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

Percentage Accurate: 65.9% → 99.7%

Time: 12.1s

Alternatives: 15

Speedup: 0.8×

• 48.0% 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: 65.9% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 76.7

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

4. Applied rewrites 76.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 10^{-290}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, x \cdot \frac{x}{y \cdot y}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 1e-290)
     (/ (/ z t) (/ t z))
     (if (<= t_1 5e+295)
       (fma (/ z (* t t)) z (* x (/ x (* y y))))
       (if (<= t_1 INFINITY)
         (* (/ x y) (* x (/ 1.0 y)))
         (fma (/ x y) (/ x y) (/ (* z z) (* t t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 1e-290) {
		tmp = (z / t) / (t / z);
	} else if (t_1 <= 5e+295) {
		tmp = fma((z / (t * t)), z, (x * (x / (y * y))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / y) * (x * (1.0 / y));
	} else {
		tmp = fma((x / y), (x / y), ((z * z) / (t * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 1e-290)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	elseif (t_1 <= 5e+295)
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(x * Float64(x / Float64(y * y))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / y) * Float64(x * Float64(1.0 / y)));
	else
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(z * z) / Float64(t * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-290], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+295], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x / y), $MachinePrecision] * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.0000000000000001e-290

   1. Initial program 71.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-add N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. unswap-sqr N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 41.9

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

   4. Applied rewrites 41.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   6. 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. lower-*.f64 N/A

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

      4. lower-/.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. lower-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   9. Applied rewrites 93.3%

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

   if 1.0000000000000001e-290 < (/.f64 (*.f64 x x) (*.f64 y y)) < 4.99999999999999991e295

   1. Initial program 78.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*r/ N/A

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

      17. associate-*l/ N/A

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

      18. lower-fma.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. associate-/r* N/A

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

      21. lower-/.f64 N/A

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

      22. lower-*.f64 86.3

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

      23. lift-/.f64 N/A

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

   8. Applied rewrites 86.6%

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

   if 4.99999999999999991e295 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. lower-*.f64 N/A

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

      4. lower-/.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. lower-*.f64 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 94.5%

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

   9. Applied rewrites 94.5%

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

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

   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. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.5

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

   4. Applied rewrites 81.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{-290}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, x \cdot \frac{x}{y \cdot y}\right)\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.7% 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 2024229 
(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))))