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

Percentage Accurate: 51.2% → 81.0%

Time: 6.6s

Alternatives: 6

Speedup: 6.2×

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: 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: 81.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ t_2 := \mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\\ \end{array} \end{array} \]

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 (fma 0.5 (* (/ x y) (/ x y)) -1.0)))
   (if (<= (* x x) 5e-156)
     t_2
     (if (<= (* x x) 1.8e+71)
       t_1
       (if (<= (* x x) 2e+87)
         t_2
         (if (<= (* x x) 4e+200) t_1 (+ 1.0 (* -8.0 (pow (/ y x) 2.0)))))))))

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 t_2 = fma(0.5, ((x / y) * (x / y)), -1.0);
	double tmp;
	if ((x * x) <= 5e-156) {
		tmp = t_2;
	} else if ((x * x) <= 1.8e+71) {
		tmp = t_1;
	} else if ((x * x) <= 2e+87) {
		tmp = t_2;
	} else if ((x * x) <= 4e+200) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * pow((y / x), 2.0));
	}
	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))
	t_2 = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0)
	tmp = 0.0
	if (Float64(x * x) <= 5e-156)
		tmp = t_2;
	elseif (Float64(x * x) <= 1.8e+71)
		tmp = t_1;
	elseif (Float64(x * x) <= 2e+87)
		tmp = t_2;
	elseif (Float64(x * x) <= 4e+200)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-8.0 * (Float64(y / x) ^ 2.0)));
	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]}, Block[{t$95$2 = N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-156], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 1.8e+71], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 2e+87], t$95$2, If[LessEqual[N[(x * x), $MachinePrecision], 4e+200], t$95$1, N[(1.0 + N[(-8.0 * N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.00000000000000007e-156 or 1.8e71 < (*.f64 x x) < 1.9999999999999999e87

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0 75.3%

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

      1. fma-neg 75.3%

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

      2. unpow2 75.3%

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

      3. unpow2 75.3%

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

      4. times-frac 85.9%

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

      5. metadata-eval 85.9%

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

   4. Simplified 85.9%

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

   if 5.00000000000000007e-156 < (*.f64 x x) < 1.8e71 or 1.9999999999999999e87 < (*.f64 x x) < 3.9999999999999999e200

   1. Initial program 84.5%

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

   if 3.9999999999999999e200 < (*.f64 x x)

   1. Initial program 7.9%

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

   2. Taylor expanded in x around inf 75.4%

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

      1. associate--l+ 75.4%

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

      2. unpow2 75.4%

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

      3. associate-*r/ 75.4%

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

      4. *-commutative 75.4%

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

      5. unpow2 75.4%

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

      6. associate-*r* 75.4%

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

      7. unpow2 75.4%

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

      8. associate-*r/ 75.4%

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

      9. *-commutative 75.4%

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

      10. unpow2 75.4%

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

      11. associate-*r* 75.4%

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

   4. Simplified 75.4%

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

   5. Taylor expanded in y around 0 75.4%

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

      1. unpow2 75.4%

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

      2. unpow2 75.4%

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

      3. times-frac 86.2%

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

      4. unpow2 86.2%

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

   7. Simplified 86.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+200}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\\ \end{array} \]

Alternative 2: 80.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+87}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\\ \end{array} \end{array} \]

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-156)
     -1.0
     (if (<= (* x x) 1.8e+71)
       t_1
       (if (<= (* x x) 2e+87)
         -1.0
         (if (<= (* x x) 4e+200) t_1 (+ 1.0 (* -8.0 (pow (/ y x) 2.0)))))))))

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-156) {
		tmp = -1.0;
	} else if ((x * x) <= 1.8e+71) {
		tmp = t_1;
	} else if ((x * x) <= 2e+87) {
		tmp = -1.0;
	} else if ((x * x) <= 4e+200) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * pow((y / x), 2.0));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    if ((x * x) <= 5d-156) then
        tmp = -1.0d0
    else if ((x * x) <= 1.8d+71) then
        tmp = t_1
    else if ((x * x) <= 2d+87) then
        tmp = -1.0d0
    else if ((x * x) <= 4d+200) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) ** 2.0d0))
    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 tmp;
	if ((x * x) <= 5e-156) {
		tmp = -1.0;
	} else if ((x * x) <= 1.8e+71) {
		tmp = t_1;
	} else if ((x * x) <= 2e+87) {
		tmp = -1.0;
	} else if ((x * x) <= 4e+200) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-8.0 * Math.pow((y / x), 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if (x * x) <= 5e-156:
		tmp = -1.0
	elif (x * x) <= 1.8e+71:
		tmp = t_1
	elif (x * x) <= 2e+87:
		tmp = -1.0
	elif (x * x) <= 4e+200:
		tmp = t_1
	else:
		tmp = 1.0 + (-8.0 * math.pow((y / x), 2.0))
	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-156)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1.8e+71)
		tmp = t_1;
	elseif (Float64(x * x) <= 2e+87)
		tmp = -1.0;
	elseif (Float64(x * x) <= 4e+200)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-8.0 * (Float64(y / x) ^ 2.0)));
	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);
	tmp = 0.0;
	if ((x * x) <= 5e-156)
		tmp = -1.0;
	elseif ((x * x) <= 1.8e+71)
		tmp = t_1;
	elseif ((x * x) <= 2e+87)
		tmp = -1.0;
	elseif ((x * x) <= 4e+200)
		tmp = t_1;
	else
		tmp = 1.0 + (-8.0 * ((y / x) ^ 2.0));
	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]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-156], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1.8e+71], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 2e+87], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 4e+200], t$95$1, N[(1.0 + N[(-8.0 * N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.00000000000000007e-156 or 1.8e71 < (*.f64 x x) < 1.9999999999999999e87

