NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.6%

Time: 12.0s

Alternatives: 3

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 78.5% 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, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. un-div-inv 81.2%

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

   2. difference-of-squares 89.0%

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

   3. associate-/r* 89.0%

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

   4. div-inv 89.0%

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

   5. metadata-eval 89.0%

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

4. Applied egg-rr 89.0%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.5%

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

6. Applied egg-rr 99.5%

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

   1. associate-/l* 99.5%

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

   2. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*r/ 99.6%

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

   5. +-commutative 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-neg-frac 99.6%

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

   8. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in a around 0 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 65.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l* 81.2%

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

   2. *-rgt-identity 81.2%

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

   3. associate-/l* 81.2%

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

   4. metadata-eval 81.2%

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

   5. associate-*l/ 81.2%

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

   6. *-lft-identity 81.2%

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

   7. sub-neg 81.2%

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

   8. distribute-neg-frac 81.2%

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

   9. metadata-eval 81.2%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.2%

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

   3. associate-*r/ 81.2%

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

   4. *-commutative 81.2%

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

   5. difference-of-squares 89.0%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 69.4%

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

7. Taylor expanded in a around 0 69.4%

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

   1. clear-num 69.4%

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

   2. inv-pow 69.4%

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

9. Applied egg-rr 69.4%

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

    1. unpow-1 69.4%

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

    2. associate-/l* 69.4%

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

11. Simplified 69.4%

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

    1. *-un-lft-identity 69.4%

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

    2. un-div-inv 69.4%

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

    3. associate-*r/ 69.4%

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

13. Applied egg-rr 69.4%

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

    1. *-lft-identity 69.4%

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

    2. associate-/l/ 69.1%

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

    3. associate-/l* 69.1%

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

    4. associate-*r* 69.2%

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

15. Simplified 69.2%

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

16. Final simplification 69.2%

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

Alternative 3: 65.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l* 81.2%

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

   2. *-rgt-identity 81.2%

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

   3. associate-/l* 81.2%

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

   4. metadata-eval 81.2%

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

   5. associate-*l/ 81.2%

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

   6. *-lft-identity 81.2%

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

   7. sub-neg 81.2%

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

   8. distribute-neg-frac 81.2%

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

   9. metadata-eval 81.2%

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

3. Simplified 81.2%

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

   1. metadata-eval 81.2%

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

   2. div-inv 81.2%

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

   3. associate-*r/ 81.2%

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

   4. *-commutative 81.2%

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

   5. difference-of-squares 89.0%

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

   6. associate-/r* 99.6%

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

6. Applied egg-rr 69.4%

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

7. Taylor expanded in a around 0 69.4%

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

8. Final simplification 69.4%

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

Reproduce

herbie shell --seed 2024095 
(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))))