NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.6%

Time: 9.2s

Alternatives: 5

Speedup: 1.1×

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: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-*r/ 77.4%

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

   2. *-rgt-identity 77.4%

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

   3. associate-*l/ 77.4%

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

   4. difference-of-squares 86.4%

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

   5. *-commutative 86.4%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*l/ 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%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-*r/ 77.4%

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

   2. *-rgt-identity 77.4%

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

   3. associate-*l/ 77.4%

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

   4. difference-of-squares 86.4%

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

   5. *-commutative 86.4%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*l/ 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%

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

5. Applied egg-rr 99.7%

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

   1. associate-/l* 98.2%

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

   2. +-commutative 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in b around 0 71.5%

   \[\leadsto \frac{\pi \cdot 0.5}{\color{blue}{{a}^{2} \cdot b + {b}^{2} \cdot \left(-1 \cdot \left(a + -1 \cdot a\right) - -1 \cdot a\right)}} \]

10. Simplified 98.3%

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

    1. times-frac 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 3: 62.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-*r/ 77.4%

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

   2. *-rgt-identity 77.4%

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

   3. associate-*l/ 77.4%

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

   4. difference-of-squares 86.4%

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

   5. *-commutative 86.4%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 64.0%

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

5. Taylor expanded in b around 0 59.5%

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

   1. associate-*r/ 59.5%

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

7. Simplified 59.5%

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

   1. associate-/l* 59.5%

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

   2. frac-times 59.2%

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

   3. metadata-eval 59.2%

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

   4. *-commutative 59.2%

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

9. Applied egg-rr 59.2%

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

    1. associate-/r* 59.5%

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

    2. *-commutative 59.5%

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

    3. div-inv 59.5%

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

    4. clear-num 59.5%

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

    5. frac-times 59.5%

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

    6. associate-/r* 59.5%

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

11. Applied egg-rr 59.5%

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

12. Final simplification 59.5%

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

Alternative 4: 62.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-*r/ 77.4%

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

   2. *-rgt-identity 77.4%

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

   3. associate-*l/ 77.4%

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

   4. difference-of-squares 86.4%

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

   5. *-commutative 86.4%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 64.0%

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

5. Taylor expanded in b around 0 59.5%

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

   1. associate-*r/ 59.5%

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

7. Simplified 59.5%

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

   1. associate-/l* 59.5%

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

   2. frac-times 59.2%

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

   3. metadata-eval 59.2%

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

   4. *-commutative 59.2%

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

9. Applied egg-rr 59.2%

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

    1. associate-*r* 59.2%

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

    2. associate-/r/ 59.2%

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

    3. associate-/l/ 59.5%

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

    4. *-un-lft-identity 59.5%

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

    5. associate-/r/ 59.5%

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

    6. times-frac 59.5%

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

    7. clear-num 59.5%

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

11. Applied egg-rr 59.5%

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

12. Final simplification 59.5%

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

Alternative 5: 62.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-*r/ 77.4%

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

   2. *-rgt-identity 77.4%

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

   3. associate-*l/ 77.4%

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

   4. difference-of-squares 86.4%

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

   5. *-commutative 86.4%

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

   6. times-frac 99.6%

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

   7. sub-neg 99.6%

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

   8. distribute-neg-frac 99.6%

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

   9. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 64.0%

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

5. Taylor expanded in b around 0 59.5%

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

   1. associate-*r/ 59.5%

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

7. Simplified 59.5%

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

   1. associate-/l* 59.5%

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

   2. associate-/r* 59.5%

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

   3. frac-times 59.5%

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

9. Applied egg-rr 59.5%

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

    1. associate-*r/ 59.5%

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

    2. metadata-eval 59.5%

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

    3. associate-*l/ 59.5%

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

    4. associate-/l* 59.5%

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

11. Simplified 59.5%

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

12. Taylor expanded in a around 0 59.5%

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

    1. associate-*r/ 59.5%

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

14. Simplified 59.5%

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

15. Final simplification 59.5%

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

Reproduce

herbie shell --seed 2023319 
(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))))