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0 84.9%

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

   if 5.00000000000000007e-156 < (*.f64 x x) < 1.8e71 or 1.9999999999999999e87 < (*.f64 x x) < 3.9999999999999999e200

   1. Initial program 84.5%

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

   if 3.9999999999999999e200 < (*.f64 x x)

   1. Initial program 7.9%

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

   2. Taylor expanded in x around inf 75.4%

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

      1. associate--l+ 75.4%

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

      2. unpow2 75.4%

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

      3. associate-*r/ 75.4%

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

      4. *-commutative 75.4%

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

      5. unpow2 75.4%

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

      6. associate-*r* 75.4%

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

      7. unpow2 75.4%

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

      8. associate-*r/ 75.4%

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

      9. *-commutative 75.4%

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

      10. unpow2 75.4%

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

      11. associate-*r* 75.4%

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

   4. Simplified 75.4%

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

   5. Taylor expanded in y around 0 75.4%

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

      1. unpow2 75.4%

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

      2. unpow2 75.4%

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

      3. times-frac 86.2%

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

      4. unpow2 86.2%

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

   7. Simplified 86.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+87}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+200}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\\ \end{array} \]

Alternative 3: 80.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+87}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+200}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8 \cdot \frac{y}{\frac{x}{y}}}{x}\\ \end{array} \end{array} \]

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-156)
     -1.0
     (if (<= (* x x) 1.8e+71)
       t_1
       (if (<= (* x x) 2e+87)
         -1.0
         (if (<= (* x x) 4e+200) t_1 (+ 1.0 (/ (* -8.0 (/ y (/ x y))) x))))))))

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-156) {
		tmp = -1.0;
	} else if ((x * x) <= 1.8e+71) {
		tmp = t_1;
	} else if ((x * x) <= 2e+87) {
		tmp = -1.0;
	} else if ((x * x) <= 4e+200) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((-8.0 * (y / (x / y))) / x);
	}
	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) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x * x) - t_0) / ((x * x) + t_0)
    if ((x * x) <= 5d-156) then
        tmp = -1.0d0
    else if ((x * x) <= 1.8d+71) then
        tmp = t_1
    else if ((x * x) <= 2d+87) then
        tmp = -1.0d0
    else if ((x * x) <= 4d+200) then
        tmp = t_1
    else
        tmp = 1.0d0 + (((-8.0d0) * (y / (x / y))) / x)
    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 tmp;
	if ((x * x) <= 5e-156) {
		tmp = -1.0;
	} else if ((x * x) <= 1.8e+71) {
		tmp = t_1;
	} else if ((x * x) <= 2e+87) {
		tmp = -1.0;
	} else if ((x * x) <= 4e+200) {
		tmp = t_1;
	} else {
		tmp = 1.0 + ((-8.0 * (y / (x / y))) / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if (x * x) <= 5e-156:
		tmp = -1.0
	elif (x * x) <= 1.8e+71:
		tmp = t_1
	elif (x * x) <= 2e+87:
		tmp = -1.0
	elif (x * x) <= 4e+200:
		tmp = t_1
	else:
		tmp = 1.0 + ((-8.0 * (y / (x / y))) / x)
	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-156)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1.8e+71)
		tmp = t_1;
	elseif (Float64(x * x) <= 2e+87)
		tmp = -1.0;
	elseif (Float64(x * x) <= 4e+200)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(Float64(-8.0 * Float64(y / Float64(x / y))) / x));
	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);
	tmp = 0.0;
	if ((x * x) <= 5e-156)
		tmp = -1.0;
	elseif ((x * x) <= 1.8e+71)
		tmp = t_1;
	elseif ((x * x) <= 2e+87)
		tmp = -1.0;
	elseif ((x * x) <= 4e+200)
		tmp = t_1;
	else
		tmp = 1.0 + ((-8.0 * (y / (x / y))) / x);
	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]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-156], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1.8e+71], t$95$1, If[LessEqual[N[(x * x), $MachinePrecision], 2e+87], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 4e+200], t$95$1, N[(1.0 + N[(N[(-8.0 * N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 5.00000000000000007e-156 or 1.8e71 < (*.f64 x x) < 1.9999999999999999e87

