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

Percentage Accurate: 50.2% → 81.1%

Time: 5.9s

Alternatives: 7

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-306}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 4.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-306)
     (+ 1.0 (* -4.0 (* (/ y x) (/ y x))))
     (if (<= t_0 4.2e+189)
       (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_0 (* x x)))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-306) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 4.2e+189) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-306)
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 4.2e+189)
		tmp = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-306], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.2e+189], N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $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) < 2.00000000000000006e-306

   1. Initial program 56.1%

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

   2. Taylor expanded in x around inf 55.4%

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

      1. unpow2 55.4%

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

   4. Simplified 55.4%

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

   5. Taylor expanded in x around inf 80.0%

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

      1. unpow2 80.0%

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

      2. unpow2 80.0%

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

      3. times-frac 91.0%

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

   7. Simplified 91.0%

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

   if 2.00000000000000006e-306 < (*.f64 (*.f64 y 4) y) < 4.19999999999999985e189

   1. Initial program 81.0%

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

      1. add-sqr-sqrt 81.0%

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

      2. difference-of-squares 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-*r* 81.0%

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

      5. sqrt-prod 81.0%

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

      6. sqrt-unprod 47.5%

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

      7. add-sqr-sqrt 62.2%

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

      8. metadata-eval 62.2%

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

      9. *-commutative 62.2%

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

      10. associate-*r* 62.2%

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

      11. sqrt-prod 62.2%

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

      12. sqrt-unprod 47.5%

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

      13. add-sqr-sqrt 81.0%

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

      14. metadata-eval 81.0%

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

   3. Applied egg-rr 81.0%

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

   if 4.19999999999999985e189 < (*.f64 (*.f64 y 4) y)

   1. Initial program 25.5%

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

   2. Taylor expanded in x around 0 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 89.9%

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

      5. metadata-eval 89.9%

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

   4. Simplified 89.9%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-306}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 4.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 2: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 10^{-147} \lor \neg \left(t_0 \leq 10^{-56}\right) \land \left(t_0 \leq 2 \cdot 10^{+50} \lor \neg \left(t_0 \leq 2 \cdot 10^{+82}\right) \land t_0 \leq 2 \cdot 10^{+134}\right):\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (or (<= t_0 1e-147)
           (and (not (<= t_0 1e-56))
                (or (<= t_0 2e+50)
                    (and (not (<= t_0 2e+82)) (<= t_0 2e+134)))))
     (+ 1.0 (* -4.0 (* (/ y x) (/ y x))))
     (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((t_0 <= 1e-147) || (!(t_0 <= 1e-56) && ((t_0 <= 2e+50) || (!(t_0 <= 2e+82) && (t_0 <= 2e+134))))) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((t_0 <= 1d-147) .or. (.not. (t_0 <= 1d-56)) .and. (t_0 <= 2d+50) .or. (.not. (t_0 <= 2d+82)) .and. (t_0 <= 2d+134)) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) * (y / x)))
    else
        tmp = (-1.0d0) + (((x / y) / (y / x)) / 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((t_0 <= 1e-147) || (!(t_0 <= 1e-56) && ((t_0 <= 2e+50) || (!(t_0 <= 2e+82) && (t_0 <= 2e+134))))) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (t_0 <= 1e-147) or (not (t_0 <= 1e-56) and ((t_0 <= 2e+50) or (not (t_0 <= 2e+82) and (t_0 <= 2e+134)))):
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if ((t_0 <= 1e-147) || (!(t_0 <= 1e-56) && ((t_0 <= 2e+50) || (!(t_0 <= 2e+82) && (t_0 <= 2e+134)))))
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y) / Float64(y / x)) / 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((t_0 <= 1e-147) || (~((t_0 <= 1e-56)) && ((t_0 <= 2e+50) || (~((t_0 <= 2e+82)) && (t_0 <= 2e+134)))))
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 1e-147], And[N[Not[LessEqual[t$95$0, 1e-56]], $MachinePrecision], Or[LessEqual[t$95$0, 2e+50], And[N[Not[LessEqual[t$95$0, 2e+82]], $MachinePrecision], LessEqual[t$95$0, 2e+134]]]]], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y 4) y) < 9.9999999999999997e-148 or 1e-56 < (*.f64 (*.f64 y 4) y) < 2.0000000000000002e50 or 1.9999999999999999e82 < (*.f64 (*.f64 y 4) y) < 1.99999999999999984e134

