

   3. times-frac 99.5%

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

   4. add-sqr-sqrt 50.1%

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

   5. sqrt-unprod 71.3%

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

   6. frac-times 71.3%

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

   7. metadata-eval 71.3%

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

   8. metadata-eval 71.3%

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

   9. frac-times 71.3%

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

   10. sqrt-unprod 30.7%

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

   11. add-sqr-sqrt 64.4%

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

   12. div-inv 64.4%

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

   13. metadata-eval 64.4%

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

6. Applied egg-rr 64.4%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. frac-times 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. +-commutative 99.4%

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

9. Applied egg-rr 99.4%

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

    1. associate-/l* 99.4%

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

    2. +-commutative 99.4%

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

    3. *-commutative 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. *-commutative 77.1%

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

   2. associate-*r* 77.1%

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

   3. associate-*r/ 77.1%

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

   4. associate-*r* 77.1%

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

   5. *-rgt-identity 77.1%

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

   6. sub-neg 77.1%

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

   7. distribute-neg-frac 77.1%

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

   8. metadata-eval 77.1%

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

3. Simplified 77.1%

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

   1. *-un-lft-identity 77.1%

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

   2. difference-of-squares 86.5%

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

   3. times-frac 99.5%

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

   4. add-sqr-sqrt 50.1%

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

   5. sqrt-unprod 71.3%

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

   6. frac-times 71.3%

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

   7. metadata-eval 71.3%

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

   8. metadata-eval 71.3%

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

   9. frac-times 71.3%

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

   10. sqrt-unprod 30.7%

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

   11. add-sqr-sqrt 64.4%

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

   12. div-inv 64.4%

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

   13. metadata-eval 64.4%

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

6. Applied egg-rr 64.4%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. frac-times 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. +-commutative 99.4%

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

9. Applied egg-rr 99.4%

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

    1. *-commutative 99.4%

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

    2. times-frac 99.7%

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

    3. *-commutative 99.7%

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

    4. +-commutative 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 4: 63.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.1%

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

   1. *-commutative 77.1%

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

   2. associate-*r* 77.1%

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

   3. associate-*r/ 77.1%

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

   4. associate-*r* 77.1%

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

   5. *-rgt-identity 77.1%

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

   6. sub-neg 77.1%

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

   7. distribute-neg-frac 77.1%

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

   8. metadata-eval 77.1%

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

3. Simplified 77.1%

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

   1. *-un-lft-identity 77.1%

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

   2. difference-of-squares 86.5%

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

   3. times-frac 99.5%

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

   4. add-sqr-sqrt 50.1%

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

   5. sqrt-unprod 71.3%

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

   6. frac-times 71.3%

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

   7. metadata-eval 71.3%

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

   8. metadata-eval 71.3%

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

   9. frac-times 71.3%

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

   10. sqrt-unprod 30.7%

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

   11. add-sqr-sqrt 64.4%

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

   12. div-inv 64.4%

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

   13. metadata-eval 64.4%

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

6. Applied egg-rr 64.4%

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

7. Taylor expanded in a around 0 99.7%

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

8. Taylor expanded in b around inf 63.4%

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

   1. associate-*r/ 63.4%

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

   2. *-commutative 63.4%

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

   3. frac-times 63.4%

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

   4. *-un-lft-identity 63.4%

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

   5. *-commutative 63.4%

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

10. Applied egg-rr 63.4%

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

11. Final simplification 63.4%

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

Alternative 5: 63.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((pi / 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[(N[(Pi / b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 77.1%

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

   1. *-commutative 77.1%

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

   2. associate-*r* 77.1%

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

   3. associate-*r/ 77.1%

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

   4. associate-*r* 77.1%

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

   5. *-rgt-identity 77.1%

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

   6. sub-neg 77.1%

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

   7. distribute-neg-frac 77.1%

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

   8. metadata-eval 77.1%

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

3. Simplified 77.1%

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

   1. *-un-lft-identity 77.1%

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

   2. difference-of-squares 86.5%

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

   3. times-frac 99.5%

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

   4. add-sqr-sqrt 50.1%

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

   5. sqrt-unprod 71.3%

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

   6. frac-times 71.3%

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

   7. metadata-eval 71.3%

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

   8. metadata-eval 71.3%

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

   9. frac-times 71.3%

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

   10. sqrt-unprod 30.7%

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

   11. add-sqr-sqrt 64.4%

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

   12. div-inv 64.4%

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

   13. metadata-eval 64.4%

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

6. Applied egg-rr 64.4%

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

7. Taylor expanded in a around 0 99.7%

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

8. Taylor expanded in b around inf 63.4%

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

   1. associate-*l/ 63.4%

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

   2. *-un-lft-identity 63.4%

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

   3. associate-*r/ 63.4%

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

   4. frac-times 63.4%

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

   5. *-commutative 63.4%

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

10. Applied egg-rr 63.4%

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

11. Final simplification 63.4%

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

Reproduce

herbie shell --seed 2024053 
(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))))