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0 84.9%

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

   if 5.00000000000000007e-156 < (*.f64 x x) < 1.8e71 or 1.9999999999999999e87 < (*.f64 x x) < 3.9999999999999999e200

   1. Initial program 84.5%

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

   if 3.9999999999999999e200 < (*.f64 x x)

   1. Initial program 7.9%

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

   2. Taylor expanded in x around inf 75.4%

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

      1. associate--l+ 75.4%

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

      2. unpow2 75.4%

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

      3. associate-*r/ 75.4%

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

      4. *-commutative 75.4%

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

      5. unpow2 75.4%

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

      6. associate-*r* 75.4%

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

      7. unpow2 75.4%

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

      8. associate-*r/ 75.4%

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

      9. *-commutative 75.4%

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

      10. unpow2 75.4%

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

      11. associate-*r* 75.4%

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

   4. Simplified 75.4%

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

   5. Taylor expanded in y around 0 75.4%

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

      1. unpow2 75.4%

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

      2. unpow2 75.4%

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

      3. times-frac 86.2%

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

      4. unpow2 86.2%

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

   7. Simplified 86.2%

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

      1. *-commutative 86.2%

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

      2. unpow2 86.2%

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

      3. times-frac 75.4%

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

      4. associate-/r* 75.8%

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

      5. associate-*l/ 75.8%

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

      6. associate-/l* 86.1%

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

   9. Applied egg-rr 86.1%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-156}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+87}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+200}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8 \cdot \frac{y}{\frac{x}{y}}}{x}\\ \end{array} \]

Alternative 4: 74.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{-8 \cdot \frac{y}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 5e+81) (+ 1.0 (/ (* -8.0 (/ y (/ x y))) x)) -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 5e+81) {
		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 ((y * (y * 4.0d0)) <= 5d+81) 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 ((y * (y * 4.0)) <= 5e+81) {
		tmp = 1.0 + ((-8.0 * (y / (x / y))) / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (y * 4.0)) <= 5e+81:
		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(y * Float64(y * 4.0)) <= 5e+81)
		tmp = Float64(1.0 + Float64(Float64(-8.0 * Float64(y / Float64(x / y))) / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (y * 4.0)) <= 5e+81)
		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[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 5e+81], N[(1.0 + N[(N[(-8.0 * N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y 4) y) < 4.9999999999999998e81

   1. Initial program 56.4%

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

   2. Taylor expanded in x around inf 72.6%

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

      1. associate--l+ 72.6%

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

      2. unpow2 72.6%

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

      3. associate-*r/ 72.6%

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

      4. *-commutative 72.6%

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

      5. unpow2 72.6%

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

      6. associate-*r* 72.6%

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

      7. unpow2 72.6%

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

      8. associate-*r/ 72.6%

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

      9. *-commutative 72.6%

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

      10. unpow2 72.6%

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

      11. associate-*r* 72.6%

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

   4. Simplified 72.6%

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

   5. Taylor expanded in y around 0 72.6%

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

      1. unpow2 72.6%

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

      2. unpow2 72.6%

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

      3. times-frac 77.1%

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

      4. unpow2 77.1%

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

   7. Simplified 77.1%

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. times-frac 72.6%

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

      4. associate-/r* 76.5%

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

      5. associate-*l/ 76.5%

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

      6. associate-/l* 77.1%

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

   9. Applied egg-rr 77.1%

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

   if 4.9999999999999998e81 < (*.f64 (*.f64 y 4) y)

   1. Initial program 32.7%

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

   2. Taylor expanded in x around 0 79.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+81}:\\ \;\;\;\;1 + \frac{-8 \cdot \frac{y}{\frac{x}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 61.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5e+40) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+40) {
		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 (y <= 5d+40) 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 (y <= 5e+40) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+40:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+40)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+40)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+40], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000003e40

   1. Initial program 48.2%

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

   2. Taylor expanded in x around inf 61.6%

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

   if 5.00000000000000003e40 < y

   1. Initial program 40.7%

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

   2. Taylor expanded in x around 0 76.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 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 46.5%

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

2. Taylor expanded in x around 0 47.8%

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

3. Final simplification 47.8%

   \[\leadsto -1 \]

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 2023279 
(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))))