Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Percentage Accurate: 66.1% → 99.7%

Time: 12.9s

Alternatives: 16

Speedup: 0.7×

• 47.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

C

double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

C

double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{y}}{\frac{y}{x}} + \frac{\frac{z}{t}}{\frac{t}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (/ x y) (/ y x)) (/ (/ z t) (/ t z))))

C

double code(double x, double y, double z, double t) {
	return ((x / y) / (y / x)) + ((z / t) / (t / z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) / (y / x)) + ((z / t) / (t / z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x / y) / (y / x)) + ((z / t) / (t / z));
}

Python

def code(x, y, z, t):
	return ((x / y) / (y / x)) + ((z / t) / (t / z))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x / y) / Float64(y / x)) + Float64(Float64(z / t) / Float64(t / z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / y) / (y / x)) + ((z / t) / (t / z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   2. clear-num N/A

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y}{x}} + \frac{z \cdot z}{t \cdot t} \]

   6. /-lowering-/.f64 82.4

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

4. Applied egg-rr 82.4%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]
5. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   2. clear-num N/A

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \frac{z}{t} \cdot \color{blue}{\frac{1}{\frac{t}{z}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \frac{\color{blue}{\frac{z}{t}}}{\frac{t}{z}} \]

   6. /-lowering-/.f64 99.7

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \frac{\frac{z}{t}}{\color{blue}{\frac{t}{z}}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]
7. Add Preprocessing

Alternative 2: 93.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+291)
     (fma (/ x y) (/ x y) t_1)
     (if (<= t_1 INFINITY)
       (/ (/ z t) (/ t z))
       (fma (/ z t) (/ z t) (/ (* x x) (* y y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+291) {
		tmp = fma((x / y), (x / y), t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = fma((z / t), (z / t), ((x * x) / (y * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+291)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x * x) / Float64(y * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+291], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000001e291

   1. Initial program 79.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{x}{y}}, \frac{z \cdot z}{t \cdot t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z \cdot z}{t \cdot t}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\color{blue}{z \cdot z}}{t \cdot t}\right) \]

      7. *-lowering-*.f64 98.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{\color{blue}{t \cdot t}}\right) \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)} \]

   if 5.0000000000000001e291 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 74.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 88.3

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 88.3%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{\frac{t}{z}} \]

      6. /-lowering-/.f64 92.0

         \[\leadsto \frac{\frac{z}{t}}{\color{blue}{\frac{t}{z}}} \]

   7. Applied egg-rr 92.0%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

   if +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 0.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} + \frac{x \cdot x}{y \cdot y} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t}}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{\frac{z}{t}}, \frac{x \cdot x}{y \cdot y}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{x \cdot x}}{y \cdot y}\right) \]

      8. *-lowering-*.f64 78.3

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{\color{blue}{y \cdot y}}\right) \]

   4. Applied egg-rr 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{-241}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{1}{y \cdot y}, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e-241)
     (/ (/ x y) (/ y x))
     (if (<= t_1 5e+291)
       (fma (* x (/ 1.0 (* y y))) x t_1)
       (/ (/ z t) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e-241) {
		tmp = (x / y) / (y / x);
	} else if (t_1 <= 5e+291) {
		tmp = fma((x * (1.0 / (y * y))), x, t_1);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e-241)
		tmp = Float64(Float64(x / y) / Float64(y / x));
	elseif (t_1 <= 5e+291)
		tmp = fma(Float64(x * Float64(1.0 / Float64(y * y))), x, t_1);
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-241], N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+291], N[(N[(x * N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999997e-242

   1. Initial program 74.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y}{x}} \]

      6. /-lowering-/.f64 97.6

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{\frac{y}{x}}} \]

   7. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

   if 9.9999999999999997e-242 < (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000001e291

   1. Initial program 86.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot y}{x \cdot x}}} + \frac{z \cdot z}{t \cdot t} \]

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 88.9

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

   4. Applied egg-rr 88.9%

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

   if 5.0000000000000001e291 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 72.2

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{\frac{t}{z}} \]

      6. /-lowering-/.f64 83.1

         \[\leadsto \frac{\frac{z}{t}}{\color{blue}{\frac{t}{z}}} \]

