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

Percentage Accurate: 67.1% → 97.9%

Time: 9.0s

Alternatives: 12

Speedup: 0.8×

• 48.5% 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: 67.1% 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: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.6 \cdot 10^{-169}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{z}{\left(-t\right) \cdot t}, {\left(\frac{x}{y\_m}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y\_m}}{y\_m}, x, {\left(\frac{z}{t}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= y_m 2.6e-169)
   (fma (- z) (/ z (* (- t) t)) (pow (/ x y_m) 2.0))
   (fma (/ (/ x y_m) y_m) x (pow (/ z t) 2.0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (y_m <= 2.6e-169) {
		tmp = fma(-z, (z / (-t * t)), pow((x / y_m), 2.0));
	} else {
		tmp = fma(((x / y_m) / y_m), x, pow((z / t), 2.0));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (y_m <= 2.6e-169)
		tmp = fma(Float64(-z), Float64(z / Float64(Float64(-t) * t)), (Float64(x / y_m) ^ 2.0));
	else
		tmp = fma(Float64(Float64(x / y_m) / y_m), x, (Float64(z / t) ^ 2.0));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[y$95$m, 2.6e-169], N[((-z) * N[(z / N[((-t) * t), $MachinePrecision]), $MachinePrecision] + N[Power[N[(x / y$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.60000000000000014e-169

   1. Initial program 63.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 67.0

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 N/A

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

      22. lower-/.f64 89.3

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

   4. Applied rewrites 89.3%

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

   if 2.60000000000000014e-169 < y

   1. Initial program 69.4%

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

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 78.0

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, {\left(\frac{z}{t}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{+292}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{z}{\left(-t\right) \cdot t}, {\left(\frac{x}{y\_m}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{x}{\left(-y\_m\right) \cdot y\_m}, {\left(\frac{z}{t}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 1e+292)
   (fma (- z) (/ z (* (- t) t)) (pow (/ x y_m) 2.0))
   (fma (- x) (/ x (* (- y_m) y_m)) (pow (/ z t) 2.0))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 1e+292) {
		tmp = fma(-z, (z / (-t * t)), pow((x / y_m), 2.0));
	} else {
		tmp = fma(-x, (x / (-y_m * y_m)), pow((z / t), 2.0));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 1e+292)
		tmp = fma(Float64(-z), Float64(z / Float64(Float64(-t) * t)), (Float64(x / y_m) ^ 2.0));
	else
		tmp = fma(Float64(-x), Float64(x / Float64(Float64(-y_m) * y_m)), (Float64(z / t) ^ 2.0));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 1e+292], N[((-z) * N[(z / N[((-t) * t), $MachinePrecision]), $MachinePrecision] + N[Power[N[(x / y$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[((-x) * N[(x / N[((-y$95$m) * y$95$m), $MachinePrecision]), $MachinePrecision] + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1e292

   1. Initial program 65.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 66.8

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. times-frac N/A

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

      20. pow2 N/A

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

      21. lower-pow.f64 N/A

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

      22. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

   if 1e292 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 66.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. sqr-neg-rev N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 68.9

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. times-frac N/A

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

      19. pow2 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-/.f64 98.1

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

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{x}{\left(-y\right) \cdot y}, {\left(\frac{z}{t}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{+292}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{y\_m} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{x}{\left(-y\_m\right) \cdot y\_m}, {\left(\frac{z}{t}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e+292)
     (+ (* (/ x y_m) (/ x y_m)) t_1)
     (fma (- x) (/ x (* (- y_m) y_m)) (pow (/ z t) 2.0)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+292) {
		tmp = ((x / y_m) * (x / y_m)) + t_1;
	} else {
		tmp = fma(-x, (x / (-y_m * y_m)), pow((z / t), 2.0));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e+292)
		tmp = Float64(Float64(Float64(x / y_m) * Float64(x / y_m)) + t_1);
	else
		tmp = fma(Float64(-x), Float64(x / Float64(Float64(-y_m) * y_m)), (Float64(z / t) ^ 2.0));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+292], N[(N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[((-x) * N[(x / N[((-y$95$m) * y$95$m), $MachinePrecision]), $MachinePrecision] + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1e292

   1. Initial program 65.5%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.4

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

   4. Applied rewrites 93.4%

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

   if 1e292 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 66.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. sqr-neg-rev N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 68.9

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. times-frac N/A

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

      19. pow2 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-/.f64 98.1

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

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{x}{\left(-y\right) \cdot y}, {\left(\frac{z}{t}\right)}^{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y\_m \cdot y\_m}\\ \mathbf{if}\;t\_1 \leq 10^{-318}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y\_m}}{y\_m} \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y_m y_m))))
   (if (<= t_1 1e-318)
     (* (/ z t) (/ z t))
     (if (<= t_1 5e+256) (fma (/ z (* t t)) z t_1) (* (/ (/ x y_m) y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (x * x) / (y_m * y_m);
	double tmp;
	if (t_1 <= 1e-318) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= 5e+256) {
		tmp = fma((z / (t * t)), z, t_1);
	} else {
		tmp = ((x / y_m) / y_m) * x;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y_m * y_m))
	tmp = 0.0
	if (t_1 <= 1e-318)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	elseif (t_1 <= 5e+256)
		tmp = fma(Float64(z / Float64(t * t)), z, t_1);
	else
		tmp = Float64(Float64(Float64(x / y_m) / y_m) * x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-318], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+256], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + t$95$1), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.9999875e-319

