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

Percentage Accurate: 51.6% → 81.2%

Time: 9.3s

Alternatives: 6

Speedup: 3.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + {\left(\frac{y}{x}\right)}^{2}}\\ t_1 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-279}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+217}:\\ \;\;\;\;\frac{x \cdot x - t\_1}{x \cdot x + t\_1}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \mathsf{fma}\left({t\_0}^{2}, t\_0, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 (pow (/ y x) 2.0)))) (t_1 (* y (* y 4.0))))
   (if (<= (* x x) 5e-279)
     -1.0
     (if (<= (* x x) 1e+217)
       (/ (- (* x x) t_1) (+ (* x x) t_1))
       (+ 1.0 (* -8.0 (fma (pow t_0 2.0) t_0 -1.0)))))))

C

double code(double x, double y) {
	double t_0 = cbrt((1.0 + pow((y / x), 2.0)));
	double t_1 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-279) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+217) {
		tmp = ((x * x) - t_1) / ((x * x) + t_1);
	} else {
		tmp = 1.0 + (-8.0 * fma(pow(t_0, 2.0), t_0, -1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = cbrt(Float64(1.0 + (Float64(y / x) ^ 2.0)))
	t_1 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-279)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+217)
		tmp = Float64(Float64(Float64(x * x) - t_1) / Float64(Float64(x * x) + t_1));
	else
		tmp = Float64(1.0 + Float64(-8.0 * fma((t_0 ^ 2.0), t_0, -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(1.0 + N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-279], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+217], N[(N[(N[(x * x), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.99999999999999969e-279

   1. Initial program 55.9%

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

   3. Taylor expanded in x around 0 92.7%

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

   if 4.99999999999999969e-279 < (*.f64 x x) < 9.9999999999999996e216

   1. Initial program 78.9%

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

   if 9.9999999999999996e216 < (*.f64 x x)

   1. Initial program 14.8%

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

   3. Taylor expanded in y around 0 68.4%

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

      1. unpow2 68.4%

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

      2. pow2 68.4%

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

      3. times-frac 84.6%

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

   5. Applied egg-rr 84.6%

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

      1. pow2 84.6%

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

      2. expm1-log1p-u 84.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)\right)} \]

      3. expm1-define 84.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} - 1\right)} \]

      4. add-cube-cbrt 84.6%

         \[\leadsto 1 + -8 \cdot \left(\color{blue}{\left(\sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}} - 1\right) \]

      5. fma-neg 84.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}, \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}, -1\right)} \]

      6. pow2 84.6%

         \[\leadsto 1 + -8 \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}\right)}^{2}}, \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}, -1\right) \]

      7. log1p-undefine 84.6%

         \[\leadsto 1 + -8 \cdot \mathsf{fma}\left({\left(\sqrt[3]{e^{\color{blue}{\log \left(1 + {\left(\frac{y}{x}\right)}^{2}\right)}}}\right)}^{2}, \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}, -1\right) \]

      8. rem-exp-log 84.6%

         \[\leadsto 1 + -8 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\color{blue}{1 + {\left(\frac{y}{x}\right)}^{2}}}\right)}^{2}, \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}, -1\right) \]

      9. +-commutative 84.6%

         \[\leadsto 1 + -8 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\color{blue}{{\left(\frac{y}{x}\right)}^{2} + 1}}\right)}^{2}, \sqrt[3]{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)}}, -1\right) \]

      10. log1p-undefine 84.6%

          \[\leadsto 1 + -8 \cdot \mathsf{fma}\left({\left(\sqrt[3]{{\left(\frac{y}{x}\right)}^{2} + 1}\right)}^{2}, \sqrt[3]{e^{\color{blue}{\log \left(1 + {\left(\frac{y}{x}\right)}^{2}\right)}}}, -1\right) \]

      11. rem-exp-log 84.6%

          \[\leadsto 1 + -8 \cdot \mathsf{fma}\left({\left(\sqrt[3]{{\left(\frac{y}{x}\right)}^{2} + 1}\right)}^{2}, \sqrt[3]{\color{blue}{1 + {\left(\frac{y}{x}\right)}^{2}}}, -1\right) \]

      12. +-commutative 84.6%

          \[\leadsto 1 + -8 \cdot \mathsf{fma}\left({\left(\sqrt[3]{{\left(\frac{y}{x}\right)}^{2} + 1}\right)}^{2}, \sqrt[3]{\color{blue}{{\left(\frac{y}{x}\right)}^{2} + 1}}, -1\right) \]

      13. metadata-eval 84.6%

          \[\leadsto 1 + -8 \cdot \mathsf{fma}\left({\left(\sqrt[3]{{\left(\frac{y}{x}\right)}^{2} + 1}\right)}^{2}, \sqrt[3]{{\left(\frac{y}{x}\right)}^{2} + 1}, \color{blue}{-1}\right) \]