   7. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{-241}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{1}{y \cdot y}, x, \frac{z \cdot z}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{-241}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e-241)
     (/ (/ x y) (/ y x))
     (if (<= t_1 5e+291) (fma (/ x (* y y)) x t_1) (/ (/ z t) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e-241) {
		tmp = (x / y) / (y / x);
	} else if (t_1 <= 5e+291) {
		tmp = fma((x / (y * y)), x, t_1);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e-241)
		tmp = Float64(Float64(x / y) / Float64(y / x));
	elseif (t_1 <= 5e+291)
		tmp = fma(Float64(x / Float64(y * y)), x, t_1);
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-241], N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+291], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999997e-242

   1. Initial program 74.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 81.3

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

   5. Simplified 81.3%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y}{x}} \]

      6. /-lowering-/.f64 97.6

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{\frac{y}{x}}} \]

   7. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

   if 9.9999999999999997e-242 < (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000001e291

   1. Initial program 86.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y} \cdot x} + \frac{z \cdot z}{t \cdot t} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z \cdot z}{t \cdot t}\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y \cdot y}}, x, \frac{z \cdot z}{t \cdot t}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{y \cdot y}}, x, \frac{z \cdot z}{t \cdot t}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y \cdot y}, x, \color{blue}{\frac{z \cdot z}{t \cdot t}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{\color{blue}{z \cdot z}}{t \cdot t}\right) \]

      8. *-lowering-*.f64 88.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z \cdot z}{\color{blue}{t \cdot t}}\right) \]

   4. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z \cdot z}{t \cdot t}\right)} \]

   if 5.0000000000000001e291 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 72.2

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{\frac{t}{z}} \]

      6. /-lowering-/.f64 83.1

         \[\leadsto \frac{\frac{z}{t}}{\color{blue}{\frac{t}{z}}} \]

   7. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (/ (/ z t) (/ t z))))
   (if (<= t_1 2e+111) t_2 (if (<= t_1 INFINITY) (/ (/ x y) (/ y x)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) / (t / z);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / y) / (y / x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) / (t / z);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x / y) / (y / x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = (z / t) / (t / z)
	tmp = 0
	if t_1 <= 2e+111:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x / y) / (y / x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(Float64(z / t) / Float64(t / z))
	tmp = 0.0
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / y) / Float64(y / x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = (z / t) / (t / z);
	tmp = 0.0;
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x / y) / (y / x);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+111], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e111 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 71.8

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 71.8%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{\frac{t}{z}} \]

      6. /-lowering-/.f64 83.6

         \[\leadsto \frac{\frac{z}{t}}{\color{blue}{\frac{t}{z}}} \]

   7. Applied egg-rr 83.6%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

   if 1.99999999999999991e111 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y}{x}} \]

      6. /-lowering-/.f64 90.4

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{\frac{y}{x}}} \]

   7. Applied egg-rr 90.4%

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

Alternative 6: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (/ z t) (/ z t))))
   (if (<= t_1 2e+111) t_2 (if (<= t_1 INFINITY) (/ (/ x y) (/ y x)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) * (z / t);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / y) / (y / x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) * (z / t);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x / y) / (y / x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = (z / t) * (z / t)
	tmp = 0
	if t_1 <= 2e+111:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x / y) / (y / x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(Float64(z / t) * Float64(z / t))
	tmp = 0.0
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / y) / Float64(y / x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = (z / t) * (z / t);
	tmp = 0.0;
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x / y) / (y / x);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+111], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e111 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 71.8

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 71.8%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      2. frac-times N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      5. /-lowering-/.f64 83.5

         \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   7. Applied egg-rr 83.5%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   if 1.99999999999999991e111 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y}{x}} \]

      6. /-lowering-/.f64 90.4

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{\frac{y}{x}}} \]

