Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 49.9% → 79.2%

Time: 5.6s

Alternatives: 7

Speedup: 19.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 49.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 79.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-259}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 100000000000:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{elif}\;x \cdot x \leq 10^{+247}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= (* x x) 5e-259)
     (fma 0.5 (* (/ x y) (/ x y)) -1.0)
     (if (<= (* x x) 5e-72)
       t_1
       (if (<= (* x x) 100000000000.0)
         (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0))
         (if (<= (* x x) 1e+247) t_1 (fma (* (/ y x) (/ y x)) -8.0 1.0)))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if ((x * x) <= 5e-259) {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	} else if ((x * x) <= 5e-72) {
		tmp = t_1;
	} else if ((x * x) <= 100000000000.0) {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	} else if ((x * x) <= 1e+247) {
		tmp = t_1;
	} else {
		tmp = fma(((y / x) * (y / x)), -8.0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-259)
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	elseif (Float64(x * x) <= 5e-72)
		tmp = t_1;
	elseif (Float64(x * x) <= 100000000000.0)
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y) / Float64(y / x)) / 4.0));
	elseif (Float64(x * x) <= 1e+247)
		tmp = t_1;
	else
		tmp = fma(Float64(Float64(y / x) * Float64(y / x)), -8.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-259], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e-72], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 100000000000.0], N[(-1.0 + N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+247], t$95$1, N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0 + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x x) < 4.99999999999999977e-259

   1. Initial program 50.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 70.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 70.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 70.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]

      3. unpow2 70.4%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 88.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. metadata-eval 88.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Simplified 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]

   if 4.99999999999999977e-259 < (*.f64 x x) < 4.9999999999999996e-72 or 1e11 < (*.f64 x x) < 9.99999999999999952e246

   1. Initial program 86.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.9999999999999996e-72 < (*.f64 x x) < 1e11

   1. Initial program 46.7%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 27.7%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 27.7%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

      2. unpow2 27.7%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot y\right)} \cdot 4} \]

      3. associate-*r* 27.7%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]

   4. Simplified 27.7%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]
   5. Step-by-step derivation

      1. div-sub 27.5%

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot \left(y \cdot 4\right)} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)}} \]

      2. associate-*r* 27.5%

         \[\leadsto \frac{x \cdot x}{\color{blue}{\left(y \cdot y\right) \cdot 4}} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      3. associate-/r* 27.5%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot x}{y \cdot y}}{4}} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      4. frac-times 27.5%

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      5. pow2 27.5%

         \[\leadsto \frac{\color{blue}{{\left(\frac{x}{y}\right)}^{2}}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      6. *-commutative 27.5%

         \[\leadsto \frac{{\left(\frac{x}{y}\right)}^{2}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y}} \]

      7. *-inverses 81.0%

         \[\leadsto \frac{{\left(\frac{x}{y}\right)}^{2}}{4} - \color{blue}{1} \]

   6. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y}\right)}^{2}}{4} - 1} \]
   7. Step-by-step derivation

      1. unpow2 81.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}}{4} - 1 \]

      2. clear-num 81.0%

         \[\leadsto \frac{\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}}{4} - 1 \]

      3. un-div-inv 81.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]

   8. Applied egg-rr 81.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]

   if 9.99999999999999952e246 < (*.f64 x x)

   1. Initial program 8.1%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 81.6%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 81.6%

         \[\leadsto \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]

      2. distribute-rgt-out-- 81.6%

         \[\leadsto 1 + \color{blue}{\frac{{y}^{2}}{{x}^{2}} \cdot \left(-4 - 4\right)} \]

      3. metadata-eval 81.6%

         \[\leadsto 1 + \frac{{y}^{2}}{{x}^{2}} \cdot \color{blue}{-8} \]

      4. *-commutative 81.6%

         \[\leadsto 1 + \color{blue}{-8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]

      5. +-commutative 81.6%

         \[\leadsto \color{blue}{-8 \cdot \frac{{y}^{2}}{{x}^{2}} + 1} \]

      6. *-commutative 81.6%

         \[\leadsto \color{blue}{\frac{{y}^{2}}{{x}^{2}} \cdot -8} + 1 \]

      7. fma-def 81.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{y}^{2}}{{x}^{2}}, -8, 1\right)} \]

      8. unpow2 81.6%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y \cdot y}}{{x}^{2}}, -8, 1\right) \]

