NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.7%

Time: 10.4s

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: 78.2% 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

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

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(0.5 * Float64(Float64(pi / 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[(0.5 * N[(N[(Pi / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(a + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.2%

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

   1. un-div-inv 74.3%

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

   2. difference-of-squares 82.1%

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

   3. associate-/r* 83.0%

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

   4. div-inv 83.0%

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

   5. metadata-eval 83.0%

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

4. Applied egg-rr 83.0%

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

   1. frac-sub 83.1%

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

   2. frac-times 99.1%

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

   3. *-un-lft-identity 99.1%

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

   4. *-commutative 99.1%

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

6. Applied egg-rr 99.1%

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

   1. *-rgt-identity 99.1%

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

   2. associate-/l* 99.1%

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

   3. *-commutative 99.1%

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

   4. +-commutative 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*l* 91.1%

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

8. Simplified 91.1%

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

   1. associate-*l/ 91.1%

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

   2. *-commutative 91.1%

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

   3. associate-*r* 99.1%

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

   4. +-commutative 99.1%

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

10. Applied egg-rr 99.1%

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

11. Taylor expanded in b around 0 99.7%

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

    1. associate-/r* 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 99.0% accurate, 1.9× speedup

Math

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

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(0.5 * pi) / Float64(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[(0.5 * Pi), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.2%

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

   1. un-div-inv 74.3%

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

   2. difference-of-squares 82.1%

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

   3. associate-/r* 83.0%

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

   4. div-inv 83.0%

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

   5. metadata-eval 83.0%

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

4. Applied egg-rr 83.0%

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

   1. frac-sub 83.1%

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

   2. frac-times 99.1%

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

   3. *-un-lft-identity 99.1%

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

   4. *-commutative 99.1%

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

6. Applied egg-rr 99.1%

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

   1. *-rgt-identity 99.1%

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

   2. associate-/l* 99.1%

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

   3. *-commutative 99.1%

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

   4. +-commutative 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*l* 91.1%

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

8. Simplified 91.1%

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

9. Taylor expanded in b around 0 99.7%

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

    1. *-commutative 99.7%

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

    2. frac-times 99.2%

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

    3. *-un-lft-identity 99.2%

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

11. Applied egg-rr 99.2%

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

Alternative 3: 65.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. un-div-inv 74.3%

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

   2. difference-of-squares 82.1%

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

   3. associate-/r* 83.0%

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

   4. div-inv 83.0%

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

   5. metadata-eval 83.0%

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

4. Applied egg-rr 83.0%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in b around 0 66.7%

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

   1. associate-*r/ 66.7%

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

   2. *-commutative 66.7%

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

9. Simplified 66.7%

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

    1. associate-/l/ 66.7%

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

    2. *-commutative 66.7%

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

    3. times-frac 66.7%

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

    4. associate-/r* 66.7%

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

11. Applied egg-rr 66.7%

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

Alternative 4: 65.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. un-div-inv 74.3%

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

   2. difference-of-squares 82.1%

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

   3. associate-/r* 83.0%

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

   4. div-inv 83.0%

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

   5. metadata-eval 83.0%

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

4. Applied egg-rr 83.0%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in b around 0 66.7%

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

   1. associate-*r/ 66.7%

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

   2. *-commutative 66.7%

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

9. Simplified 66.7%

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

    1. associate-/l/ 66.7%

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

    2. times-frac 66.7%

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

11. Applied egg-rr 66.7%

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

Alternative 5: 59.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. un-div-inv 74.3%

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

   2. difference-of-squares 82.1%

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

   3. associate-/r* 83.0%

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

   4. div-inv 83.0%

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

   5. metadata-eval 83.0%

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

4. Applied egg-rr 83.0%

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

   1. associate-*l/ 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in b around 0 66.7%

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

   1. associate-*r/ 66.7%

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

   2. *-commutative 66.7%

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

9. Simplified 66.7%

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

    1. associate-/l/ 66.7%

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

    2. *-commutative 66.7%

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

    3. *-commutative 66.7%

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

    4. associate-*l* 58.8%

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

    5. times-frac 58.7%

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

11. Applied egg-rr 58.7%

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

Reproduce

herbie shell --seed 2024137 
(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))))