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

Percentage Accurate: 50.3% → 80.1%

Time: 6.6s

Alternatives: 6

Speedup: 19.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ t_1 := y \cdot \left(y \cdot 4\right)\\ t_2 := \frac{x \cdot x - t_1}{t_1 + x \cdot x}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+223}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{x}{y}\right)}^{2} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (* y (/ y x)) (/ 8.0 x))))
        (t_1 (* y (* y 4.0)))
        (t_2 (/ (- (* x x) t_1) (+ t_1 (* x x)))))
   (if (<= t_1 2e-296)
     t_0
     (if (<= t_1 1e-134)
       t_2
       (if (<= t_1 1e-65)
         t_0
         (if (<= t_1 2e+223) t_2 (+ (* 0.5 (pow (/ x y) 2.0)) -1.0)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - ((y * (y / x)) * (8.0 / x));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 2e-296) {
		tmp = t_0;
	} else if (t_1 <= 1e-134) {
		tmp = t_2;
	} else if (t_1 <= 1e-65) {
		tmp = t_0;
	} else if (t_1 <= 2e+223) {
		tmp = t_2;
	} else {
		tmp = (0.5 * pow((x / y), 2.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - ((y * (y / x)) * (8.0d0 / x))
    t_1 = y * (y * 4.0d0)
    t_2 = ((x * x) - t_1) / (t_1 + (x * x))
    if (t_1 <= 2d-296) then
        tmp = t_0
    else if (t_1 <= 1d-134) then
        tmp = t_2
    else if (t_1 <= 1d-65) then
        tmp = t_0
    else if (t_1 <= 2d+223) then
        tmp = t_2
    else
        tmp = (0.5d0 * ((x / y) ** 2.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - ((y * (y / x)) * (8.0 / x));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 2e-296) {
		tmp = t_0;
	} else if (t_1 <= 1e-134) {
		tmp = t_2;
	} else if (t_1 <= 1e-65) {
		tmp = t_0;
	} else if (t_1 <= 2e+223) {
		tmp = t_2;
	} else {
		tmp = (0.5 * Math.pow((x / y), 2.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - ((y * (y / x)) * (8.0 / x))
	t_1 = y * (y * 4.0)
	t_2 = ((x * x) - t_1) / (t_1 + (x * x))
	tmp = 0
	if t_1 <= 2e-296:
		tmp = t_0
	elif t_1 <= 1e-134:
		tmp = t_2
	elif t_1 <= 1e-65:
		tmp = t_0
	elif t_1 <= 2e+223:
		tmp = t_2
	else:
		tmp = (0.5 * math.pow((x / y), 2.0)) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(y * Float64(y / x)) * Float64(8.0 / x)))
	t_1 = Float64(y * Float64(y * 4.0))
	t_2 = Float64(Float64(Float64(x * x) - t_1) / Float64(t_1 + Float64(x * x)))
	tmp = 0.0
	if (t_1 <= 2e-296)
		tmp = t_0;
	elseif (t_1 <= 1e-134)
		tmp = t_2;
	elseif (t_1 <= 1e-65)
		tmp = t_0;
	elseif (t_1 <= 2e+223)
		tmp = t_2;
	else
		tmp = Float64(Float64(0.5 * (Float64(x / y) ^ 2.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - ((y * (y / x)) * (8.0 / x));
	t_1 = y * (y * 4.0);
	t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	tmp = 0.0;
	if (t_1 <= 2e-296)
		tmp = t_0;
	elseif (t_1 <= 1e-134)
		tmp = t_2;
	elseif (t_1 <= 1e-65)
		tmp = t_0;
	elseif (t_1 <= 2e+223)
		tmp = t_2;
	else
		tmp = (0.5 * ((x / y) ^ 2.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(8.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * x), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(t$95$1 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-296], t$95$0, If[LessEqual[t$95$1, 1e-134], t$95$2, If[LessEqual[t$95$1, 1e-65], t$95$0, If[LessEqual[t$95$1, 2e+223], t$95$2, N[(N[(0.5 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e-296 or 1.00000000000000004e-134 < (*.f64 (*.f64 y 4) y) < 9.99999999999999923e-66