      9. unpow2 81.6%

         \[\leadsto \mathsf{fma}\left(\frac{y \cdot y}{\color{blue}{x \cdot x}}, -8, 1\right) \]

      10. times-frac 92.3%

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, -8, 1\right) \]

   4. Simplified 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-259}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 100000000000:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{elif}\;x \cdot x \leq 10^{+247}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \]

Alternative 2: 64.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;x \leq 7.5 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 260000:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= x 7.5e-130)
     (fma 0.5 (* (/ x y) (/ x y)) -1.0)
     (if (<= x 1.1e-34)
       t_1
       (if (<= x 260000.0)
         (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0))
         (if (<= x 2.9e+124) t_1 (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (x <= 7.5e-130) {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	} else if (x <= 1.1e-34) {
		tmp = t_1;
	} else if (x <= 260000.0) {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	} else if (x <= 2.9e+124) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (x <= 7.5e-130)
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	elseif (x <= 1.1e-34)
		tmp = t_1;
	elseif (x <= 260000.0)
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y) / Float64(y / x)) / 4.0));
	elseif (x <= 2.9e+124)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7.5e-130], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[x, 1.1e-34], t$95$1, If[LessEqual[x, 260000.0], N[(-1.0 + N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e+124], t$95$1, N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.4999999999999994e-130

   1. Initial program 45.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 50.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 50.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 50.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]

      3. unpow2 50.2%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 60.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. metadata-eval 60.6%

         \[\leadsto \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, \color{blue}{-1}\right) \]

   4. Simplified 60.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]

   if 7.4999999999999994e-130 < x < 1.0999999999999999e-34 or 2.6e5 < x < 2.90000000000000021e124

   1. Initial program 87.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 1.0999999999999999e-34 < x < 2.6e5

   1. Initial program 50.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 31.0%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 31.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

      2. unpow2 31.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot y\right)} \cdot 4} \]

      3. associate-*r* 31.0%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]

   4. Simplified 31.0%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]
   5. Step-by-step derivation

      1. div-sub 30.7%

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot \left(y \cdot 4\right)} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)}} \]

      2. associate-*r* 30.7%

         \[\leadsto \frac{x \cdot x}{\color{blue}{\left(y \cdot y\right) \cdot 4}} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      3. associate-/r* 30.7%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot x}{y \cdot y}}{4}} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      4. frac-times 30.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      5. pow2 30.7%

         \[\leadsto \frac{\color{blue}{{\left(\frac{x}{y}\right)}^{2}}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      6. *-commutative 30.7%

         \[\leadsto \frac{{\left(\frac{x}{y}\right)}^{2}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y}} \]

      7. *-inverses 81.0%

         \[\leadsto \frac{{\left(\frac{x}{y}\right)}^{2}}{4} - \color{blue}{1} \]

   6. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y}\right)}^{2}}{4} - 1} \]
   7. Step-by-step derivation

      1. unpow2 81.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}}{4} - 1 \]

      2. clear-num 81.0%

         \[\leadsto \frac{\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}}{4} - 1 \]

      3. un-div-inv 81.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]

   8. Applied egg-rr 81.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]

   if 2.90000000000000021e124 < x

   1. Initial program 12.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 13.0%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   3. Step-by-step derivation

      1. unpow2 13.0%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Simplified 13.0%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. Taylor expanded in x around inf 79.0%

      \[\leadsto \color{blue}{1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 79.0%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 79.0%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 91.4%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      4. unpow2 91.4%

         \[\leadsto 1 + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \]

