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

Percentage Accurate: 51.2% → 80.8%

Time: 4.9s

Alternatives: 5

Speedup: 3.7×

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: 51.2% 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: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-315}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 10^{+117}:\\ \;\;\;\;\frac{x \cdot x - t_0}{\mathsf{fma}\left(x, x, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 4e-315)
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     (if (<= t_0 1e+117) (/ (- (* x x) t_0) (fma x x t_0)) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 4e-315) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 1e+117) {
		tmp = ((x * x) - t_0) / fma(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 (t_0 <= 4e-315)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 1e+117)
		tmp = Float64(Float64(Float64(x * x) - t_0) / fma(x, x, t_0));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-315], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+117], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(x * x + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 3.9999999989e-315

   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. Step-by-step derivation

      1. *-commutative 50.0%

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

      2. fma-def 50.0%

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

      3. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in x around inf 82.1%

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

      1. associate--l+ 82.1%

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

      2. unpow2 82.1%

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

      3. unpow2 82.1%

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

      4. unpow2 82.1%

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

      5. unpow2 82.1%

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

      6. associate-*r/ 82.1%

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

      7. *-commutative 82.1%

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

      8. associate-*r* 82.1%

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

   6. Simplified 82.1%

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

      1. associate-*r/ 82.1%

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

      2. sub-div 82.1%

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

      3. *-commutative 82.1%

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

      4. metadata-eval 82.1%

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

      5. distribute-rgt-neg-in 82.1%

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

      6. associate-*r* 82.1%

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

      7. distribute-rgt-neg-in 82.1%

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

      8. distribute-rgt-neg-in 82.1%

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

      9. metadata-eval 82.1%

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

      10. associate-*r* 82.1%

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

      11. *-commutative 82.1%

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

   8. Applied egg-rr 82.1%

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

   9. Taylor expanded in y around 0 82.1%

      \[\leadsto 1 + \color{blue}{-8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 82.1%

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

       2. unpow2 82.1%

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

       3. times-frac 90.1%

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

   11. Simplified 90.1%

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

   if 3.9999999989e-315 < (*.f64 (*.f64 y 4) y) < 1.00000000000000005e117

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. fma-def 77.9%

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

      3. *-commutative 77.9%

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

   3. Simplified 77.9%

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

   if 1.00000000000000005e117 < (*.f64 (*.f64 y 4) y)

   1. Initial program 23.6%

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

      1. *-commutative 23.6%

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

      2. fma-def 23.6%

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

      3. *-commutative 23.6%

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

   3. Simplified 23.6%

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

   4. Taylor expanded in x around 0 83.2%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-315}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+117}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-315}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 10^{+117}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 4e-315)
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     (if (<= t_0 1e+117) (/ (- (* x x) t_0) (+ t_0 (* x x))) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 4e-315) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 1e+117) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} 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 (t_0 <= 4d-315) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else if (t_0 <= 1d+117) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    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 (t_0 <= 4e-315) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 1e+117) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 4e-315:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	elif t_0 <= 1e+117:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 4e-315)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 1e+117)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	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 (t_0 <= 4e-315)
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	elseif (t_0 <= 1e+117)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	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[t$95$0, 4e-315], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+117], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 3.9999999989e-315

   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. Step-by-step derivation

      1. *-commutative 50.0%

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

      2. fma-def 50.0%

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

      3. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in x around inf 82.1%

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

      1. associate--l+ 82.1%

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

      2. unpow2 82.1%

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

      3. unpow2 82.1%

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

      4. unpow2 82.1%

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

      5. unpow2 82.1%

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

      6. associate-*r/ 82.1%

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

      7. *-commutative 82.1%

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

      8. associate-*r* 82.1%

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

   6. Simplified 82.1%

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

      1. associate-*r/ 82.1%

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

      2. sub-div 82.1%

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

      3. *-commutative 82.1%

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

      4. metadata-eval 82.1%

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

      5. distribute-rgt-neg-in 82.1%

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

      6. associate-*r* 82.1%

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

      7. distribute-rgt-neg-in 82.1%

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

      8. distribute-rgt-neg-in 82.1%

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

      9. metadata-eval 82.1%

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

      10. associate-*r* 82.1%

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

      11. *-commutative 82.1%

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

   8. Applied egg-rr 82.1%

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

   9. Taylor expanded in y around 0 82.1%

      \[\leadsto 1 + \color{blue}{-8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 82.1%

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

       2. unpow2 82.1%

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

       3. times-frac 90.1%

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

   11. Simplified 90.1%

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

   if 3.9999999989e-315 < (*.f64 (*.f64 y 4) y) < 1.00000000000000005e117

   1. Initial program 77.9%

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

   if 1.00000000000000005e117 < (*.f64 (*.f64 y 4) y)