   1. Initial program 57.5%

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

      1. frac-2neg 57.5%

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

      2. div-inv 57.4%

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

      3. sub-neg 57.4%

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

      4. +-commutative 57.4%

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

      5. *-commutative 57.4%

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

      6. distribute-rgt-neg-in 57.4%

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

      7. fma-def 57.4%

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

      8. distribute-rgt-neg-in 57.4%

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

      9. metadata-eval 57.4%

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

      10. +-commutative 57.4%

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

      11. distribute-neg-in 57.4%

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

      12. *-commutative 57.4%

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

      13. distribute-rgt-neg-in 57.4%

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

      14. fma-def 57.4%

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

      15. distribute-rgt-neg-in 57.4%

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

      16. metadata-eval 57.4%

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

   3. Applied egg-rr 57.4%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, -x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-neg-in 57.4%

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

   5. Simplified 57.4%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, x \cdot \left(-x\right)\right)}} \]
   6. Step-by-step derivation

      1. fma-udef 57.4%

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

   7. Applied egg-rr 57.4%

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

   8. Taylor expanded in x around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. unsub-neg 82.5%

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

      3. unpow2 82.5%

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

      4. unpow2 82.5%

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

      5. distribute-rgt-out-- 82.5%

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

      6. metadata-eval 82.5%

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

      7. unpow2 82.5%

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

      8. times-frac 90.2%

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

      9. associate-*r/ 90.7%

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

   10. Simplified 90.7%

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

   if 2e-296 < (*.f64 (*.f64 y 4) y) < 1.00000000000000004e-134 or 9.99999999999999923e-66 < (*.f64 (*.f64 y 4) y) < 2.00000000000000009e223

   1. Initial program 81.6%

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

   if 2.00000000000000009e223 < (*.f64 (*.f64 y 4) y)

   1. Initial program 14.1%

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

      1. frac-2neg 14.1%

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

      2. div-inv 14.1%

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

      3. sub-neg 14.1%

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

      4. +-commutative 14.1%

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

      5. *-commutative 14.1%

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

      6. distribute-rgt-neg-in 14.1%

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

      7. fma-def 14.1%

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

      8. distribute-rgt-neg-in 14.1%

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

      9. metadata-eval 14.1%

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

      10. +-commutative 14.1%

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

      11. distribute-neg-in 14.1%

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

      12. *-commutative 14.1%

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

      13. distribute-rgt-neg-in 14.1%

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

      14. fma-def 14.1%

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

      15. distribute-rgt-neg-in 14.1%

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

      16. metadata-eval 14.1%

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

   3. Applied egg-rr 14.1%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, -x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-neg-in 14.1%

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

   5. Simplified 14.1%

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

   6. Taylor expanded in x around 0 62.8%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + 0.5 \cdot \frac{{x}^{2}}{{y}^{2}}\right) - 1} \]
   7. Step-by-step derivation

      1. associate--l+ 62.8%

         \[\leadsto \color{blue}{-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} + \left(0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right)} \]

      2. fma-def 62.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right)} \]

      3. metadata-eval 62.8%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{{x}^{\color{blue}{\left(2 \cdot 2\right)}}}{{y}^{4}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      4. pow-sqr 62.8%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{{x}^{2} \cdot {x}^{2}}}{{y}^{4}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      5. unpow2 62.8%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}}{{y}^{4}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      6. unpow2 62.8%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}}{{y}^{4}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      7. metadata-eval 62.8%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{{y}^{\color{blue}{\left(2 \cdot 2\right)}}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      8. pow-sqr 62.8%

         \[\leadsto \mathsf{fma}\left(-0.125, \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{{y}^{2} \cdot {y}^{2}}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      9. times-frac 82.1%