   7. Simplified 91.4%

      \[\leadsto \color{blue}{1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 91.4%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 91.4%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 91.4%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   9. Applied egg-rr 91.4%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \leq 260000:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+124}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 3: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ t_2 := -1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-259}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 100000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 10^{+247}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0)))
        (t_2 (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0))))
   (if (<= (* x x) 5e-259)
     t_2
     (if (<= (* x x) 5e-72)
       t_1
       (if (<= (* x x) 100000000000.0)
         t_2
         (if (<= (* x x) 1e+247) t_1 (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = -1.0 + (((x / y) / (y / x)) / 4.0);
	double tmp;
	if ((x * x) <= 5e-259) {
		tmp = t_2;
	} else if ((x * x) <= 5e-72) {
		tmp = t_1;
	} else if ((x * x) <= 100000000000.0) {
		tmp = t_2;
	} else if ((x * x) <= 1e+247) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    t_2 = (-1.0d0) + (((x / y) / (y / x)) / 4.0d0)
    if ((x * x) <= 5d-259) then
        tmp = t_2
    else if ((x * x) <= 5d-72) then
        tmp = t_1
    else if ((x * x) <= 100000000000.0d0) then
        tmp = t_2
    else if ((x * x) <= 1d+247) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = -1.0 + (((x / y) / (y / x)) / 4.0);
	double tmp;
	if ((x * x) <= 5e-259) {
		tmp = t_2;
	} else if ((x * x) <= 5e-72) {
		tmp = t_1;
	} else if ((x * x) <= 100000000000.0) {
		tmp = t_2;
	} else if ((x * x) <= 1e+247) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	t_2 = -1.0 + (((x / y) / (y / x)) / 4.0)
	tmp = 0
	if (x * x) <= 5e-259:
		tmp = t_2
	elif (x * x) <= 5e-72:
		tmp = t_1
	elif (x * x) <= 100000000000.0:
		tmp = t_2
	elif (x * x) <= 1e+247:
		tmp = t_1
	else:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	t_2 = Float64(-1.0 + Float64(Float64(Float64(x / y) / Float64(y / x)) / 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-259)
		tmp = t_2;
	elseif (Float64(x * x) <= 5e-72)
		tmp = t_1;
	elseif (Float64(x * x) <= 100000000000.0)
		tmp = t_2;
	elseif (Float64(x * x) <= 1e+247)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	t_2 = -1.0 + (((x / y) / (y / x)) / 4.0);
	tmp = 0.0;
	if ((x * x) <= 5e-259)
		tmp = t_2;
	elseif ((x * x) <= 5e-72)
		tmp = t_1;
	elseif ((x * x) <= 100000000000.0)
		tmp = t_2;
	elseif ((x * x) <= 1e+247)
		tmp = t_1;
	else
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 + N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-259], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 5e-72], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 100000000000.0], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 1e+247], t$95$1, N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.99999999999999977e-259 or 4.9999999999999996e-72 < (*.f64 x x) < 1e11

   1. Initial program 49.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 45.2%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 45.2%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

      2. unpow2 45.2%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot y\right)} \cdot 4} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]

   4. Simplified 45.2%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]
   5. Step-by-step derivation

      1. div-sub 45.2%

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot \left(y \cdot 4\right)} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)}} \]

      2. associate-*r* 45.2%

         \[\leadsto \frac{x \cdot x}{\color{blue}{\left(y \cdot y\right) \cdot 4}} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      3. associate-/r* 45.2%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot x}{y \cdot y}}{4}} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      4. frac-times 45.2%

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      5. pow2 45.2%

         \[\leadsto \frac{\color{blue}{{\left(\frac{x}{y}\right)}^{2}}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      6. *-commutative 45.2%

         \[\leadsto \frac{{\left(\frac{x}{y}\right)}^{2}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y}} \]

      7. *-inverses 86.8%

         \[\leadsto \frac{{\left(\frac{x}{y}\right)}^{2}}{4} - \color{blue}{1} \]

   6. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y}\right)}^{2}}{4} - 1} \]
   7. Step-by-step derivation

      1. unpow2 86.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}}{4} - 1 \]

      2. clear-num 86.8%

         \[\leadsto \frac{\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}}{4} - 1 \]

      3. un-div-inv 86.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]

   8. Applied egg-rr 86.8%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]

   if 4.99999999999999977e-259 < (*.f64 x x) < 4.9999999999999996e-72 or 1e11 < (*.f64 x x) < 9.99999999999999952e246

   1. Initial program 86.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 9.99999999999999952e246 < (*.f64 x x)

   1. Initial program 8.1%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 8.3%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   3. Step-by-step derivation

      1. unpow2 8.3%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Simplified 8.3%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. Taylor expanded in x around inf 81.6%

      \[\leadsto \color{blue}{1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 81.6%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 81.6%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 92.0%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      4. unpow2 92.0%

         \[\leadsto 1 + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \]