   1. Initial program 68.3%

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

   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 75.9%

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

      1. unpow2 75.9%

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

      2. unpow2 75.9%

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

      3. times-frac 81.0%

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

   7. Simplified 81.0%

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

   if 9.9999999999999997e-148 < (*.f64 (*.f64 y 4) y) < 1e-56 or 2.0000000000000002e50 < (*.f64 (*.f64 y 4) y) < 1.9999999999999999e82 or 1.99999999999999984e134 < (*.f64 (*.f64 y 4) y)

   1. Initial program 42.8%

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

   2. Taylor expanded in x around 0 38.1%

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

      1. unpow2 38.1%

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

      2. *-commutative 38.1%

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

      3. associate-*r* 38.1%

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

   4. Simplified 38.1%

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

      1. div-sub 38.1%

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

      2. associate-*r* 38.1%

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

      3. associate-/r* 38.1%

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

      4. frac-times 38.2%

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

      5. pow2 38.2%

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

      6. *-commutative 38.2%

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

      7. *-inverses 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. unpow2 83.2%

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

      2. clear-num 83.2%

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

      3. un-div-inv 83.2%

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

   8. Applied egg-rr 83.2%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-147} \lor \neg \left(y \cdot \left(y \cdot 4\right) \leq 10^{-56}\right) \land \left(y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+50} \lor \neg \left(y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+82}\right) \land y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+134}\right):\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \]

Alternative 3: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-306}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 4.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-306)
     (+ 1.0 (* -4.0 (* (/ y x) (/ y x))))
     (if (<= t_0 4.2e+189)
       (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_0 (* x x)))
       (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-306) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 4.2e+189) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 2d-306) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) * (y / x)))
    else if (t_0 <= 4.2d+189) then
        tmp = ((x + (y * 2.0d0)) * (x - (y * 2.0d0))) / (t_0 + (x * x))
    else
        tmp = (-1.0d0) + (((x / y) / (y / x)) / 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-306) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 4.2e+189) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 2e-306:
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)))
	elif t_0 <= 4.2e+189:
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x))
	else:
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-306)
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 4.2e+189)
		tmp = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y) / Float64(y / x)) / 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 2e-306)
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	elseif (t_0 <= 4.2e+189)
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	else
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-306], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.2e+189], N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2.00000000000000006e-306

   1. Initial program 56.1%

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

   2. Taylor expanded in x around inf 55.4%

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

      1. unpow2 55.4%

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

   4. Simplified 55.4%

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

   5. Taylor expanded in x around inf 80.0%

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

      1. unpow2 80.0%

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

      2. unpow2 80.0%

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

      3. times-frac 91.0%

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

   7. Simplified 91.0%

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

   if 2.00000000000000006e-306 < (*.f64 (*.f64 y 4) y) < 4.19999999999999985e189

   1. Initial program 81.0%

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

      1. add-sqr-sqrt 81.0%

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

      2. difference-of-squares 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-*r* 81.0%

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

      5. sqrt-prod 81.0%

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

      6. sqrt-unprod 47.5%

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

      7. add-sqr-sqrt 62.2%

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

      8. metadata-eval 62.2%

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

      9. *-commutative 62.2%

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

      10. associate-*r* 62.2%

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

      11. sqrt-prod 62.2%

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

      12. sqrt-unprod 47.5%

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

      13. add-sqr-sqrt 81.0%

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

      14. metadata-eval 81.0%

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

   3. Applied egg-rr 81.0%

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

   if 4.19999999999999985e189 < (*.f64 (*.f64 y 4) y)

   1. Initial program 25.5%

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

   2. Taylor expanded in x around 0 25.6%

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

      1. unpow2 25.6%

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

      2. *-commutative 25.6%

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

      3. associate-*r* 25.6%

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

   4. Simplified 25.6%

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

      1. div-sub 25.6%

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

      2. associate-*r* 25.6%

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

      3. associate-/r* 25.6%

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

      4. frac-times 25.7%

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

      5. pow2 25.7%

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

      6. *-commutative 25.7%

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

      7. *-inverses 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. unpow2 89.4%