   1. Initial program 67.0%

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

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 70.5

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   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. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 92.7

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

   7. Applied rewrites 92.7%

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

   if 9.9999875e-319 < (/.f64 (*.f64 x x) (*.f64 y y)) < 5.00000000000000015e256

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 94.7%

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

   9. Applied rewrites 90.7%

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

   if 5.00000000000000015e256 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 55.8%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.0

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

   4. Applied rewrites 88.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{x}{y} \cdot {\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + \frac{z \cdot z}{t \cdot t} \]

      3. sqrt-pow1 N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 16.7

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

   6. Applied rewrites 16.7%

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

   7. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)}\right)}{{y}^{2}} \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({x}^{2}\right)}{{y}^{2}}\right)} \]

      8. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{x}^{2}}{{y}^{2}}\right)\right)}\right) \]

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 77.4

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

   9. Applied rewrites 77.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{-318}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x \cdot x}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y\_m \cdot y\_m}\\ \mathbf{if}\;t\_1 \leq 10^{-318}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y\_m \cdot y\_m} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y\_m}}{y\_m} \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y_m y_m))))
   (if (<= t_1 1e-318)
     (* (/ z t) (/ z t))
     (if (<= t_1 5e+256)
       (fma (/ z (* t t)) z (* (/ x (* y_m y_m)) x))
       (* (/ (/ x y_m) y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (x * x) / (y_m * y_m);
	double tmp;
	if (t_1 <= 1e-318) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= 5e+256) {
		tmp = fma((z / (t * t)), z, ((x / (y_m * y_m)) * x));
	} else {
		tmp = ((x / y_m) / y_m) * x;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y_m * y_m))
	tmp = 0.0
	if (t_1 <= 1e-318)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	elseif (t_1 <= 5e+256)
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(x / Float64(y_m * y_m)) * x));
	else
		tmp = Float64(Float64(Float64(x / y_m) / y_m) * x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-318], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+256], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.9999875e-319

   1. Initial program 67.0%

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

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 70.5

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   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. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 92.7

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

   7. Applied rewrites 92.7%

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

   if 9.9999875e-319 < (/.f64 (*.f64 x x) (*.f64 y y)) < 5.00000000000000015e256

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 93.3%

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

   9. Applied rewrites 90.7%

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

   if 5.00000000000000015e256 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 55.8%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.0

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

   4. Applied rewrites 88.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{x}{y} \cdot {\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + \frac{z \cdot z}{t \cdot t} \]

      3. sqrt-pow1 N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 16.7

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

   6. Applied rewrites 16.7%

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

   7. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)}\right)}{{y}^{2}} \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({x}^{2}\right)}{{y}^{2}}\right)} \]

      8. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{x}^{2}}{{y}^{2}}\right)\right)}\right) \]

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 77.4

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

   9. Applied rewrites 77.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{-318}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{x}{y \cdot y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{y\_m} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{\frac{x}{y\_m}}{y\_m} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-113)
     (+ (* (/ x y_m) (/ x y_m)) t_1)
     (fma (/ (/ z t) t) z (* (/ (/ x y_m) y_m) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-113) {
		tmp = ((x / y_m) * (x / y_m)) + t_1;
	} else {
		tmp = fma(((z / t) / t), z, (((x / y_m) / y_m) * x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e-113)
		tmp = Float64(Float64(Float64(x / y_m) * Float64(x / y_m)) + t_1);
	else
		tmp = fma(Float64(Float64(z / t) / t), z, Float64(Float64(Float64(x / y_m) / y_m) * x));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-113], N[(N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision] * z + N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.99999999999999996e-113

   1. Initial program 63.8%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 91.9

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

   4. Applied rewrites 91.9%

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

   if 1.99999999999999996e-113 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 96.7

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

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{-39} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{\frac{x}{y\_m}}{y\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (or (<= t_1 1e-39) (not (<= t_1 INFINITY)))
     (* (/ (/ x y_m) y_m) x)
     (* (/ z (* t t)) z))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if ((t_1 <= 1e-39) || !(t_1 <= ((double) INFINITY))) {
		tmp = ((x / y_m) / y_m) * x;
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if ((t_1 <= 1e-39) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = ((x / y_m) / y_m) * x;
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if (t_1 <= 1e-39) or not (t_1 <= math.inf):
		tmp = ((x / y_m) / y_m) * x
	else:
		tmp = (z / (t * t)) * z
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if ((t_1 <= 1e-39) || !(t_1 <= Inf))
		tmp = Float64(Float64(Float64(x / y_m) / y_m) * x);
	else
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if ((t_1 <= 1e-39) || ~((t_1 <= Inf)))
		tmp = ((x / y_m) / y_m) * x;
	else
		tmp = (z / (t * t)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 1e-39], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.99999999999999929e-40 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 53.5%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{x}{y} \cdot {\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + \frac{z \cdot z}{t \cdot t} \]