   7. Simplified 92.0%

      \[\leadsto \color{blue}{1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 92.0%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 92.0%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 92.0%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   9. Applied egg-rr 92.0%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-259}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-72}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 100000000000:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{elif}\;x \cdot x \leq 10^{+247}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+173}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e+173)
   (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0))
   (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+173) {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 1d+173) then
        tmp = (-1.0d0) + (((x / y) / (y / x)) / 4.0d0)
    else
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+173) {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1e+173:
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0)
	else:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e+173)
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y) / Float64(y / x)) / 4.0));
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 1e+173)
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	else
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+173], N[(-1.0 + N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e173

   1. Initial program 64.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 44.5%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{4 \cdot {y}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 44.5%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{{y}^{2} \cdot 4}} \]

      2. unpow2 44.5%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot y\right)} \cdot 4} \]

      3. associate-*r* 44.5%

         \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]

   4. Simplified 44.5%

      \[\leadsto \frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{\color{blue}{y \cdot \left(y \cdot 4\right)}} \]
   5. Step-by-step derivation

      1. div-sub 44.1%

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot \left(y \cdot 4\right)} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)}} \]

      2. associate-*r* 44.1%

         \[\leadsto \frac{x \cdot x}{\color{blue}{\left(y \cdot y\right) \cdot 4}} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      3. associate-/r* 44.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot x}{y \cdot y}}{4}} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      4. frac-times 44.1%

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      5. pow2 44.1%

         \[\leadsto \frac{\color{blue}{{\left(\frac{x}{y}\right)}^{2}}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{y \cdot \left(y \cdot 4\right)} \]

      6. *-commutative 44.1%

         \[\leadsto \frac{{\left(\frac{x}{y}\right)}^{2}}{4} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right) \cdot y}} \]

      7. *-inverses 75.0%

         \[\leadsto \frac{{\left(\frac{x}{y}\right)}^{2}}{4} - \color{blue}{1} \]

   6. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{y}\right)}^{2}}{4} - 1} \]
   7. Step-by-step derivation

      1. unpow2 75.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}}{4} - 1 \]

      2. clear-num 75.0%

         \[\leadsto \frac{\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}}}{4} - 1 \]

      3. un-div-inv 75.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]

   8. Applied egg-rr 75.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]

   if 1e173 < (*.f64 x x)

   1. Initial program 22.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 20.7%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   3. Step-by-step derivation

      1. unpow2 20.7%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Simplified 20.7%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. Taylor expanded in x around inf 80.8%

      \[\leadsto \color{blue}{1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 80.8%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 80.8%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 89.4%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      4. unpow2 89.4%

         \[\leadsto 1 + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \]

   7. Simplified 89.4%

      \[\leadsto \color{blue}{1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 89.4%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 89.4%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 89.4%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   9. Applied egg-rr 89.4%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+173}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 5: 61.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+86}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.2e+86) -1.0 (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.2e+86) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.2d+86) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.2e+86) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.2e+86:
		tmp = -1.0
	else:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.2e+86)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.2e+86)
		tmp = -1.0;
	else
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.2e+86], -1.0, N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2e86

   1. Initial program 53.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 59.1%

      \[\leadsto \color{blue}{-1} \]

   if 3.2e86 < x

   1. Initial program 26.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 23.9%

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   3. Step-by-step derivation

      1. unpow2 23.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Simplified 23.9%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. Taylor expanded in x around inf 78.4%

      \[\leadsto \color{blue}{1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 78.4%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 78.4%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 88.7%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      4. unpow2 88.7%

         \[\leadsto 1 + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \]

   7. Simplified 88.7%

      \[\leadsto \color{blue}{1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 88.7%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 88.7%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 88.7%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   9. Applied egg-rr 88.7%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+86}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 6: 61.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+86}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 9.5e+86) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.5e+86) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 9.5d+86) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9.5e+86) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9.5e+86:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.5e+86)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9.5e+86)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.5e+86], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.50000000000000028e86

   1. Initial program 53.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 59.1%

      \[\leadsto \color{blue}{-1} \]

   if 9.50000000000000028e86 < x

   1. Initial program 26.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 88.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+86}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 50.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 47.3%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

2. Taylor expanded in x around 0 48.6%

   \[\leadsto \color{blue}{-1} \]

3. Final simplification 48.6%

   \[\leadsto -1 \]

Developer target: 50.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t_0\\ t_2 := \frac{t_0}{t_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t_3}{x \cdot x + t_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t_1} - t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t_1}}\right)}^{2} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2023213 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))