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

Percentage Accurate: 50.6% → 80.1%

Time: 3.7s

Alternatives: 4

Speedup: 3.2×

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: 50.6% 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: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+239}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4e-189)
   -1.0
   (if (<= (* x x) 4e+239)
     (/ (fma x x (* y (* y -4.0))) (fma x x (* y (* y 4.0))))
     1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4e-189) {
		tmp = -1.0;
	} else if ((x * x) <= 4e+239) {
		tmp = fma(x, x, (y * (y * -4.0))) / fma(x, x, (y * (y * 4.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 4e-189)
		tmp = -1.0;
	elseif (Float64(x * x) <= 4e+239)
		tmp = Float64(fma(x, x, Float64(y * Float64(y * -4.0))) / fma(x, x, Float64(y * Float64(y * 4.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e-189], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 4e+239], N[(N[(x * x + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.00000000000000027e-189

   1. Initial program 61.6%

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

      1. sub-neg 61.6%

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

      2. distribute-rgt-neg-in 61.6%

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

      3. cancel-sign-sub 61.6%

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

      4. distribute-lft-neg-out 61.6%

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

      5. remove-double-neg 61.6%

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

      6. distribute-lft-neg-out 61.6%

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

      7. distribute-lft-neg-in 61.6%

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

      8. distribute-rgt-neg-out 61.6%

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

   3. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 87.1%

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

   if 4.00000000000000027e-189 < (*.f64 x x) < 3.99999999999999996e239

   1. Initial program 86.1%

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

      1. sub-neg 86.1%

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

      2. distribute-rgt-neg-in 86.1%

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

      3. cancel-sign-sub 86.1%

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

      4. distribute-lft-neg-out 86.1%

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

      5. remove-double-neg 86.1%

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

      6. distribute-lft-neg-out 86.1%

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

      7. distribute-lft-neg-in 86.1%

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

      8. distribute-rgt-neg-out 86.1%

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

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   if 3.99999999999999996e239 < (*.f64 x x)

   1. Initial program 11.0%

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

      1. sub-neg 11.0%

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

      2. distribute-rgt-neg-in 11.0%

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

      3. cancel-sign-sub 11.0%

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

      4. distribute-lft-neg-out 11.0%

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

      5. remove-double-neg 11.0%

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

      6. distribute-lft-neg-out 11.0%

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

      7. distribute-lft-neg-in 11.0%

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

      8. distribute-rgt-neg-out 11.0%

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

   3. Simplified 11.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 85.9%

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

Alternative 2: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 2.5 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.7 \cdot 10^{+240}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 2.5e-189)
     -1.0
     (if (<= (* x x) 1.7e+240) (/ (- (* x x) t_0) (+ (* x x) t_0)) 1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2.5e-189) {
		tmp = -1.0;
	} else if ((x * x) <= 1.7e+240) {
		tmp = ((x * x) - t_0) / ((x * x) + t_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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 2.5d-189) then
        tmp = -1.0d0
    else if ((x * x) <= 1.7d+240) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2.5e-189) {
		tmp = -1.0;
	} else if ((x * x) <= 1.7e+240) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 2.5e-189:
		tmp = -1.0
	elif (x * x) <= 1.7e+240:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 2.5e-189)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1.7e+240)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 2.5e-189)
		tmp = -1.0;
	elseif ((x * x) <= 1.7e+240)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2.5e-189], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1.7e+240], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.4999999999999999e-189

   1. Initial program 61.6%

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

      1. sub-neg 61.6%

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

      2. distribute-rgt-neg-in 61.6%

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

      3. cancel-sign-sub 61.6%

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

      4. distribute-lft-neg-out 61.6%

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

      5. remove-double-neg 61.6%

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

      6. distribute-lft-neg-out 61.6%

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

      7. distribute-lft-neg-in 61.6%

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

      8. distribute-rgt-neg-out 61.6%

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

   3. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 87.1%

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

   if 2.4999999999999999e-189 < (*.f64 x x) < 1.70000000000000004e240

   1. Initial program 86.1%

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

   if 1.70000000000000004e240 < (*.f64 x x)

   1. Initial program 11.0%

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

      1. sub-neg 11.0%

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

      2. distribute-rgt-neg-in 11.0%

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

      3. cancel-sign-sub 11.0%

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

      4. distribute-lft-neg-out 11.0%

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

      5. remove-double-neg 11.0%

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

      6. distribute-lft-neg-out 11.0%

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

      7. distribute-lft-neg-in 11.0%

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

      8. distribute-rgt-neg-out 11.0%

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

   3. Simplified 11.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 85.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.5 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.7 \cdot 10^{+240}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-70}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 6.5e-70) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.5e-70) {
		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 <= 6.5d-70) 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 <= 6.5e-70) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.5e-70:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.5e-70)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.5e-70)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.5e-70], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.5000000000000005e-70

   1. Initial program 56.0%

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

      1. sub-neg 56.0%

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

      2. distribute-rgt-neg-in 56.0%

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

      3. cancel-sign-sub 56.0%

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

      4. distribute-lft-neg-out 56.0%

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

      5. remove-double-neg 56.0%

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

      6. distribute-lft-neg-out 56.0%

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

      7. distribute-lft-neg-in 56.0%

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

      8. distribute-rgt-neg-out 56.0%

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

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 51.7%

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

   if 6.5000000000000005e-70 < x

   1. Initial program 38.9%

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

      1. sub-neg 38.9%

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

      2. distribute-rgt-neg-in 38.9%

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

      3. cancel-sign-sub 38.9%

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

      4. distribute-lft-neg-out 38.9%

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

      5. remove-double-neg 38.9%

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

      6. distribute-lft-neg-out 38.9%

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

      7. distribute-lft-neg-in 38.9%

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

      8. distribute-rgt-neg-out 38.9%

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

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 71.3%

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

Alternative 4: 49.9% 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 51.2%

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

   1. sub-neg 51.2%

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

   2. distribute-rgt-neg-in 51.2%

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

   3. cancel-sign-sub 51.2%

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

   4. distribute-lft-neg-out 51.2%

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

   5. remove-double-neg 51.2%

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

   6. distribute-lft-neg-out 51.2%

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

   7. distribute-lft-neg-in 51.2%

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

   8. distribute-rgt-neg-out 51.2%

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

3. Simplified 51.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 45.2%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 51.1% 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 2024170 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 9743233849626781/10000000000000000) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4))))))

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