   7. Applied egg-rr 90.4%

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

Alternative 7: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{+150}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t} + \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e+150)
     (+ (/ (/ x y) (/ y x)) t_1)
     (+ (* (/ z t) (/ z t)) (/ (* x (/ x y)) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+150) {
		tmp = ((x / y) / (y / x)) + t_1;
	} else {
		tmp = ((z / t) * (z / t)) + ((x * (x / y)) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) / (t * t)
    if (t_1 <= 1d+150) then
        tmp = ((x / y) / (y / x)) + t_1
    else
        tmp = ((z / t) * (z / t)) + ((x * (x / y)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+150) {
		tmp = ((x / y) / (y / x)) + t_1;
	} else {
		tmp = ((z / t) * (z / t)) + ((x * (x / y)) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 1e+150:
		tmp = ((x / y) / (y / x)) + t_1
	else:
		tmp = ((z / t) * (z / t)) + ((x * (x / y)) / y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e+150)
		tmp = Float64(Float64(Float64(x / y) / Float64(y / x)) + t_1);
	else
		tmp = Float64(Float64(Float64(z / t) * Float64(z / t)) + Float64(Float64(x * Float64(x / y)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 1e+150)
		tmp = ((x / y) / (y / x)) + t_1;
	else
		tmp = ((z / t) * (z / t)) + ((x * (x / y)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+150], N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.99999999999999981e149

   1. Initial program 77.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      2. clear-num N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y}{x}} + \frac{z \cdot z}{t \cdot t} \]

      6. /-lowering-/.f64 99.0

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

   4. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

   if 9.99999999999999981e149 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 61.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      5. /-lowering-/.f64 82.7

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t}} \cdot z}{t} \]

   4. Applied egg-rr 82.7%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   5. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

      5. /-lowering-/.f64 93.7

         \[\leadsto \frac{\color{blue}{\frac{x}{y}} \cdot x}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

   6. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      4. /-lowering-/.f64 99.0

         \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   8. Applied egg-rr 99.0%

      \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{+150}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t} + \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t} + \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e+150)
     (fma (/ x y) (/ x y) t_1)
     (+ (* (/ z t) (/ z t)) (/ (* x (/ x y)) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+150) {
		tmp = fma((x / y), (x / y), t_1);
	} else {
		tmp = ((z / t) * (z / t)) + ((x * (x / y)) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e+150)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	else
		tmp = Float64(Float64(Float64(z / t) * Float64(z / t)) + Float64(Float64(x * Float64(x / y)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+150], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.99999999999999981e149

   1. Initial program 77.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{x}{y}}, \frac{z \cdot z}{t \cdot t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z \cdot z}{t \cdot t}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\color{blue}{z \cdot z}}{t \cdot t}\right) \]

      7. *-lowering-*.f64 98.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{\color{blue}{t \cdot t}}\right) \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)} \]

   if 9.99999999999999981e149 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 61.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      5. /-lowering-/.f64 82.7

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t}} \cdot z}{t} \]

   4. Applied egg-rr 82.7%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   5. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x}}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

      5. /-lowering-/.f64 93.7

         \[\leadsto \frac{\color{blue}{\frac{x}{y}} \cdot x}{y} + \frac{\frac{z}{t} \cdot z}{t} \]

   6. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y} \cdot x}{y}} + \frac{\frac{z}{t} \cdot z}{t} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      4. /-lowering-/.f64 99.0

         \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   8. Applied egg-rr 99.0%

      \[\leadsto \frac{\frac{x}{y} \cdot x}{y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t} + \frac{x \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (/ z t) (/ z t))))
   (if (<= t_1 2e+111) t_2 (if (<= t_1 INFINITY) (* (/ x y) (/ x y)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) * (z / t);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) * (z / t);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = (z / t) * (z / t)
	tmp = 0
	if t_1 <= 2e+111:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x / y) * (x / y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(Float64(z / t) * Float64(z / t))
	tmp = 0.0
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = (z / t) * (z / t);
	tmp = 0.0;
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x / y) * (x / y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+111], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e111 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 71.8

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 71.8%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} \]

      2. frac-times N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

      5. /-lowering-/.f64 83.5

         \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   7. Applied egg-rr 83.5%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   if 1.99999999999999991e111 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 90.4

