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

Percentage Accurate: 50.6% → 80.8%

Time: 8.5s

Alternatives: 8

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: 50.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{log1p}\left(\frac{0.5}{{\left(\frac{x}{y}\right)}^{2}}\right), -8, 1\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= y 1.4e-97)
     (fma (* 2.0 (log1p (/ 0.5 (pow (/ x y) 2.0)))) -8.0 1.0)
     (if (<= y 1.75e+112)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (y <= 1.4e-97) {
		tmp = fma((2.0 * log1p((0.5 / pow((x / y), 2.0)))), -8.0, 1.0);
	} else if (y <= 1.75e+112) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (y <= 1.4e-97)
		tmp = fma(Float64(2.0 * log1p(Float64(0.5 / (Float64(x / y) ^ 2.0)))), -8.0, 1.0);
	elseif (y <= 1.75e+112)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.4e-97], N[(N[(2.0 * N[Log[1 + N[(0.5 / N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -8.0 + 1.0), $MachinePrecision], If[LessEqual[y, 1.75e+112], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.4000000000000001e-97

   1. Initial program 53.5%

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

   2. Taylor expanded in x around inf 54.1%

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

      1. associate--l+ 54.1%

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

      2. distribute-rgt-out-- 54.1%

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

      3. metadata-eval 54.1%

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

      4. *-commutative 54.1%

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

      5. +-commutative 54.1%

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

      6. *-commutative 54.1%

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

      7. fma-def 54.1%

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

      8. unpow2 54.1%

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

      9. unpow2 54.1%

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

      10. times-frac 58.4%

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

   4. Simplified 58.4%

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

      1. add-log-exp 57.9%

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

      2. add-sqr-sqrt 57.9%

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

      3. log-prod 57.9%

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

      4. pow2 57.9%

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

      5. pow2 57.9%

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

   6. Applied egg-rr 57.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right) + \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)}, -8, 1\right) \]
   7. Step-by-step derivation

      1. count-2 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in y around 0 54.5%

      \[\leadsto \mathsf{fma}\left(2 \cdot \log \color{blue}{\left(1 + 0.5 \cdot \frac{{y}^{2}}{{x}^{2}}\right)}, -8, 1\right) \]
   10. Step-by-step derivation

       1. unpow2 54.5%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + 0.5 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}}\right), -8, 1\right) \]

       2. unpow2 54.5%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + 0.5 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}}\right), -8, 1\right) \]

       3. times-frac 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + 0.5 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}\right), -8, 1\right) \]

       4. unpow2 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + 0.5 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}}\right), -8, 1\right) \]

   11. Simplified 60.0%

       \[\leadsto \mathsf{fma}\left(2 \cdot \log \color{blue}{\left(1 + 0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)}, -8, 1\right) \]
   12. Step-by-step derivation

       1. *-un-lft-identity 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \color{blue}{\left(1 \cdot \left(1 + 0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)\right)}, -8, 1\right) \]

       2. log-prod 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\left(\log 1 + \log \left(1 + 0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)\right)}, -8, 1\right) \]

       3. metadata-eval 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \left(\color{blue}{0} + \log \left(1 + 0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)\right), -8, 1\right) \]

       4. log1p-def 60.1%

          \[\leadsto \mathsf{fma}\left(2 \cdot \left(0 + \color{blue}{\mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)}\right), -8, 1\right) \]

       5. unpow2 60.1%

          \[\leadsto \mathsf{fma}\left(2 \cdot \left(0 + \mathsf{log1p}\left(0.5 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}\right)\right), -8, 1\right) \]

       6. clear-num 60.1%

          \[\leadsto \mathsf{fma}\left(2 \cdot \left(0 + \mathsf{log1p}\left(0.5 \cdot \left(\color{blue}{\frac{1}{\frac{x}{y}}} \cdot \frac{y}{x}\right)\right)\right), -8, 1\right) \]

       7. associate-/r/ 60.1%

          \[\leadsto \mathsf{fma}\left(2 \cdot \left(0 + \mathsf{log1p}\left(0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{x}{y}}{\frac{y}{x}}}}\right)\right), -8, 1\right) \]

