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

Percentage Accurate: 50.3% → 79.3%

Time: 10.9s

Alternatives: 6

Speedup: 19.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-95}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+182}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-95)
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
   (if (<= (* x x) 5e+182)
     (+
      (exp
       (log1p
        (/ (fma y (* y -4.0) (pow x 2.0)) (fma 4.0 (pow y 2.0) (pow x 2.0)))))
      -1.0)
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-95) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 5e+182) {
		tmp = exp(log1p((fma(y, (y * -4.0), pow(x, 2.0)) / fma(4.0, pow(y, 2.0), pow(x, 2.0))))) + -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-95)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	elseif (Float64(x * x) <= 5e+182)
		tmp = Float64(exp(log1p(Float64(fma(y, Float64(y * -4.0), (x ^ 2.0)) / fma(4.0, (y ^ 2.0), (x ^ 2.0))))) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-95], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+182], N[(N[Exp[N[Log[1 + N[(N[(y * N[(y * -4.0), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[Power[y, 2.0], $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999989e-96

   1. Initial program 57.4%

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

   3. Taylor expanded in x around 0 75.4%

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

      1. unpow2 75.4%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 75.4%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} - 1 \]

      3. times-frac 85.2%

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

   5. Applied egg-rr 85.2%

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

   if 9.99999999999999989e-96 < (*.f64 x x) < 4.99999999999999973e182

   1. Initial program 78.4%

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

      1. expm1-log1p-u 78.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right)} \]

      2. expm1-undefine 78.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)} - 1} \]

   4. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)} - 1} \]

   if 4.99999999999999973e182 < (*.f64 x x)

   1. Initial program 25.3%

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

   3. Taylor expanded in y around 0 83.2%

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

      1. unpow2 83.2%

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

      2. unpow2 83.2%

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

      3. times-frac 88.7%

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

   5. Applied egg-rr 88.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-95}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+182}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}{\mathsf{fma}\left(4, {y}^{2}, {x}^{2}\right)}\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-95}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-95)
     (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
     (if (<= (* x x) 5e+182)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-95) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 5e+182) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} 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) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 1d-95) then
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    else if ((x * x) <= 5d+182) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    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 tmp;
	if ((x * x) <= 1e-95) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 5e+182) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-95:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	elif (x * x) <= 5e+182:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	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))
	tmp = 0.0
	if (Float64(x * x) <= 1e-95)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	elseif (Float64(x * x) <= 5e+182)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 1e-95)
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	elseif ((x * x) <= 5e+182)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	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]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-95], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+182], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999989e-96

   1. Initial program 57.4%

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

   3. Taylor expanded in x around 0 75.4%

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

      1. unpow2 75.4%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 75.4%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} - 1 \]

      3. times-frac 85.2%

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

   5. Applied egg-rr 85.2%

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

   if 9.99999999999999989e-96 < (*.f64 x x) < 4.99999999999999973e182

   1. Initial program 78.4%

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

   if 4.99999999999999973e182 < (*.f64 x x)

   1. Initial program 25.3%

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

   3. Taylor expanded in y around 0 83.2%

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

      1. unpow2 83.2%

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

      2. unpow2 83.2%

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

      3. times-frac 88.7%

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

   5. Applied egg-rr 88.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-95}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-49}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-21} \lor \neg \left(x \leq 1.8 \cdot 10^{+16}\right):\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.4e-49)
   -1.0
   (if (or (<= x 7.5e-21) (not (<= x 1.8e+16)))
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.4e-49) {
		tmp = -1.0;
	} else if ((x <= 7.5e-21) || !(x <= 1.8e+16)) {
		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 <= 2.4d-49) then
        tmp = -1.0d0
    else if ((x <= 7.5d-21) .or. (.not. (x <= 1.8d+16))) 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 <= 2.4e-49) {
		tmp = -1.0;
	} else if ((x <= 7.5e-21) || !(x <= 1.8e+16)) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.4e-49:
		tmp = -1.0
	elif (x <= 7.5e-21) or not (x <= 1.8e+16):
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.4e-49)
		tmp = -1.0;
	elseif ((x <= 7.5e-21) || !(x <= 1.8e+16))
		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 <= 2.4e-49)
		tmp = -1.0;
	elseif ((x <= 7.5e-21) || ~((x <= 1.8e+16)))
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.4e-49], -1.0, If[Or[LessEqual[x, 7.5e-21], N[Not[LessEqual[x, 1.8e+16]], $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 x < 2.39999999999999992e-49 or 7.50000000000000072e-21 < x < 1.8e16