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

   7. Applied egg-rr 90.4%

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

Alternative 10: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* z (/ (/ z t) t))))
   (if (<= t_1 2e+111) t_2 (if (<= t_1 INFINITY) (* (/ x y) (/ x y)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * ((z / t) / t);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * ((z / t) / t);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = z * ((z / t) / t)
	tmp = 0
	if t_1 <= 2e+111:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x / y) * (x / y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(z * Float64(Float64(z / t) / t))
	tmp = 0.0
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = z * ((z / t) / t);
	tmp = 0.0;
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x / y) * (x / y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+111], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e111 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 71.8

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 71.8%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto z \cdot \color{blue}{\frac{\frac{z}{t}}{t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{\frac{z}{t}}{t}} \]

      3. /-lowering-/.f64 80.5

         \[\leadsto z \cdot \frac{\color{blue}{\frac{z}{t}}}{t} \]

   7. Applied egg-rr 80.5%

      \[\leadsto z \cdot \color{blue}{\frac{\frac{z}{t}}{t}} \]

   if 1.99999999999999991e111 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 90.4

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

   7. Applied egg-rr 90.4%

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

Alternative 11: 77.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := z \cdot \frac{\frac{z}{t}}{t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* z (/ (/ z t) t))))
   (if (<= t_1 2e+111) t_2 (if (<= t_1 INFINITY) (* x (/ x (* y y))) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * ((z / t) / t);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x * (x / (y * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * ((z / t) / t);
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x * (x / (y * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = z * ((z / t) / t)
	tmp = 0
	if t_1 <= 2e+111:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = x * (x / (y * y))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(z * Float64(Float64(z / t) / t))
	tmp = 0.0
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(x * Float64(x / Float64(y * y)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = z * ((z / t) / t);
	tmp = 0.0;
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = x * (x / (y * y));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+111], t$95$2, If[LessEqual[t$95$1, Infinity], N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e111 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 71.8

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 71.8%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto z \cdot \color{blue}{\frac{\frac{z}{t}}{t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{\frac{z}{t}}{t}} \]

      3. /-lowering-/.f64 80.5

         \[\leadsto z \cdot \frac{\color{blue}{\frac{z}{t}}}{t} \]

   7. Applied egg-rr 80.5%

      \[\leadsto z \cdot \color{blue}{\frac{\frac{z}{t}}{t}} \]

   if 1.99999999999999991e111 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

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

Alternative 12: 92.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{z}{t}}{t} + x \cdot \frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e+140)
     (fma (/ x y) (/ x y) t_1)
     (+ (/ (* z (/ z t)) t) (* x (/ x (* y y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e+140) {
		tmp = fma((x / y), (x / y), t_1);
	} else {
		tmp = ((z * (z / t)) / t) + (x * (x / (y * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e+140)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	else
		tmp = Float64(Float64(Float64(z * Float64(z / t)) / t) + Float64(x * Float64(x / Float64(y * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+140], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(z * N[(z / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.00000000000000012e140

   1. Initial program 77.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{x}{y}}, \frac{z \cdot z}{t \cdot t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z \cdot z}{t \cdot t}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\color{blue}{z \cdot z}}{t \cdot t}\right) \]

      7. *-lowering-*.f64 98.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{\color{blue}{t \cdot t}}\right) \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)} \]

   if 2.00000000000000012e140 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 62.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t} \cdot z}}{t} \]

      5. /-lowering-/.f64 82.8

         \[\leadsto \frac{x \cdot x}{y \cdot y} + \frac{\color{blue}{\frac{z}{t}} \cdot z}{t} \]

   4. Applied egg-rr 82.8%

      \[\leadsto \frac{x \cdot x}{y \cdot y} + \color{blue}{\frac{\frac{z}{t} \cdot z}{t}} \]
   5. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y} \cdot x} + \frac{\frac{z}{t} \cdot z}{t} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y} \cdot x} + \frac{\frac{z}{t} \cdot z}{t} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

      4. *-lowering-*.f64 92.3

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \cdot x + \frac{\frac{z}{t} \cdot z}{t} \]

   6. Applied egg-rr 92.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y} \cdot x} + \frac{\frac{z}{t} \cdot z}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{z}{t}}{t} + x \cdot \frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+291) (fma (/ x y) (/ x y) t_1) (/ (/ z t) (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+291) {
		tmp = fma((x / y), (x / y), t_1);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+291)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+291], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000001e291