       8. un-div-inv 60.1%

          \[\leadsto \mathsf{fma}\left(2 \cdot \left(0 + \mathsf{log1p}\left(\color{blue}{\frac{0.5}{\frac{\frac{x}{y}}{\frac{y}{x}}}}\right)\right), -8, 1\right) \]

       9. div-inv 60.1%

          \[\leadsto \mathsf{fma}\left(2 \cdot \left(0 + \mathsf{log1p}\left(\frac{0.5}{\color{blue}{\frac{x}{y} \cdot \frac{1}{\frac{y}{x}}}}\right)\right), -8, 1\right) \]

       10. clear-num 60.1%

           \[\leadsto \mathsf{fma}\left(2 \cdot \left(0 + \mathsf{log1p}\left(\frac{0.5}{\frac{x}{y} \cdot \color{blue}{\frac{x}{y}}}\right)\right), -8, 1\right) \]

       11. pow2 60.1%

           \[\leadsto \mathsf{fma}\left(2 \cdot \left(0 + \mathsf{log1p}\left(\frac{0.5}{\color{blue}{{\left(\frac{x}{y}\right)}^{2}}}\right)\right), -8, 1\right) \]

   13. Applied egg-rr 60.1%

       \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\left(0 + \mathsf{log1p}\left(\frac{0.5}{{\left(\frac{x}{y}\right)}^{2}}\right)\right)}, -8, 1\right) \]
   14. Step-by-step derivation

       1. +-lft-identity 60.1%

          \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\mathsf{log1p}\left(\frac{0.5}{{\left(\frac{x}{y}\right)}^{2}}\right)}, -8, 1\right) \]

   15. Simplified 60.1%

       \[\leadsto \mathsf{fma}\left(2 \cdot \color{blue}{\mathsf{log1p}\left(\frac{0.5}{{\left(\frac{x}{y}\right)}^{2}}\right)}, -8, 1\right) \]

   if 1.4000000000000001e-97 < y < 1.74999999999999998e112

   1. Initial program 76.3%

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

   if 1.74999999999999998e112 < y

   1. Initial program 15.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. unpow2 65.5%

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

      4. times-frac 73.5%

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

      5. metadata-eval 73.5%

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

   4. Simplified 73.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \mathsf{log1p}\left(\frac{0.5}{{\left(\frac{x}{y}\right)}^{2}}\right), -8, 1\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 2: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;y \leq 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \log \left(1 + \frac{0.5 \cdot \frac{y}{x}}{\frac{x}{y}}\right), -8, 1\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= y 1e-97)
     (fma (* 2.0 (log (+ 1.0 (/ (* 0.5 (/ y x)) (/ x y))))) -8.0 1.0)
     (if (<= y 1.6e+112)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (y <= 1e-97) {
		tmp = fma((2.0 * log((1.0 + ((0.5 * (y / x)) / (x / y))))), -8.0, 1.0);
	} else if (y <= 1.6e+112) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (y <= 1e-97)
		tmp = fma(Float64(2.0 * log(Float64(1.0 + Float64(Float64(0.5 * Float64(y / x)) / Float64(x / y))))), -8.0, 1.0);
	elseif (y <= 1.6e+112)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1e-97], N[(N[(2.0 * N[Log[N[(1.0 + N[(N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -8.0 + 1.0), $MachinePrecision], If[LessEqual[y, 1.6e+112], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.00000000000000004e-97

   1. Initial program 53.5%

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

   2. Taylor expanded in x around inf 54.1%

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

      1. associate--l+ 54.1%

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

      2. distribute-rgt-out-- 54.1%

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

      3. metadata-eval 54.1%

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

      4. *-commutative 54.1%

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

      5. +-commutative 54.1%

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

      6. *-commutative 54.1%

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

      7. fma-def 54.1%

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

      8. unpow2 54.1%

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

      9. unpow2 54.1%

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

      10. times-frac 58.4%

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

   4. Simplified 58.4%

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

      1. add-log-exp 57.9%

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

      2. add-sqr-sqrt 57.9%

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

      3. log-prod 57.9%

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

      4. pow2 57.9%

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

      5. pow2 57.9%

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

   6. Applied egg-rr 57.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right) + \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)}, -8, 1\right) \]
   7. Step-by-step derivation

