NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.7%

Time: 11.1s

Alternatives: 6

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: 78.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-*l* 79.4%

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

   2. associate-*l/ 79.3%

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

   3. *-lft-identity 79.3%

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

   4. difference-of-squares 87.1%

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

   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%

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

   4. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.7%

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

   1. expm1-log1p-u 72.8%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{a \cdot b}\right)\right)}}{a + b} \]

   2. expm1-udef 59.5%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{a \cdot b}\right)} - 1\right)}}{a + b} \]

9. Applied egg-rr 59.5%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{a \cdot b}\right)} - 1\right)}}{a + b} \]
10. Step-by-step derivation

    1. expm1-def 72.8%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{a \cdot b}\right)\right)}}{a + b} \]

    2. expm1-log1p 99.7%

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

    3. associate-/l/ 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 94.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-*l* 79.4%

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

   2. associate-*l/ 79.3%

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

   3. *-lft-identity 79.3%

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

   4. difference-of-squares 87.1%

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

   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%

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

   4. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.7%

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

   1. expm1-log1p-u 72.8%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{a \cdot b}\right)\right)}}{a + b} \]

   2. expm1-udef 59.5%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{a \cdot b}\right)} - 1\right)}}{a + b} \]

9. Applied egg-rr 59.5%

   \[\leadsto \frac{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\pi}{a \cdot b}\right)} - 1\right)}}{a + b} \]
10. Step-by-step derivation

    1. expm1-def 72.8%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{a \cdot b}\right)\right)}}{a + b} \]

    2. expm1-log1p 99.7%

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

    3. associate-/l/ 99.7%

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

11. Simplified 99.7%

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

    1. associate-/l* 98.7%

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

    2. associate-/l/ 98.7%

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

    3. associate-/r* 98.7%

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

    4. associate-/r/ 99.6%

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

    5. clear-num 99.6%

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

    6. times-frac 92.4%

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

    7. associate-/r/ 92.4%

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

    8. associate-*r/ 92.4%

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

    9. associate-/l/ 91.9%

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

    10. associate-*r/ 91.9%

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

    11. times-frac 92.3%

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

    12. clear-num 92.4%

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

    13. +-commutative 92.4%

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

13. Applied egg-rr 92.4%

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

14. Final simplification 92.4%

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

Alternative 3: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.3%

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

   1. associate-*l* 79.4%

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

   2. associate-*l/ 79.3%

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

   3. *-lft-identity 79.3%

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

   4. difference-of-squares 87.1%

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

   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. expm1-log1p-u 72.8%

      \[\leadsto \frac{\pi}{2} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a \cdot b}\right)\right)}}{b + a} \]

   2. expm1-udef 59.4%

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

7. Applied egg-rr 59.4%

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

   1. expm1-def 72.8%

      \[\leadsto \frac{\pi}{2} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a \cdot b}\right)\right)}}{b + a} \]

   2. expm1-log1p 99.6%

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

   3. associate-/r* 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. associate-*r/ 99.5%

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

   6. associate-*l/ 99.6%

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

   7. *-lft-identity 99.6%

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

9. Simplified 99.6%

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

    1. frac-times 99.6%

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

    2. associate-/l/ 99.6%

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

    3. div-inv 99.7%

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

    4. *-un-lft-identity 99.7%

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

    5. +-commutative 99.7%

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

    6. times-frac 99.7%

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

    7. metadata-eval 99.7%

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

    8. metadata-eval 99.7%

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

    9. times-frac 99.7%

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

    10. *-un-lft-identity 99.7%

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

    11. associate-/l* 98.7%

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

11. Applied egg-rr 98.7%

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

12. Final simplification 98.7%

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

Alternative 4: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(Pi / N[(N[(N[(b + a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. associate-*l* 79.4%

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

   2. associate-*l/ 79.3%

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

   3. *-lft-identity 79.3%

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

   4. difference-of-squares 87.1%

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

   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. *-commutative 99.6%

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

   2. associate-/l/ 98.7%

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

   3. frac-times 98.8%

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

   4. *-un-lft-identity 98.8%

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

   5. +-commutative 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

   \[\leadsto \frac{\pi}{\left(\left(b + a\right) \cdot \left(b \cdot a\right)\right) \cdot 2} \]
9. Add Preprocessing

Alternative 5: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-*l* 79.4%

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

   2. associate-*l/ 79.3%

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

   3. *-lft-identity 79.3%

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

   4. difference-of-squares 87.1%

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

   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%

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

   4. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.7%

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

8. Final simplification 99.7%

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

Alternative 6: 64.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.3%

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

   1. associate-*l* 79.4%

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

   2. associate-*l/ 79.3%

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

   3. *-lft-identity 79.3%

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

   4. difference-of-squares 87.1%

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

   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%

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

   4. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a around 0 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. times-frac 99.7%

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

   3. metadata-eval 99.7%

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

   4. metadata-eval 99.7%

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

   5. times-frac 99.7%

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

   6. *-un-lft-identity 99.7%

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

   7. div-inv 99.6%

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

   8. associate-/l/ 99.6%

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

   9. +-commutative 99.6%

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

   10. frac-times 99.6%

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

   11. expm1-log1p-u 78.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{b}}{a}}{b + a}\right)\right)} \]

   12. expm1-udef 51.7%

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

9. Applied egg-rr 51.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{a + b}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 78.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\pi}{a} \cdot \frac{0.5}{b}}{a + b}\right)\right)} \]

    2. expm1-log1p 99.7%

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

    3. associate-/l* 92.4%

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

11. Simplified 92.4%

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

12. Taylor expanded in a around inf 63.6%

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

    1. *-commutative 63.6%

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

    2. *-commutative 63.6%

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

    3. *-commutative 63.6%

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

    4. associate-*r* 63.9%

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

14. Simplified 63.9%

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

15. Final simplification 63.9%

    \[\leadsto \frac{\frac{\pi}{a}}{a \cdot \left(b \cdot 2\right)} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))