NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.3% → 99.7%

Time: 10.1s

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: 79.3% 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: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{\frac{\frac{\pi}{b}}{2 \cdot a}}{b + a} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ (/ PI b) (* 2.0 a)) (+ b a)))

C

assert(a < b);
double code(double a, double b) {
	return ((((double) M_PI) / b) / (2.0 * a)) / (b + a);
}

Java

assert a < b;
public static double code(double a, double b) {
	return ((Math.PI / b) / (2.0 * a)) / (b + a);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return ((math.pi / b) / (2.0 * a)) / (b + a)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(pi / b) / Float64(2.0 * a)) / Float64(b + a))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((pi / b) / (2.0 * a)) / (b + a);
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[(2.0 * a), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 80.2%

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

   2. associate-*l/ 80.2%

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

   3. *-lft-identity 80.2%

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

   4. difference-of-squares 90.4%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.7%

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

   2. div-inv 99.7%

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

   3. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.6%

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

   1. associate-/r* 99.7%

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

   2. metadata-eval 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. *-lft-identity 99.7%

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

   6. associate-*l/ 99.7%

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

   7. distribute-lft-neg-in 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. metadata-eval 99.7%

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

   11. distribute-neg-frac 99.7%

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

   12. distribute-rgt-neg-in 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-*l/ 99.7%

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

   15. *-lft-identity 99.7%

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

   16. distribute-neg-frac 99.7%

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

   17. distribute-neg-frac 99.7%

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

   18. metadata-eval 99.7%

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

9. Simplified 99.7%

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

    1. metadata-eval 99.7%

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

    2. div-inv 99.7%

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

    3. frac-times 99.7%

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

    4. div-inv 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

    \[\leadsto \frac{\frac{\frac{\pi}{b}}{2 \cdot a}}{b + a} \]
13. Add Preprocessing

Alternative 2: 98.9% accurate, 1.9× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* 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 80.2%

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

   1. associate-*l* 80.2%

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

   2. associate-*l/ 80.2%

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

   3. *-lft-identity 80.2%

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

   4. difference-of-squares 90.4%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.7%

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

   2. div-inv 99.7%

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

   3. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.6%

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

   1. associate-/r* 99.7%

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

   2. metadata-eval 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. *-lft-identity 99.7%

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

   6. associate-*l/ 99.7%

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

   7. distribute-lft-neg-in 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. metadata-eval 99.7%

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

   11. distribute-neg-frac 99.7%

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

   12. distribute-rgt-neg-in 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-*l/ 99.7%

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

   15. *-lft-identity 99.7%

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

   16. distribute-neg-frac 99.7%

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

   17. distribute-neg-frac 99.7%

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

   18. metadata-eval 99.7%

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

9. Simplified 99.7%

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

    1. metadata-eval 99.7%

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

    2. div-inv 99.7%

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

    3. frac-times 99.7%

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

    4. div-inv 99.7%

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

11. Applied egg-rr 99.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\pi}{b}}{2 \cdot a}}}{b + a} \]
12. Step-by-step derivation

    1. expm1-log1p-u 80.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{\pi}{b}}{2 \cdot a}}{b + a}\right)\right)} \]

    2. expm1-udef 57.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\frac{\pi}{b}}{2 \cdot a}}{b + a}\right)} - 1} \]

    3. associate-/l/ 57.8%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\pi}{b}}{\left(b + a\right) \cdot \left(2 \cdot a\right)}}\right)} - 1 \]

13. Applied egg-rr 57.8%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{b}}{\left(b + a\right) \cdot \left(2 \cdot a\right)}\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 74.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{b}}{\left(b + a\right) \cdot \left(2 \cdot a\right)}\right)\right)} \]