      1. count-2 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in y around 0 54.5%

      \[\leadsto \mathsf{fma}\left(2 \cdot \log \color{blue}{\left(1 + 0.5 \cdot \frac{{y}^{2}}{{x}^{2}}\right)}, -8, 1\right) \]
   10. Step-by-step derivation

       1. unpow2 54.5%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + 0.5 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}}\right), -8, 1\right) \]

       2. unpow2 54.5%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + 0.5 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}}\right), -8, 1\right) \]

       3. times-frac 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + 0.5 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}\right), -8, 1\right) \]

       4. unpow2 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + 0.5 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}}\right), -8, 1\right) \]

   11. Simplified 60.0%

       \[\leadsto \mathsf{fma}\left(2 \cdot \log \color{blue}{\left(1 + 0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)}, -8, 1\right) \]
   12. Step-by-step derivation

       1. *-commutative 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + \color{blue}{{\left(\frac{y}{x}\right)}^{2} \cdot 0.5}\right), -8, 1\right) \]

       2. unpow2 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot 0.5\right), -8, 1\right) \]

       3. clear-num 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \cdot 0.5\right), -8, 1\right) \]

       4. div-inv 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \cdot 0.5\right), -8, 1\right) \]

       5. associate-*l/ 60.0%

          \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + \color{blue}{\frac{\frac{y}{x} \cdot 0.5}{\frac{x}{y}}}\right), -8, 1\right) \]

   13. Applied egg-rr 60.0%

       \[\leadsto \mathsf{fma}\left(2 \cdot \log \left(1 + \color{blue}{\frac{\frac{y}{x} \cdot 0.5}{\frac{x}{y}}}\right), -8, 1\right) \]

   if 1.00000000000000004e-97 < y < 1.59999999999999993e112

   1. Initial program 76.3%

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

   if 1.59999999999999993e112 < y

   1. Initial program 15.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. unpow2 65.5%

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

      4. times-frac 73.5%

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

      5. metadata-eval 73.5%

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

   4. Simplified 73.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(2 \cdot \log \left(1 + \frac{0.5 \cdot \frac{y}{x}}{\frac{x}{y}}\right), -8, 1\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 3: 80.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-194}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+216}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 5e-194)
     (fma (* (/ y x) (/ y x)) -8.0 1.0)
     (if (<= t_0 5e+216)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 5e-194) {
		tmp = fma(((y / x) * (y / x)), -8.0, 1.0);
	} else if (t_0 <= 5e+216) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 5e-194)
		tmp = fma(Float64(Float64(y / x) * Float64(y / x)), -8.0, 1.0);
	elseif (t_0 <= 5e+216)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-194], N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0 + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+216], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 5.0000000000000002e-194

   1. Initial program 59.2%

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

   2. Taylor expanded in x around inf 80.7%

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

      1. associate--l+ 80.7%

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

      2. distribute-rgt-out-- 80.7%

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

      3. metadata-eval 80.7%

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

      4. *-commutative 80.7%

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

      5. +-commutative 80.7%

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

      6. *-commutative 80.7%

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

      7. fma-def 80.7%

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

      8. unpow2 80.7%

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

      9. unpow2 80.7%

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

      10. times-frac 86.9%

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

   4. Simplified 86.9%

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

   if 5.0000000000000002e-194 < (*.f64 (*.f64 y 4) y) < 4.9999999999999998e216

   1. Initial program 79.2%

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

   if 4.9999999999999998e216 < (*.f64 (*.f64 y 4) y)

   1. Initial program 15.1%

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

   2. Taylor expanded in x around 0 75.7%

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

      1. fma-neg 75.7%

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

      2. unpow2 75.7%

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

      3. unpow2 75.7%

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

      4. times-frac 84.7%

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

      5. metadata-eval 84.7%

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

   4. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{-194}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+216}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 4: 80.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;y \leq 1.6 \cdot 10^{-94}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= y 1.6e-94)
     (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))
     (if (<= y 5.2e+112)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (y <= 1.6e-94) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else if (y <= 5.2e+112) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (y <= 1.6e-94)
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (y <= 5.2e+112)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.6e-94], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+112], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.59999999999999998e-94

