NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.2% → 99.7%

Time: 10.5s

Alternatives: 3

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: 79.2% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. un-div-inv 78.7%

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

   2. difference-of-squares 88.1%

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

   3. associate-/r* 89.4%

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

   4. div-inv 89.4%

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

   5. metadata-eval 89.4%

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

4. Applied egg-rr 89.4%

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

5. Taylor expanded in a around inf 62.0%

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

6. Taylor expanded in b around 0 95.5%

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

   1. neg-mul-1 95.5%

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

8. Simplified 95.5%

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

   1. associate-*l/ 93.4%

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

   2. frac-2neg 93.4%

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

   3. *-commutative 93.4%

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

   4. frac-2neg 93.4%

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

   5. metadata-eval 93.4%

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

   6. frac-times 92.4%

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

   7. *-un-lft-identity 92.4%

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

   8. add-sqr-sqrt 49.1%

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

   9. sqrt-unprod 55.9%

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

   10. sqr-neg 55.9%

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

   11. sqrt-unprod 14.5%

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

   12. add-sqr-sqrt 31.5%

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

   13. add-sqr-sqrt 17.1%

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

   14. sqrt-unprod 55.8%

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

   15. sqr-neg 55.8%

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

   16. sqrt-unprod 43.2%

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

   17. add-sqr-sqrt 92.4%

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

10. Applied egg-rr 92.4%

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

    1. *-commutative 92.4%

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

    2. associate-/r* 93.5%

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

    3. distribute-frac-neg2 93.5%

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

    4. remove-double-neg 93.5%

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

    5. associate-/r* 99.7%

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

    6. *-commutative 99.7%

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

    7. +-commutative 99.7%

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

    8. *-commutative 99.7%

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

12. Simplified 99.7%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. *-commutative 78.7%

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

   2. associate-*r* 78.7%

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

   3. associate-*r/ 78.7%

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

   4. associate-*r* 78.7%

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

   5. *-rgt-identity 78.7%

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

   6. sub-neg 78.7%

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

   7. distribute-neg-frac 78.7%

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

   8. metadata-eval 78.7%

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

3. Simplified 78.7%

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

5. Taylor expanded in a around 0 56.4%

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

   1. difference-of-squares 63.4%

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

7. Applied egg-rr 63.4%

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

   1. times-frac 70.6%

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

9. Applied egg-rr 70.6%

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

10. Taylor expanded in a around 0 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. un-div-inv 78.7%

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

   2. difference-of-squares 88.1%

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

   3. associate-/r* 89.4%

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

   4. div-inv 89.4%

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

   5. metadata-eval 89.4%

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

4. Applied egg-rr 89.4%

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

5. Taylor expanded in a around inf 62.0%

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

   1. *-commutative 62.0%

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

   2. frac-2neg 62.0%

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

   3. metadata-eval 62.0%

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

   4. associate-/l/ 61.7%

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

   5. *-commutative 61.7%

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

   6. frac-times 61.8%

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

   7. *-un-lft-identity 61.8%

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

7. Applied egg-rr 61.8%

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

   1. associate-/l* 61.7%

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

   2. *-commutative 61.7%

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

   3. associate-*l* 65.7%

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

   4. +-commutative 65.7%

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

   5. *-commutative 65.7%

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

   6. distribute-lft-neg-out 65.7%

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

   7. distribute-rgt-neg-out 65.7%

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

9. Simplified 65.7%

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

10. Taylor expanded in b around 0 99.1%

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

Reproduce

herbie shell --seed 2024157 
(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))))