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

      2. clear-num 89.4%

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

      3. un-div-inv 89.4%

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

   8. Applied egg-rr 89.4%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-306}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 4.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \]

Alternative 4: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-306}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 4.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-306)
     (+ 1.0 (* -4.0 (* (/ y x) (/ y x))))
     (if (<= t_0 4.2e+189)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-306) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 4.2e+189) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 2d-306) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) * (y / x)))
    else if (t_0 <= 4.2d+189) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (-1.0d0) + (((x / y) / (y / x)) / 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-306) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 4.2e+189) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 2e-306:
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)))
	elif t_0 <= 4.2e+189:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-306)
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 4.2e+189)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y) / Float64(y / x)) / 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 2e-306)
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	elseif (t_0 <= 4.2e+189)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-306], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.2e+189], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2.00000000000000006e-306

   1. Initial program 56.1%

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

   2. Taylor expanded in x around inf 55.4%

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

      1. unpow2 55.4%

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

   4. Simplified 55.4%

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

   5. Taylor expanded in x around inf 80.0%

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

      1. unpow2 80.0%

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

      2. unpow2 80.0%

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

      3. times-frac 91.0%

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

   7. Simplified 91.0%

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

   if 2.00000000000000006e-306 < (*.f64 (*.f64 y 4) y) < 4.19999999999999985e189

   1. Initial program 81.0%

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

   if 4.19999999999999985e189 < (*.f64 (*.f64 y 4) y)

   1. Initial program 25.5%

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

   2. Taylor expanded in x around 0 25.6%

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

      1. unpow2 25.6%

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

      2. *-commutative 25.6%

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

      3. associate-*r* 25.6%

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

   4. Simplified 25.6%

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

      1. div-sub 25.6%

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

      2. associate-*r* 25.6%

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

      3. associate-/r* 25.6%

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

      4. frac-times 25.7%

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

      5. pow2 25.7%

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

      6. *-commutative 25.7%

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

      7. *-inverses 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. unpow2 89.4%

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

      2. clear-num 89.4%

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

      3. un-div-inv 89.4%

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

   8. Applied egg-rr 89.4%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}}{4} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-306}:\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 4.2 \cdot 10^{+189}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \]

Alternative 5: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.3 \cdot 10^{-52} \lor \neg \left(y \leq 4.7 \cdot 10^{-29}\right) \land \left(y \leq 5.8 \cdot 10^{+24} \lor \neg \left(y \leq 3 \cdot 10^{+41}\right) \land y \leq 1.1 \cdot 10^{+78}\right):\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 6.3e-52)
         (and (not (<= y 4.7e-29))
              (or (<= y 5.8e+24) (and (not (<= y 3e+41)) (<= y 1.1e+78)))))
   (+ 1.0 (* -4.0 (* (/ y x) (/ y x))))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 6.3e-52) || (!(y <= 4.7e-29) && ((y <= 5.8e+24) || (!(y <= 3e+41) && (y <= 1.1e+78))))) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 6.3d-52) .or. (.not. (y <= 4.7d-29)) .and. (y <= 5.8d+24) .or. (.not. (y <= 3d+41)) .and. (y <= 1.1d+78)) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) * (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 6.3e-52) || (!(y <= 4.7e-29) && ((y <= 5.8e+24) || (!(y <= 3e+41) && (y <= 1.1e+78))))) {
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 6.3e-52) or (not (y <= 4.7e-29) and ((y <= 5.8e+24) or (not (y <= 3e+41) and (y <= 1.1e+78)))):
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 6.3e-52) || (!(y <= 4.7e-29) && ((y <= 5.8e+24) || (!(y <= 3e+41) && (y <= 1.1e+78)))))
		tmp = Float64(1.0 + Float64(-4.0 * 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 ((y <= 6.3e-52) || (~((y <= 4.7e-29)) && ((y <= 5.8e+24) || (~((y <= 3e+41)) && (y <= 1.1e+78)))))
		tmp = 1.0 + (-4.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 6.3e-52], And[N[Not[LessEqual[y, 4.7e-29]], $MachinePrecision], Or[LessEqual[y, 5.8e+24], And[N[Not[LessEqual[y, 3e+41]], $MachinePrecision], LessEqual[y, 1.1e+78]]]]], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 6.3000000000000003e-52 or 4.6999999999999998e-29 < y < 5.79999999999999958e24 or 2.9999999999999998e41 < y < 1.10000000000000007e78