   1. Initial program 53.5%

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

   2. Taylor expanded in x around inf 36.4%

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

      1. unpow2 36.4%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in x around inf 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. times-frac 58.2%

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

      4. unpow2 58.2%

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

   7. Simplified 58.2%

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

      1. unpow2 58.2%

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

      2. clear-num 58.2%

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

      3. un-div-inv 58.2%

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

   9. Applied egg-rr 58.2%

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

   if 1.59999999999999998e-94 < y < 5.2000000000000001e112

   1. Initial program 76.3%

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

   if 5.2000000000000001e112 < y

   1. Initial program 15.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. unpow2 65.5%

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

      4. times-frac 73.5%

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

      5. metadata-eval 73.5%

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

   4. Simplified 73.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-94}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 5: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;y \leq 9.5 \cdot 10^{-98}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= y 9.5e-98)
     (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))
     (if (<= y 1.15e+111) (/ (- (* x x) t_0) (+ (* x x) t_0)) -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (y <= 9.5e-98) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else if (y <= 1.15e+111) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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 (y <= 9.5d-98) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    else if (y <= 1.15d+111) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (y <= 9.5e-98) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else if (y <= 1.15e+111) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if y <= 9.5e-98:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	elif y <= 1.15e+111:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (y <= 9.5e-98)
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (y <= 1.15e+111)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (y <= 9.5e-98)
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	elseif (y <= 1.15e+111)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9.5e-98], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+111], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.5000000000000001e-98

   1. Initial program 53.5%

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

   2. Taylor expanded in x around inf 36.4%

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

      1. unpow2 36.4%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in x around inf 53.9%

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

      1. unpow2 53.9%

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

      2. unpow2 53.9%

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

      3. times-frac 58.2%

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

      4. unpow2 58.2%

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

   7. Simplified 58.2%

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

      1. unpow2 58.2%

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

      2. clear-num 58.2%

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

      3. un-div-inv 58.2%

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

   9. Applied egg-rr 58.2%

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

   if 9.5000000000000001e-98 < y < 1.15000000000000001e111

   1. Initial program 76.3%

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

   if 1.15000000000000001e111 < y

   1. Initial program 15.2%

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

   2. Taylor expanded in x around 0 72.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-98}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 74.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-21}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 1e-21) (+ 1.0 (* -4.0 (/ (/ y x) (/ x y)))) -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 1e-21) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (y * 4.0d0)) <= 1d-21) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 1e-21) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if (y * (y * 4.0)) <= 1e-21:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(y * 4.0)) <= 1e-21)
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (y * 4.0)) <= 1e-21)
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 1e-21], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y 4) y) < 9.99999999999999908e-22

   1. Initial program 64.3%

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

   2. Taylor expanded in x around inf 52.9%

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

      1. unpow2 52.9%

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

   4. Simplified 52.9%

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

   5. Taylor expanded in x around inf 76.2%

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

      1. unpow2 76.2%

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

      2. unpow2 76.2%

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

      3. times-frac 80.9%

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

      4. unpow2 80.9%

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

   7. Simplified 80.9%

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

      1. unpow2 80.9%

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

      2. clear-num 80.9%

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

      3. un-div-inv 80.9%

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

   9. Applied egg-rr 80.9%

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

   if 9.99999999999999908e-22 < (*.f64 (*.f64 y 4) y)

   1. Initial program 35.4%

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

   2. Taylor expanded in x around 0 79.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-21}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 73.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 2e-11) 1.0 -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2e-11) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d-11) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-11) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2e-11:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2e-11)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-11)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 2e-11], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999988e-11

   1. Initial program 56.3%

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

   2. Taylor expanded in x around inf 57.5%

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

   if 1.99999999999999988e-11 < y

   1. Initial program 31.8%

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

   2. Taylor expanded in x around 0 68.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 50.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ -1 \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

y = abs(y);
double code(double x, double y) {
	return -1.0;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	return -1.0;
}

Python

y = abs(y)
def code(x, y):
	return -1.0

Julia

y = abs(y)
function code(x, y)
	return -1.0
end

MATLAB

y = abs(y)
function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 50.0%

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

2. Taylor expanded in x around 0 49.8%

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

3. Final simplification 49.8%

   \[\leadsto -1 \]

Developer target: 51.2% 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 2023229 
(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))))