NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.7%

Time: 9.6s

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: 78.0% 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: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\pi}{a}}{2 \cdot b}}{a + b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. associate-*l* 78.5%

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

   2. associate-*l/ 78.5%

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

   3. *-lft-identity 78.5%

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

   4. difference-of-squares 86.3%

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

   5. associate-/l/ 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a around inf 68.5%

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

6. Taylor expanded in b around 0 99.6%

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

   1. associate-/r* 99.6%

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

8. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

   2. frac-times 99.7%

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

   3. div-inv 99.7%

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

   4. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{\frac{\frac{\pi}{a}}{2 \cdot b}}{a + b} \]
12. Add Preprocessing

Alternative 2: 74.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{a \cdot b}\\ \mathbf{if}\;a \leq -4 \cdot 10^{-42}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ PI (* a b))))
   (if (<= a -4e-42) (* t_0 (/ 0.5 a)) (* t_0 (/ 0.5 b)))))

C

double code(double a, double b) {
	double t_0 = ((double) M_PI) / (a * b);
	double tmp;
	if (a <= -4e-42) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.PI / (a * b);
	double tmp;
	if (a <= -4e-42) {
		tmp = t_0 * (0.5 / a);
	} else {
		tmp = t_0 * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pi / (a * b)
	tmp = 0
	if a <= -4e-42:
		tmp = t_0 * (0.5 / a)
	else:
		tmp = t_0 * (0.5 / b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(pi / Float64(a * b))
	tmp = 0.0
	if (a <= -4e-42)
		tmp = Float64(t_0 * Float64(0.5 / a));
	else
		tmp = Float64(t_0 * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = pi / (a * b);
	tmp = 0.0;
	if (a <= -4e-42)
		tmp = t_0 * (0.5 / a);
	else
		tmp = t_0 * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(Pi / N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4e-42], N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.00000000000000015e-42

   1. Initial program 81.3%

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

      1. associate-*l* 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. associate-*l/ 81.2%

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

      3. *-lft-identity 81.2%

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

      4. difference-of-squares 90.7%

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

      5. associate-/l/ 99.8%

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

      6. sub-neg 99.8%

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

      7. distribute-neg-frac 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around 0 99.7%

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

      1. expm1-log1p-u 91.6%

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

      2. expm1-udef 58.5%

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

      3. *-commutative 58.5%

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

      4. associate-/l/ 58.5%

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

      5. frac-times 58.5%

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

      6. *-un-lft-identity 58.5%

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

      7. +-commutative 58.5%

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

   7. Applied egg-rr 58.5%

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

      1. expm1-def 91.2%

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

      2. expm1-log1p 99.5%

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

      3. *-rgt-identity 99.5%

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

      4. *-commutative 99.5%

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

      5. times-frac 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-/r* 99.7%

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

      8. times-frac 99.7%

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

      9. associate-*r/ 99.7%

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

      10. *-rgt-identity 99.7%

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

   9. Simplified 99.7%

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

       1. expm1-log1p-u 91.5%

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

       2. expm1-udef 58.5%

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

       3. *-un-lft-identity 58.5%

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

       4. times-frac 58.5%

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

       5. metadata-eval 58.5%

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

   11. Applied egg-rr 58.5%

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

       1. expm1-def 91.5%

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

       2. expm1-log1p 99.7%

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

       3. associate-/r* 99.5%

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

       4. associate-*r/ 99.5%

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

       5. *-commutative 99.5%

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

       6. times-frac 99.7%

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

   13. Simplified 99.7%

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

   14. Taylor expanded in a around inf 86.3%

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

   if -4.00000000000000015e-42 < a

   1. Initial program 77.6%

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

      1. associate-*l* 77.6%

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

      2. associate-*l/ 77.6%

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

      3. *-lft-identity 77.6%

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

      4. difference-of-squares 84.9%

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

      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 99.5%

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

      1. expm1-log1p-u 75.0%

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

      2. expm1-udef 49.0%

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

      3. *-commutative 49.0%

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

      4. associate-/l/ 49.0%

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

      5. frac-times 49.0%

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

      6. *-un-lft-identity 49.0%

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

      7. +-commutative 49.0%

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

   7. Applied egg-rr 49.0%

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

      1. expm1-def 74.2%

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

      2. expm1-log1p 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. *-commutative 98.8%

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

      5. times-frac 98.8%

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

      6. *-commutative 98.8%

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

      7. associate-/r* 99.5%

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

      8. times-frac 99.6%

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

      9. associate-*r/ 99.6%

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

      10. *-rgt-identity 99.6%

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

   9. Simplified 99.6%

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

       1. expm1-log1p-u 75.0%

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

       2. expm1-udef 49.0%

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

       3. *-un-lft-identity 49.0%

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

       4. times-frac 49.0%

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

       5. metadata-eval 49.0%

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

   11. Applied egg-rr 49.0%

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

       1. expm1-def 75.0%

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

       2. expm1-log1p 99.6%

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

       3. associate-/r* 98.8%

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

       4. associate-*r/ 98.8%

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

       5. *-commutative 98.8%

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

       6. times-frac 99.6%

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

   13. Simplified 99.6%

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

   14. Taylor expanded in a around 0 70.1%

       \[\leadsto \frac{\pi}{a \cdot b} \cdot \color{blue}{\frac{0.5}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{a \cdot b} \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