   1. Initial program 58.5%

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

   2. Taylor expanded in x around inf 36.3%

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

      1. unpow2 36.3%

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

   4. Simplified 36.3%

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

   5. Taylor expanded in x around inf 55.0%

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

      1. unpow2 55.0%

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

      2. unpow2 55.0%

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

      3. times-frac 59.4%

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

   7. Simplified 59.4%

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

   if 6.3000000000000003e-52 < y < 4.6999999999999998e-29 or 5.79999999999999958e24 < y < 2.9999999999999998e41 or 1.10000000000000007e78 < y

   1. Initial program 42.8%

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

   2. Taylor expanded in x around 0 88.7%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.3 \cdot 10^{-52} \lor \neg \left(y \leq 4.7 \cdot 10^{-29}\right) \land \left(y \leq 5.8 \cdot 10^{+24} \lor \neg \left(y \leq 3 \cdot 10^{+41}\right) \land y \leq 1.1 \cdot 10^{+78}\right):\\ \;\;\;\;1 + -4 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 61.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-54}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+70}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e-54)
   1.0
   (if (<= y 5e-29)
     -1.0
     (if (<= y 5.6e+24)
       1.0
       (if (<= y 6.2e+41) -1.0 (if (<= y 5e+70) 1.0 -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-54) {
		tmp = 1.0;
	} else if (y <= 5e-29) {
		tmp = -1.0;
	} else if (y <= 5.6e+24) {
		tmp = 1.0;
	} else if (y <= 6.2e+41) {
		tmp = -1.0;
	} else if (y <= 5e+70) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d-54) then
        tmp = 1.0d0
    else if (y <= 5d-29) then
        tmp = -1.0d0
    else if (y <= 5.6d+24) then
        tmp = 1.0d0
    else if (y <= 6.2d+41) then
        tmp = -1.0d0
    else if (y <= 5d+70) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e-54) {
		tmp = 1.0;
	} else if (y <= 5e-29) {
		tmp = -1.0;
	} else if (y <= 5.6e+24) {
		tmp = 1.0;
	} else if (y <= 6.2e+41) {
		tmp = -1.0;
	} else if (y <= 5e+70) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-54:
		tmp = 1.0
	elif y <= 5e-29:
		tmp = -1.0
	elif y <= 5.6e+24:
		tmp = 1.0
	elif y <= 6.2e+41:
		tmp = -1.0
	elif y <= 5e+70:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-54)
		tmp = 1.0;
	elseif (y <= 5e-29)
		tmp = -1.0;
	elseif (y <= 5.6e+24)
		tmp = 1.0;
	elseif (y <= 6.2e+41)
		tmp = -1.0;
	elseif (y <= 5e+70)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e-54)
		tmp = 1.0;
	elseif (y <= 5e-29)
		tmp = -1.0;
	elseif (y <= 5.6e+24)
		tmp = 1.0;
	elseif (y <= 6.2e+41)
		tmp = -1.0;
	elseif (y <= 5e+70)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-54], 1.0, If[LessEqual[y, 5e-29], -1.0, If[LessEqual[y, 5.6e+24], 1.0, If[LessEqual[y, 6.2e+41], -1.0, If[LessEqual[y, 5e+70], 1.0, -1.0]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1e-54 or 4.99999999999999986e-29 < y < 5.6000000000000003e24 or 6.2e41 < y < 5.0000000000000002e70

   1. Initial program 58.3%

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

   2. Taylor expanded in x around inf 58.2%

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

   if 1e-54 < y < 4.99999999999999986e-29 or 5.6000000000000003e24 < y < 6.2e41 or 5.0000000000000002e70 < y

   1. Initial program 43.8%

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

   2. Taylor expanded in x around 0 87.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-54}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+70}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 50.1% 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 55.1%

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

2. Taylor expanded in x around 0 52.8%

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

3. Final simplification 52.8%

   \[\leadsto -1 \]

Developer target: 50.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t_0\\ t_2 := \frac{t_0}{t_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t_3}{x \cdot x + t_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t_1} - t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t_1}}\right)}^{2} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2023207 
(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))))