         \[\leadsto \mathsf{fma}\left(-0.125, \color{blue}{\frac{x \cdot x}{{y}^{2}} \cdot \frac{x \cdot x}{{y}^{2}}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      10. unpow2 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, \frac{x \cdot x}{\color{blue}{y \cdot y}} \cdot \frac{x \cdot x}{{y}^{2}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      11. times-frac 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} \cdot \frac{x \cdot x}{{y}^{2}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      12. unpow2 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, \color{blue}{{\left(\frac{x}{y}\right)}^{2}} \cdot \frac{x \cdot x}{{y}^{2}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      13. unpow2 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{2} \cdot \frac{x \cdot x}{\color{blue}{y \cdot y}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      14. times-frac 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{2} \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      15. unpow2 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{2} \cdot \color{blue}{{\left(\frac{x}{y}\right)}^{2}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      16. pow-sqr 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, \color{blue}{{\left(\frac{x}{y}\right)}^{\left(2 \cdot 2\right)}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      17. metadata-eval 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{\color{blue}{4}}, 0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1\right) \]

      18. fma-neg 82.1%

          \[\leadsto \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{{y}^{2}}, -1\right)}\right) \]

   8. Simplified 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, \mathsf{fma}\left(0.5, {\left(\frac{x}{y}\right)}^{2}, -1\right)\right)} \]
   9. Step-by-step derivation

      1. fma-udef 87.3%

         \[\leadsto \color{blue}{-0.125 \cdot {\left(\frac{x}{y}\right)}^{4} + \mathsf{fma}\left(0.5, {\left(\frac{x}{y}\right)}^{2}, -1\right)} \]

      2. fma-udef 87.3%

         \[\leadsto -0.125 \cdot {\left(\frac{x}{y}\right)}^{4} + \color{blue}{\left(0.5 \cdot {\left(\frac{x}{y}\right)}^{2} + -1\right)} \]

      3. associate-+r+ 87.3%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {\left(\frac{x}{y}\right)}^{4} + 0.5 \cdot {\left(\frac{x}{y}\right)}^{2}\right) + -1} \]

      4. *-commutative 87.3%

         \[\leadsto \left(-0.125 \cdot {\left(\frac{x}{y}\right)}^{4} + \color{blue}{{\left(\frac{x}{y}\right)}^{2} \cdot 0.5}\right) + -1 \]

   10. Applied egg-rr 87.3%

       \[\leadsto \color{blue}{\left(-0.125 \cdot {\left(\frac{x}{y}\right)}^{4} + {\left(\frac{x}{y}\right)}^{2} \cdot 0.5\right) + -1} \]

   11. Taylor expanded in x around 0 82.3%

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

       1. unpow2 82.3%

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

       2. unpow2 82.3%

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

       3. times-frac 87.8%

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

       4. unpow2 87.8%

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

   13. Simplified 87.8%

       \[\leadsto \color{blue}{0.5 \cdot {\left(\frac{x}{y}\right)}^{2}} + -1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-296}:\\ \;\;\;\;1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{-134}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{-65}:\\ \;\;\;\;1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{x}{y}\right)}^{2} + -1\\ \end{array} \]