   7. Applied egg-rr 84.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-279}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+217}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \mathsf{fma}\left({\left(\sqrt[3]{1 + {\left(\frac{y}{x}\right)}^{2}}\right)}^{2}, \sqrt[3]{1 + {\left(\frac{y}{x}\right)}^{2}}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-279}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+217}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-8 \cdot \left(1 + {\left(\frac{y}{x}\right)}^{2}\right) + 8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-279)
     -1.0
     (if (<= (* x x) 1e+217)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (+ (* -8.0 (+ 1.0 (pow (/ y x) 2.0))) 8.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-279) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+217) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + ((-8.0 * (1.0 + pow((y / x), 2.0))) + 8.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 5d-279) then
        tmp = -1.0d0
    else if ((x * x) <= 1d+217) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + (((-8.0d0) * (1.0d0 + ((y / x) ** 2.0d0))) + 8.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-279) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+217) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + ((-8.0 * (1.0 + Math.pow((y / x), 2.0))) + 8.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 5e-279:
		tmp = -1.0
	elif (x * x) <= 1e+217:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + ((-8.0 * (1.0 + math.pow((y / x), 2.0))) + 8.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-279)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+217)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(Float64(-8.0 * Float64(1.0 + (Float64(y / x) ^ 2.0))) + 8.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 5e-279)
		tmp = -1.0;
	elseif ((x * x) <= 1e+217)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + ((-8.0 * (1.0 + ((y / x) ^ 2.0))) + 8.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-279], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+217], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-8.0 * N[(1.0 + N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.99999999999999969e-279

   1. Initial program 55.9%

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

   3. Taylor expanded in x around 0 92.7%

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

   if 4.99999999999999969e-279 < (*.f64 x x) < 9.9999999999999996e216

   1. Initial program 78.9%

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

   if 9.9999999999999996e216 < (*.f64 x x)

   1. Initial program 14.8%

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

   3. Taylor expanded in y around 0 68.4%

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

      1. unpow2 68.4%

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

      2. pow2 68.4%

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

      3. times-frac 84.6%

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

   5. Applied egg-rr 84.6%

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

      1. pow2 84.6%

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

      2. expm1-log1p-u 84.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)\right)} \]

      3. expm1-define 84.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} - 1\right)} \]

      4. sub-neg 84.6%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} + \left(-1\right)\right)} \]

      5. distribute-rgt-in 84.6%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} \cdot -8 + \left(-1\right) \cdot -8\right)} \]

      6. log1p-undefine 84.6%

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

      7. rem-exp-log 84.6%

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

      8. +-commutative 84.6%

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

      9. metadata-eval 84.6%

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

      10. metadata-eval 84.6%

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

   7. Applied egg-rr 84.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-279}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+217}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-8 \cdot \left(1 + {\left(\frac{y}{x}\right)}^{2}\right) + 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-279}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+217}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-279)
     -1.0
     (if (<= (* x x) 1e+217)
       (/ (- (* 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) <= 5e-279) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+217) {
		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) <= 5d-279) then
        tmp = -1.0d0
    else if ((x * x) <= 1d+217) 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) <= 5e-279) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+217) {
		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) <= 5e-279:
		tmp = -1.0
	elif (x * x) <= 1e+217:
		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) <= 5e-279)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+217)
		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) <= 5e-279)
		tmp = -1.0;
	elseif ((x * x) <= 1e+217)
		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], 5e-279], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+217], 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) < 4.99999999999999969e-279

   1. Initial program 55.9%

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

   3. Taylor expanded in x around 0 92.7%

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

   if 4.99999999999999969e-279 < (*.f64 x x) < 9.9999999999999996e216

   1. Initial program 78.9%

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

   if 9.9999999999999996e216 < (*.f64 x x)

   1. Initial program 14.8%

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

   3. Taylor expanded in y around 0 68.4%

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

      1. unpow2 68.4%

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

      2. pow2 68.4%

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

      3. times-frac 84.6%

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

   5. Applied egg-rr 84.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-279}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+217}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.115 \lor \neg \left(x \leq 10^{+49}\right):\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.65e-30)
   -1.0
   (if (or (<= x 0.115) (not (<= x 1e+49)))
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.65e-30) {
		tmp = -1.0;
	} else if ((x <= 0.115) || !(x <= 1e+49)) {
		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 <= 2.65d-30) then
        tmp = -1.0d0
    else if ((x <= 0.115d0) .or. (.not. (x <= 1d+49))) 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 <= 2.65e-30) {
		tmp = -1.0;
	} else if ((x <= 0.115) || !(x <= 1e+49)) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.65e-30:
		tmp = -1.0
	elif (x <= 0.115) or not (x <= 1e+49):
		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 <= 2.65e-30)
		tmp = -1.0;
	elseif ((x <= 0.115) || !(x <= 1e+49))
		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 <= 2.65e-30)
		tmp = -1.0;
	elseif ((x <= 0.115) || ~((x <= 1e+49)))
		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, 2.65e-30], -1.0, If[Or[LessEqual[x, 0.115], N[Not[LessEqual[x, 1e+49]], $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 < 2.64999999999999987e-30 or 0.115000000000000005 < x < 9.99999999999999946e48

   1. Initial program 58.2%

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

   3. Taylor expanded in x around 0 59.6%

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

   if 2.64999999999999987e-30 < x < 0.115000000000000005 or 9.99999999999999946e48 < x

   1. Initial program 27.4%

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

   3. Taylor expanded in y around 0 62.2%

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

      1. unpow2 62.2%

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

      2. pow2 62.2%

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

      3. times-frac 75.6%

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

   5. Applied egg-rr 75.6%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.115 \lor \neg \left(x \leq 10^{+49}\right):\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.9e+45) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.9e+45) {
		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 <= 2.9d+45) 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 <= 2.9e+45) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.9e+45:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.9e+45)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.9e+45)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.9e+45], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.8999999999999997e45

   1. Initial program 58.4%

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

   3. Taylor expanded in x around 0 59.8%

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

   if 2.8999999999999997e45 < x

   1. Initial program 25.4%

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

   3. Taylor expanded in x around inf 76.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.3% 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 50.8%

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

3. Taylor expanded in x around 0 51.3%

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

4. Final simplification 51.3%

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

Developer target: 52.0% 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 2024080 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (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))))