    2. expm1-log1p 94.2%

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

    3. *-lft-identity 94.2%

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

    4. associate-*l/ 94.2%

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

    5. associate-/l/ 95.0%

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

    6. *-commutative 95.0%

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

    7. associate-*r/ 99.7%

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

    8. associate-/r* 99.7%

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

    9. metadata-eval 99.7%

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

    10. times-frac 99.7%

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

    11. associate-/r* 99.7%

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

    12. associate-*r/ 99.7%

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

    13. associate-/r* 94.2%

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

    14. times-frac 95.9%

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

    15. associate-/r* 95.9%

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

    16. associate-*l/ 95.8%

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

    17. associate-*r/ 95.8%

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

    18. associate-/l/ 95.4%

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

    19. remove-double-neg 95.4%

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

15. Simplified 99.0%

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

16. Final simplification 99.0%

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

Alternative 3: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 80.2%

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

   2. associate-*l/ 80.2%

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

   3. *-lft-identity 80.2%

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

   4. difference-of-squares 90.4%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a around 0 99.6%

   \[\leadsto \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{1}{a \cdot b}}}{b + a} \]
6. Step-by-step derivation

   1. associate-/r* 99.7%

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

   2. metadata-eval 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. *-lft-identity 99.7%

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

   6. associate-*l/ 99.7%

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

   7. distribute-lft-neg-in 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. metadata-eval 99.7%

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

   11. distribute-neg-frac 99.7%

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

   12. distribute-rgt-neg-in 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-*l/ 99.7%

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

   15. *-lft-identity 99.7%

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

   16. distribute-neg-frac 99.7%

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

   17. distribute-neg-frac 99.7%

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

   18. metadata-eval 99.7%

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

7. Simplified 99.6%

   \[\leadsto \frac{\pi}{2} \cdot \frac{\color{blue}{\frac{\frac{1}{b}}{a}}}{b + a} \]
8. Step-by-step derivation

   1. associate-*r/ 99.7%

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

   2. div-inv 99.7%

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

   3. metadata-eval 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-/l/ 99.6%

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

   6. div-inv 99.7%

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

   7. div-inv 99.6%

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

   8. associate-/l* 99.5%

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

   9. frac-times 99.0%

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

   10. metadata-eval 99.0%

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

   11. *-commutative 99.0%

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

   12. *-un-lft-identity 99.0%

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

   13. times-frac 99.0%

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

   14. /-rgt-identity 99.0%

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

9. Applied egg-rr 99.0%

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

10. Final simplification 99.0%

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

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 80.2%

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

   2. associate-*l/ 80.2%

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

   3. *-lft-identity 80.2%

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

   4. difference-of-squares 90.4%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.7%

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

   2. div-inv 99.7%

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

   3. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.7%

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

   1. associate-*r/ 99.7%

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

9. Simplified 99.7%

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

    1. *-commutative 99.7%

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

    2. times-frac 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 5: 62.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 80.2%

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

   2. associate-*l/ 80.2%

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

   3. *-lft-identity 80.2%

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

   4. difference-of-squares 90.4%

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

   5. associate-/l/ 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.7%

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

   2. div-inv 99.7%

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

   3. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.6%

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

   1. associate-/r* 99.7%

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

   2. metadata-eval 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. *-lft-identity 99.7%

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

   6. associate-*l/ 99.7%

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

   7. distribute-lft-neg-in 99.7%

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

   8. distribute-neg-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. metadata-eval 99.7%

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

   11. distribute-neg-frac 99.7%

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

   12. distribute-rgt-neg-in 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-*l/ 99.7%

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

   15. *-lft-identity 99.7%

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

   16. distribute-neg-frac 99.7%

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

   17. distribute-neg-frac 99.7%

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

   18. metadata-eval 99.7%

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

9. Simplified 99.7%

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

    1. metadata-eval 99.7%

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

    2. div-inv 99.7%

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

    3. associate-*r/ 99.6%

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

    4. div-inv 99.5%

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

    5. associate-/l/ 99.6%

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

    6. frac-times 99.0%

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

    7. metadata-eval 99.0%

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

    8. +-commutative 99.0%

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

    9. associate-*r* 95.4%

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

    10. div-inv 95.4%

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

    11. div-inv 95.4%

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

    12. metadata-eval 95.4%

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

    13. times-frac 95.9%

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

    14. +-commutative 95.9%

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

11. Applied egg-rr 95.9%

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

12. Taylor expanded in b around 0 64.8%

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

13. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2024019 
(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))))