Alternative 2: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ t_1 := y \cdot \left(y \cdot 4\right)\\ t_2 := \frac{x \cdot x - t_1}{t_1 + x \cdot x}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+223}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (* y (/ y x)) (/ 8.0 x))))
        (t_1 (* y (* y 4.0)))
        (t_2 (/ (- (* x x) t_1) (+ t_1 (* x x)))))
   (if (<= t_1 2e-296)
     t_0
     (if (<= t_1 1e-134)
       t_2
       (if (<= t_1 1e-65)
         t_0
         (if (<= t_1 2e+223) t_2 (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - ((y * (y / x)) * (8.0 / x));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 2e-296) {
		tmp = t_0;
	} else if (t_1 <= 1e-134) {
		tmp = t_2;
	} else if (t_1 <= 1e-65) {
		tmp = t_0;
	} else if (t_1 <= 2e+223) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - ((y * (y / x)) * (8.0d0 / x))
    t_1 = y * (y * 4.0d0)
    t_2 = ((x * x) - t_1) / (t_1 + (x * x))
    if (t_1 <= 2d-296) then
        tmp = t_0
    else if (t_1 <= 1d-134) then
        tmp = t_2
    else if (t_1 <= 1d-65) then
        tmp = t_0
    else if (t_1 <= 2d+223) then
        tmp = t_2
    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 = 1.0 - ((y * (y / x)) * (8.0 / x));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 2e-296) {
		tmp = t_0;
	} else if (t_1 <= 1e-134) {
		tmp = t_2;
	} else if (t_1 <= 1e-65) {
		tmp = t_0;
	} else if (t_1 <= 2e+223) {
		tmp = t_2;
	} else {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - ((y * (y / x)) * (8.0 / x))
	t_1 = y * (y * 4.0)
	t_2 = ((x * x) - t_1) / (t_1 + (x * x))
	tmp = 0
	if t_1 <= 2e-296:
		tmp = t_0
	elif t_1 <= 1e-134:
		tmp = t_2
	elif t_1 <= 1e-65:
		tmp = t_0
	elif t_1 <= 2e+223:
		tmp = t_2
	else:
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(Float64(y * Float64(y / x)) * Float64(8.0 / x)))
	t_1 = Float64(y * Float64(y * 4.0))
	t_2 = Float64(Float64(Float64(x * x) - t_1) / Float64(t_1 + Float64(x * x)))
	tmp = 0.0
	if (t_1 <= 2e-296)
		tmp = t_0;
	elseif (t_1 <= 1e-134)
		tmp = t_2;
	elseif (t_1 <= 1e-65)
		tmp = t_0;
	elseif (t_1 <= 2e+223)
		tmp = t_2;
	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 = 1.0 - ((y * (y / x)) * (8.0 / x));
	t_1 = y * (y * 4.0);
	t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	tmp = 0.0;
	if (t_1 <= 2e-296)
		tmp = t_0;
	elseif (t_1 <= 1e-134)
		tmp = t_2;
	elseif (t_1 <= 1e-65)
		tmp = t_0;
	elseif (t_1 <= 2e+223)
		tmp = t_2;
	else
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(8.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * x), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(t$95$1 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-296], t$95$0, If[LessEqual[t$95$1, 1e-134], t$95$2, If[LessEqual[t$95$1, 1e-65], t$95$0, If[LessEqual[t$95$1, 2e+223], t$95$2, 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) < 2e-296 or 1.00000000000000004e-134 < (*.f64 (*.f64 y 4) y) < 9.99999999999999923e-66

   1. Initial program 57.5%

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

      1. frac-2neg 57.5%

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

      2. div-inv 57.4%

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

      3. sub-neg 57.4%

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

      4. +-commutative 57.4%

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

      5. *-commutative 57.4%

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

      6. distribute-rgt-neg-in 57.4%

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

      7. fma-def 57.4%

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

      8. distribute-rgt-neg-in 57.4%

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

      9. metadata-eval 57.4%

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

      10. +-commutative 57.4%

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

      11. distribute-neg-in 57.4%

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

      12. *-commutative 57.4%

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

      13. distribute-rgt-neg-in 57.4%

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

      14. fma-def 57.4%

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

      15. distribute-rgt-neg-in 57.4%

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

      16. metadata-eval 57.4%

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

   3. Applied egg-rr 57.4%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, -x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-neg-in 57.4%

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

   5. Simplified 57.4%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, x \cdot \left(-x\right)\right)}} \]
   6. Step-by-step derivation

      1. fma-udef 57.4%

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

   7. Applied egg-rr 57.4%

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

   8. Taylor expanded in x around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. unsub-neg 82.5%

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

      3. unpow2 82.5%

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

      4. unpow2 82.5%

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

      5. distribute-rgt-out-- 82.5%

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

      6. metadata-eval 82.5%

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

      7. unpow2 82.5%

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

      8. times-frac 90.2%

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

      9. associate-*r/ 90.7%

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

   10. Simplified 90.7%

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

   if 2e-296 < (*.f64 (*.f64 y 4) y) < 1.00000000000000004e-134 or 9.99999999999999923e-66 < (*.f64 (*.f64 y 4) y) < 2.00000000000000009e223

   1. Initial program 81.6%

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

   if 2.00000000000000009e223 < (*.f64 (*.f64 y 4) y)