   1. Initial program 53.8%

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

   3. Taylor expanded in x around 0 61.4%

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

   if 2.39999999999999992e-49 < x < 7.50000000000000072e-21 or 1.8e16 < x

   1. Initial program 48.6%

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

   3. Taylor expanded in y around 0 72.0%

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

      1. unpow2 72.0%

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

      2. unpow2 72.0%

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

      3. times-frac 75.3%

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

   5. Applied egg-rr 75.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-49}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-21} \lor \neg \left(x \leq 1.8 \cdot 10^{+16}\right):\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-48} \lor \neg \left(x \leq 9.5 \cdot 10^{-21}\right) \land x \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 1.5e-48) (and (not (<= x 9.5e-21)) (<= x 6.5e+15)))
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 1.5e-48) || (!(x <= 9.5e-21) && (x <= 6.5e+15))) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} 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) :: tmp
    if ((x <= 1.5d-48) .or. (.not. (x <= 9.5d-21)) .and. (x <= 6.5d+15)) then
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    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 tmp;
	if ((x <= 1.5e-48) || (!(x <= 9.5e-21) && (x <= 6.5e+15))) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 1.5e-48) or (not (x <= 9.5e-21) and (x <= 6.5e+15)):
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 1.5e-48) || (!(x <= 9.5e-21) && (x <= 6.5e+15)))
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 1.5e-48) || (~((x <= 9.5e-21)) && (x <= 6.5e+15)))
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 1.5e-48], And[N[Not[LessEqual[x, 9.5e-21]], $MachinePrecision], LessEqual[x, 6.5e+15]]], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e-48 or 9.4999999999999994e-21 < x < 6.5e15

   1. Initial program 53.8%

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

   3. Taylor expanded in x around 0 55.6%

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

      1. unpow2 55.6%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 55.6%

         \[\leadsto 0.5 \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}} - 1 \]

      3. times-frac 62.6%

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

   5. Applied egg-rr 62.6%

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

   if 1.5e-48 < x < 9.4999999999999994e-21 or 6.5e15 < x

   1. Initial program 48.6%

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

   3. Taylor expanded in y around 0 72.0%

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

      1. unpow2 72.0%

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

      2. unpow2 72.0%

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

      3. times-frac 75.3%

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

   5. Applied egg-rr 75.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-48} \lor \neg \left(x \leq 9.5 \cdot 10^{-21}\right) \land x \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-40}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+18}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.4e-40) -1.0 (if (<= x 5e-22) 1.0 (if (<= x 1e+18) -1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.4e-40) {
		tmp = -1.0;
	} else if (x <= 5e-22) {
		tmp = 1.0;
	} else if (x <= 1e+18) {
		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 <= 3.4d-40) then
        tmp = -1.0d0
    else if (x <= 5d-22) then
        tmp = 1.0d0
    else if (x <= 1d+18) 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 <= 3.4e-40) {
		tmp = -1.0;
	} else if (x <= 5e-22) {
		tmp = 1.0;
	} else if (x <= 1e+18) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.4e-40:
		tmp = -1.0
	elif x <= 5e-22:
		tmp = 1.0
	elif x <= 1e+18:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.4e-40)
		tmp = -1.0;
	elseif (x <= 5e-22)
		tmp = 1.0;
	elseif (x <= 1e+18)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.4e-40)
		tmp = -1.0;
	elseif (x <= 5e-22)
		tmp = 1.0;
	elseif (x <= 1e+18)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.4e-40], -1.0, If[LessEqual[x, 5e-22], 1.0, If[LessEqual[x, 1e+18], -1.0, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999984e-40 or 4.99999999999999954e-22 < x < 1e18

   1. Initial program 53.8%

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

   3. Taylor expanded in x around 0 61.9%

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

   if 3.39999999999999984e-40 < x < 4.99999999999999954e-22 or 1e18 < x

   1. Initial program 48.6%

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

   3. Taylor expanded in x around inf 75.9%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-40}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-22}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+18}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.8% 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 52.3%

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

3. Taylor expanded in x around 0 51.1%

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

4. Final simplification 51.1%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 50.8% 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 2024039 
(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))))