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 78.5%

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

   1. associate-*l* 78.5%

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

   2. associate-*l/ 78.5%

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

   3. *-lft-identity 78.5%

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

   4. difference-of-squares 86.3%

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

   5. associate-/l/ 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{b - a}}{b + a}} \]
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 79.1%

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

   2. expm1-udef 51.3%

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

   3. *-commutative 51.3%

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

   4. associate-/l/ 51.3%

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

   5. frac-times 51.3%

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

   6. *-un-lft-identity 51.3%

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

   7. +-commutative 51.3%

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

7. Applied egg-rr 51.3%

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

   1. expm1-def 78.4%

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

   2. expm1-log1p 99.0%

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

   3. *-rgt-identity 99.0%

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

   4. *-commutative 99.0%

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

   5. times-frac 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 99.6%

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

   8. times-frac 99.6%

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

   9. associate-*r/ 99.7%

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

   10. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

    1. expm1-log1p-u 79.1%

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

    2. expm1-udef 51.3%

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

    3. *-un-lft-identity 51.3%

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

    4. times-frac 51.3%

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

    5. metadata-eval 51.3%

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

11. Applied egg-rr 51.3%

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

    1. expm1-def 79.1%

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

    2. expm1-log1p 99.7%

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

    3. associate-/r* 99.0%

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

    4. associate-*r/ 99.0%

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

    5. *-commutative 99.0%

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

    6. times-frac 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. associate-*l* 78.5%

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

   2. associate-*l/ 78.5%

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

   3. *-lft-identity 78.5%

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

   4. difference-of-squares 86.3%

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

   5. associate-/l/ 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{b - a}}{b + a}} \]
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 79.1%

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

   2. expm1-udef 51.3%

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

   3. *-commutative 51.3%

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

   4. associate-/l/ 51.3%

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

   5. frac-times 51.3%

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

   6. *-un-lft-identity 51.3%

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

   7. +-commutative 51.3%

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

7. Applied egg-rr 51.3%

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

   1. expm1-def 78.4%

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

   2. expm1-log1p 99.0%

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

   3. *-rgt-identity 99.0%

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

   4. *-commutative 99.0%

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

   5. times-frac 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 99.6%

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

   8. times-frac 99.6%

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

   9. associate-*r/ 99.7%

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

   10. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

    1. expm1-log1p-u 79.1%

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

    2. expm1-udef 51.3%

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

    3. *-un-lft-identity 51.3%

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

    4. times-frac 51.3%

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

    5. metadata-eval 51.3%

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

11. Applied egg-rr 51.3%

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

    1. expm1-def 79.1%

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

    2. expm1-log1p 99.7%

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

    3. associate-/r* 99.0%

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

    4. associate-*r/ 99.0%

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

    5. *-commutative 99.0%

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

    6. times-frac 99.6%

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

13. Simplified 99.6%

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

    1. associate-*l/ 99.6%

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

15. Applied egg-rr 99.6%

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

16. Final simplification 99.6%

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

Alternative 5: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. associate-*l* 78.5%

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

   2. associate-*l/ 78.5%

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

   3. *-lft-identity 78.5%

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

   4. difference-of-squares 86.3%

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

   5. associate-/l/ 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{b - a}}{b + a}} \]
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 79.1%

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

   2. expm1-udef 51.3%

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

   3. *-commutative 51.3%

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

   4. associate-/l/ 51.3%

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

   5. frac-times 51.3%

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

   6. *-un-lft-identity 51.3%

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

   7. +-commutative 51.3%

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

7. Applied egg-rr 51.3%

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

   1. expm1-def 78.4%

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

   2. expm1-log1p 99.0%

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

   3. *-rgt-identity 99.0%

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

   4. *-commutative 99.0%

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

   5. times-frac 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 99.6%

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

   8. times-frac 99.6%

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

   9. associate-*r/ 99.7%

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

   10. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 6: 63.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. associate-*l* 78.5%

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

   2. associate-*l/ 78.5%

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

   3. *-lft-identity 78.5%

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

   4. difference-of-squares 86.3%

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

   5. associate-/l/ 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-neg-frac 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} + \frac{-1}{b}}{b - a}}{b + a}} \]
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 79.1%

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

   2. expm1-udef 51.3%

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

   3. *-commutative 51.3%

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

   4. associate-/l/ 51.3%

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

   5. frac-times 51.3%

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

   6. *-un-lft-identity 51.3%

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

   7. +-commutative 51.3%

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

7. Applied egg-rr 51.3%

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

   1. expm1-def 78.4%

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

   2. expm1-log1p 99.0%

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

   3. *-rgt-identity 99.0%

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

   4. *-commutative 99.0%

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

   5. times-frac 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 99.6%

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

   8. times-frac 99.6%

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

   9. associate-*r/ 99.7%

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

   10. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

    1. expm1-log1p-u 79.1%

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

    2. expm1-udef 51.3%

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

    3. *-un-lft-identity 51.3%

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

    4. times-frac 51.3%

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

    5. metadata-eval 51.3%

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

11. Applied egg-rr 51.3%

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

    1. expm1-def 79.1%

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

    2. expm1-log1p 99.7%

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

    3. associate-/r* 99.0%

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

    4. associate-*r/ 99.0%

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

    5. *-commutative 99.0%

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

    6. times-frac 99.6%

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

13. Simplified 99.6%

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

14. Taylor expanded in a around inf 61.1%

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

15. Final simplification 61.1%

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

Reproduce

herbie shell --seed 2024018 
(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))))