   1. Initial program 14.1%

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

   2. Taylor expanded in x around 0 14.3%

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

      1. *-commutative 14.3%

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

      2. unpow2 14.3%

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

      3. associate-*r* 14.3%

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

   4. Simplified 14.3%

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

      1. div-sub 14.3%

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

      2. associate-*r* 14.3%

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

      3. associate-/r* 14.3%

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

      4. frac-times 14.4%

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

      5. pow2 14.4%

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

      6. *-commutative 14.4%

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

      7. *-inverses 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. unpow2 87.8%

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

      2. clear-num 87.8%

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

      3. un-div-inv 87.8%

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

   8. Applied egg-rr 87.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-296}:\\ \;\;\;\;1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{-134}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{-65}:\\ \;\;\;\;1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \]

Alternative 3: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-65}:\\ \;\;\;\;1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 1e-65)
   (- 1.0 (* (* y (/ y x)) (/ 8.0 x)))
   (+ -1.0 (/ (/ (/ x y) (/ y x)) 4.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 1e-65) {
		tmp = 1.0 - ((y * (y / x)) * (8.0 / 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) :: tmp
    if ((y * (y * 4.0d0)) <= 1d-65) then
        tmp = 1.0d0 - ((y * (y / x)) * (8.0d0 / 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 tmp;
	if ((y * (y * 4.0)) <= 1e-65) {
		tmp = 1.0 - ((y * (y / x)) * (8.0 / x));
	} else {
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (y * 4.0)) <= 1e-65:
		tmp = 1.0 - ((y * (y / x)) * (8.0 / x))
	else:
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(y * 4.0)) <= 1e-65)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(y / x)) * Float64(8.0 / 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)
	tmp = 0.0;
	if ((y * (y * 4.0)) <= 1e-65)
		tmp = 1.0 - ((y * (y / x)) * (8.0 / x));
	else
		tmp = -1.0 + (((x / y) / (y / x)) / 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 1e-65], N[(1.0 - N[(N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(8.0 / 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 2 regimes

2. if (*.f64 (*.f64 y 4) y) < 9.99999999999999923e-66

   1. Initial program 67.2%

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

      1. frac-2neg 67.2%

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

      2. div-inv 67.1%

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

      3. sub-neg 67.1%

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

      4. +-commutative 67.1%

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

      5. *-commutative 67.1%

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

      6. distribute-rgt-neg-in 67.1%

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

      7. fma-def 67.1%

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

      8. distribute-rgt-neg-in 67.1%

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

      9. metadata-eval 67.1%

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

      10. +-commutative 67.1%

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

      11. distribute-neg-in 67.1%

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

      12. *-commutative 67.1%

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

      13. distribute-rgt-neg-in 67.1%

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

      14. fma-def 67.1%

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

      15. distribute-rgt-neg-in 67.1%

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

      16. metadata-eval 67.1%

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

   3. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, -x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-neg-in 67.1%

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

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, x \cdot \left(-x\right)\right)}} \]
   6. Step-by-step derivation

      1. fma-udef 67.1%

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

   7. Applied egg-rr 67.1%

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

   8. Taylor expanded in x around inf 78.1%

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

      1. mul-1-neg 78.1%

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

      2. unsub-neg 78.1%

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

      3. unpow2 78.1%

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

      4. unpow2 78.1%

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

      5. distribute-rgt-out-- 78.1%

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

      6. metadata-eval 78.1%

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

      7. unpow2 78.1%

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

      8. times-frac 83.4%

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

      9. associate-*r/ 83.7%

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

   10. Simplified 83.7%

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

   if 9.99999999999999923e-66 < (*.f64 (*.f64 y 4) y)

   1. Initial program 42.1%

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

   2. Taylor expanded in x around 0 36.3%

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

      1. *-commutative 36.3%

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

      2. unpow2 36.3%

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

      3. associate-*r* 36.3%

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

   4. Simplified 36.3%

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

      1. div-sub 36.3%

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

      2. associate-*r* 36.3%

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

      3. associate-/r* 36.3%

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

      4. frac-times 36.3%

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

      5. pow2 36.3%

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

      6. *-commutative 36.3%

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

      7. *-inverses 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. unpow2 77.2%

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

      2. clear-num 77.2%

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

      3. un-div-inv 77.2%

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

   8. Applied egg-rr 77.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-65}:\\ \;\;\;\;1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{\frac{x}{y}}{\frac{y}{x}}}{4}\\ \end{array} \]