   1. Initial program 23.6%

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

      1. *-commutative 23.6%

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

      2. fma-def 23.6%

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

      3. *-commutative 23.6%

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

   3. Simplified 23.6%

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

   4. Taylor expanded in x around 0 83.2%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-315}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+117}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+55}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+120} \lor \neg \left(x \cdot x \leq 5 \cdot 10^{+217}\right):\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e+55)
   -1.0
   (if (or (<= (* x x) 5e+120) (not (<= (* x x) 5e+217)))
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+55) {
		tmp = -1.0;
	} else if (((x * x) <= 5e+120) || !((x * x) <= 5e+217)) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} 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 * x) <= 1d+55) then
        tmp = -1.0d0
    else if (((x * x) <= 5d+120) .or. (.not. ((x * x) <= 5d+217))) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+55) {
		tmp = -1.0;
	} else if (((x * x) <= 5e+120) || !((x * x) <= 5e+217)) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1e+55:
		tmp = -1.0
	elif ((x * x) <= 5e+120) or not ((x * x) <= 5e+217):
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e+55)
		tmp = -1.0;
	elseif ((Float64(x * x) <= 5e+120) || !(Float64(x * x) <= 5e+217))
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 1e+55)
		tmp = -1.0;
	elseif (((x * x) <= 5e+120) || ~(((x * x) <= 5e+217)))
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+55], -1.0, If[Or[LessEqual[N[(x * x), $MachinePrecision], 5e+120], N[Not[LessEqual[N[(x * x), $MachinePrecision], 5e+217]], $MachinePrecision]], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000001e55 or 5.00000000000000019e120 < (*.f64 x x) < 5.00000000000000041e217

   1. Initial program 60.1%

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

      1. *-commutative 60.1%

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

      2. fma-def 60.1%

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

      3. *-commutative 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in x around 0 79.9%

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

   if 1.00000000000000001e55 < (*.f64 x x) < 5.00000000000000019e120 or 5.00000000000000041e217 < (*.f64 x x)

   1. Initial program 20.4%

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

      1. *-commutative 20.4%

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

      2. fma-def 20.4%

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

      3. *-commutative 20.4%

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

   3. Simplified 20.4%

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

   4. Taylor expanded in x around inf 73.4%

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

      1. associate--l+ 73.4%

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

      2. unpow2 73.4%

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

      3. unpow2 73.4%

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

      4. unpow2 73.4%

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

      5. unpow2 73.4%

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

      6. associate-*r/ 73.4%

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

      7. *-commutative 73.4%

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

      8. associate-*r* 73.4%

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

   6. Simplified 73.4%

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

      1. associate-*r/ 73.4%

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

      2. sub-div 73.4%

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

      3. *-commutative 73.4%

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

      4. metadata-eval 73.4%

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

      5. distribute-rgt-neg-in 73.4%

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

      6. associate-*r* 73.4%

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

      7. distribute-rgt-neg-in 73.4%

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

      8. distribute-rgt-neg-in 73.4%

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

      9. metadata-eval 73.4%

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

      10. associate-*r* 73.4%

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

      11. *-commutative 73.4%

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

   8. Applied egg-rr 73.4%

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

   9. Taylor expanded in y around 0 73.4%

      \[\leadsto 1 + \color{blue}{-8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 73.4%

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

       2. unpow2 73.4%

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

       3. times-frac 81.6%

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

   11. Simplified 81.6%

       \[\leadsto 1 + \color{blue}{-8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+55}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+120} \lor \neg \left(x \cdot x \leq 5 \cdot 10^{+217}\right):\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 72.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+115}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8e+29) 1.0 (if (<= x 5e+115) -1.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8e+29) {
		tmp = 1.0;
	} else if (x <= 5e+115) {
		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 <= (-8d+29)) then
        tmp = 1.0d0
    else if (x <= 5d+115) 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 <= -8e+29) {
		tmp = 1.0;
	} else if (x <= 5e+115) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8e+29:
		tmp = 1.0
	elif x <= 5e+115:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8e+29)
		tmp = 1.0;
	elseif (x <= 5e+115)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e+29)
		tmp = 1.0;
	elseif (x <= 5e+115)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8e+29], 1.0, If[LessEqual[x, 5e+115], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999931e29 or 5.00000000000000008e115 < x

   1. Initial program 19.4%

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

      1. *-commutative 19.4%

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

      2. fma-def 19.4%

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

      3. *-commutative 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in x around inf 78.6%

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

   if -7.99999999999999931e29 < x < 5.00000000000000008e115

   1. Initial program 60.7%

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

      1. *-commutative 60.7%

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

      2. fma-def 60.7%

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

      3. *-commutative 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in x around 0 78.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+115}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 50.9% 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 45.7%

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

   1. *-commutative 45.7%

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

   2. fma-def 45.7%

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

   3. *-commutative 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in x around 0 58.3%

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

5. Final simplification 58.3%

   \[\leadsto -1 \]

Developer target: 51.7% 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 2023195 
(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))))