   1. Initial program 79.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \color{blue}{\frac{x}{y}}, \frac{z \cdot z}{t \cdot t}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \color{blue}{\frac{z \cdot z}{t \cdot t}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\color{blue}{z \cdot z}}{t \cdot t}\right) \]

      7. *-lowering-*.f64 98.9

         \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{\color{blue}{t \cdot t}}\right) \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)} \]

   if 5.0000000000000001e291 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.9%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 72.2

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{t \cdot t} \cdot z} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{\frac{t}{z}} \]

      6. /-lowering-/.f64 83.1

         \[\leadsto \frac{\frac{z}{t}}{\color{blue}{\frac{t}{z}}} \]

   7. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{\frac{t}{z}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := z \cdot \frac{z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* z (/ z (* t t)))))
   (if (<= t_1 2e+111) t_2 (if (<= t_1 INFINITY) (* x (/ x (* y y))) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * (z / (t * t));
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x * (x / (y * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = z * (z / (t * t));
	double tmp;
	if (t_1 <= 2e+111) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x * (x / (y * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = z * (z / (t * t))
	tmp = 0
	if t_1 <= 2e+111:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = x * (x / (y * y))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(z * Float64(z / Float64(t * t)))
	tmp = 0.0
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(x * Float64(x / Float64(y * y)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = z * (z / (t * t));
	tmp = 0.0;
	if (t_1 <= 2e+111)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = x * (x / (y * y));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+111], t$95$2, If[LessEqual[t$95$1, Infinity], N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e111 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{z}{{t}^{2}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{z}{{t}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

      6. *-lowering-*.f64 71.8

         \[\leadsto z \cdot \frac{z}{\color{blue}{t \cdot t}} \]

   5. Simplified 71.8%

      \[\leadsto \color{blue}{z \cdot \frac{z}{t \cdot t}} \]

   if 1.99999999999999991e111 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

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

Alternative 15: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{y}}{\frac{y}{x}} + \frac{z}{t} \cdot \frac{z}{t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (/ x y) (/ y x)) (* (/ z t) (/ z t))))

C

double code(double x, double y, double z, double t) {
	return ((x / y) / (y / x)) + ((z / t) * (z / t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) / (y / x)) + ((z / t) * (z / t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x / y) / (y / x)) + ((z / t) * (z / t));
}

Python

def code(x, y, z, t):
	return ((x / y) / (y / x)) + ((z / t) * (z / t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x / y) / Float64(y / x)) + Float64(Float64(z / t) * Float64(z / t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / y) / (y / x)) + ((z / t) * (z / t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   2. clear-num N/A

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y}{x}} + \frac{z \cdot z}{t \cdot t} \]

   6. /-lowering-/.f64 82.4

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]

4. Applied egg-rr 82.4%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + \frac{z \cdot z}{t \cdot t} \]
5. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \color{blue}{\frac{z}{t}} \cdot \frac{z}{t} \]

   4. /-lowering-/.f64 99.7

      \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \frac{\frac{x}{y}}{\frac{y}{x}} + \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} \]
7. Add Preprocessing

Alternative 16: 50.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{y \cdot y} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (/ x (* y y))))

C

double code(double x, double y, double z, double t) {
	return x * (x / (y * y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * (x / (y * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * (x / (y * y));
}

Python

def code(x, y, z, t):
	return x * (x / (y * y))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(x / Float64(y * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (x / (y * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 50.8

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

5. Simplified 50.8%

   \[\leadsto \color{blue}{x \cdot \frac{x}{y \cdot y}} \]
6. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0)))

C

double code(double x, double y, double z, double t) {
	return pow((x / y), 2.0) + pow((z / t), 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) ** 2.0d0) + ((z / t) ** 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow((x / y), 2.0) + Math.pow((z / t), 2.0);
}

Python

def code(x, y, z, t):
	return math.pow((x / y), 2.0) + math.pow((z / t), 2.0)

Julia

function code(x, y, z, t)
	return Float64((Float64(x / y) ^ 2.0) + (Float64(z / t) ^ 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / y) ^ 2.0) + ((z / t) ^ 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (pow (/ x y) 2) (pow (/ z t) 2)))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))