Alternative 4: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-31}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05e-31)
   -1.0
   (if (<= y 1.9e-6) (- 1.0 (* (* y (/ y x)) (/ 8.0 x))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-31) {
		tmp = -1.0;
	} else if (y <= 1.9e-6) {
		tmp = 1.0 - ((y * (y / x)) * (8.0 / 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 <= (-1.05d-31)) then
        tmp = -1.0d0
    else if (y <= 1.9d-6) then
        tmp = 1.0d0 - ((y * (y / x)) * (8.0d0 / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05e-31) {
		tmp = -1.0;
	} else if (y <= 1.9e-6) {
		tmp = 1.0 - ((y * (y / x)) * (8.0 / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05e-31:
		tmp = -1.0
	elif y <= 1.9e-6:
		tmp = 1.0 - ((y * (y / x)) * (8.0 / x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05e-31)
		tmp = -1.0;
	elseif (y <= 1.9e-6)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(y / x)) * Float64(8.0 / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.05e-31)
		tmp = -1.0;
	elseif (y <= 1.9e-6)
		tmp = 1.0 - ((y * (y / x)) * (8.0 / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05e-31], -1.0, If[LessEqual[y, 1.9e-6], N[(1.0 - N[(N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(8.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.04999999999999996e-31 or 1.9e-6 < y

   1. Initial program 41.9%

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

   2. Taylor expanded in x around 0 77.2%

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

   if -1.04999999999999996e-31 < y < 1.9e-6

   1. Initial program 66.7%

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

      1. frac-2neg 66.7%

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

      2. div-inv 66.5%

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

      3. sub-neg 66.5%

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

      4. +-commutative 66.5%

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

      5. *-commutative 66.5%

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

      6. distribute-rgt-neg-in 66.5%

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

      7. fma-def 66.5%

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

      8. distribute-rgt-neg-in 66.5%

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

      9. metadata-eval 66.5%

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

      10. +-commutative 66.5%

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

      11. distribute-neg-in 66.5%

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

      12. *-commutative 66.5%

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

      13. distribute-rgt-neg-in 66.5%

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

      14. fma-def 66.5%

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

      15. distribute-rgt-neg-in 66.5%

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

      16. metadata-eval 66.5%

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

   3. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, -x \cdot x\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-neg-in 66.5%

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

   5. Simplified 66.5%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(y, y \cdot -4, x \cdot \left(-x\right)\right)}} \]
   6. Step-by-step derivation

      1. fma-udef 66.5%

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

   7. Applied egg-rr 66.5%

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

   8. Taylor expanded in x around inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. unsub-neg 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

      5. distribute-rgt-out-- 77.2%

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

      6. metadata-eval 77.2%

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

      7. unpow2 77.2%

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

      8. times-frac 82.4%

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

      9. associate-*r/ 82.7%

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

   10. Simplified 82.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-31}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;1 - \left(y \cdot \frac{y}{x}\right) \cdot \frac{8}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 74.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e-29) -1.0 (if (<= y 2e-32) 1.0 -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e-29) {
		tmp = -1.0;
	} else if (y <= 2e-32) {
		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-29)) then
        tmp = -1.0d0
    else if (y <= 2d-32) 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-29) {
		tmp = -1.0;
	} else if (y <= 2e-32) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e-29:
		tmp = -1.0
	elif y <= 2e-32:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e-29)
		tmp = -1.0;
	elseif (y <= 2e-32)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e-29)
		tmp = -1.0;
	elseif (y <= 2e-32)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e-29], -1.0, If[LessEqual[y, 2e-32], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999943e-30 or 2.00000000000000011e-32 < y

   1. Initial program 42.1%

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

   2. Taylor expanded in x around 0 76.5%

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

   if -9.99999999999999943e-30 < y < 2.00000000000000011e-32

   1. Initial program 67.2%

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

   2. Taylor expanded in x around inf 82.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 50.9% 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 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 0 50.0%

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

3. Final simplification 50.0%

   \[\leadsto -1 \]

Developer target: 50.9% 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 2023185 
(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))))