      3. sqrt-pow1 N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 24.5

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

   6. Applied rewrites 24.5%

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

   7. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)}\right)}{{y}^{2}} \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({x}^{2}\right)}{{y}^{2}}\right)} \]

      8. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{x}^{2}}{{y}^{2}}\right)\right)}\right) \]

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 71.5

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

   9. Applied rewrites 71.5%

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

   if 9.99999999999999929e-40 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 80.7%

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

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.3

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

   5. Applied rewrites 88.3%

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

   7. Applied rewrites 86.2%

      \[\leadsto \frac{z}{t \cdot t} \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{-39} \lor \neg \left(\frac{z \cdot z}{t \cdot t} \leq \infty\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{x}{y\_m} \cdot \frac{x}{y\_m} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{x}{y\_m \cdot y\_m} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e+227)
     (+ (* (/ x y_m) (/ x y_m)) t_1)
     (fma (/ (/ z t) t) z (* (/ x (* y_m y_m)) x)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e+227) {
		tmp = ((x / y_m) * (x / y_m)) + t_1;
	} else {
		tmp = fma(((z / t) / t), z, ((x / (y_m * y_m)) * x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e+227)
		tmp = Float64(Float64(Float64(x / y_m) * Float64(x / y_m)) + t_1);
	else
		tmp = fma(Float64(Float64(z / t) / t), z, Float64(Float64(x / Float64(y_m * y_m)) * x));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+227], N[(N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision] * z + N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.0000000000000002e227

   1. Initial program 66.4%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.2

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

   4. Applied rewrites 93.2%

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

   if 2.0000000000000002e227 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 95.9

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

   5. Applied rewrites 95.9%

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

   7. Applied rewrites 95.0%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{x}{y \cdot y} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y\_m \cdot y\_m} \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{x}{y\_m \cdot y\_m} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{\frac{x}{y\_m}}{y\_m} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= (/ (* x x) (* y_m y_m)) 2e+295)
   (fma (/ (/ z t) t) z (* (/ x (* y_m y_m)) x))
   (fma (/ z (* t t)) z (* (/ (/ x y_m) y_m) x))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((x * x) / (y_m * y_m)) <= 2e+295) {
		tmp = fma(((z / t) / t), z, ((x / (y_m * y_m)) * x));
	} else {
		tmp = fma((z / (t * t)), z, (((x / y_m) / y_m) * x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y_m * y_m)) <= 2e+295)
		tmp = fma(Float64(Float64(z / t) / t), z, Float64(Float64(x / Float64(y_m * y_m)) * x));
	else
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(Float64(x / y_m) / y_m) * x));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], 2e+295], N[(N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision] * z + N[(N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 2e295

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 94.2

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

   5. Applied rewrites 94.2%

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

   7. Applied rewrites 92.3%

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

   if 2e295 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 56.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 90.2%

      \[\leadsto \mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{\frac{x}{y}}{y} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{x}{y \cdot y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y\_m \cdot y\_m} \leq 10^{-318}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{\frac{x}{y\_m}}{y\_m} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= (/ (* x x) (* y_m y_m)) 1e-318)
   (* (/ z t) (/ z t))
   (fma (/ z (* t t)) z (* (/ (/ x y_m) y_m) x))))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((x * x) / (y_m * y_m)) <= 1e-318) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = fma((z / (t * t)), z, (((x / y_m) / y_m) * x));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y_m * y_m)) <= 1e-318)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = fma(Float64(z / Float64(t * t)), z, Float64(Float64(Float64(x / y_m) / y_m) * x));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], 1e-318], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.9999875e-319

   1. Initial program 67.0%

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

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 70.5

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   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. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 92.7

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

   7. Applied rewrites 92.7%

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

   if 9.9999875e-319 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 64.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 92.8

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 89.9%

      \[\leadsto \mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{\frac{x}{y}}{y} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10^{-318}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y\_m \cdot y\_m} \leq 6 \cdot 10^{+122}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y\_m}}{y\_m} \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t)
 :precision binary64
 (if (<= (/ (* x x) (* y_m y_m)) 6e+122)
   (* (/ z t) (/ z t))
   (* (/ (/ x y_m) y_m) x)))

C

y_m = fabs(y);
double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((x * x) / (y_m * y_m)) <= 6e+122) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = ((x / y_m) / y_m) * x;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y_m * y_m)) <= 6d+122) then
        tmp = (z / t) * (z / t)
    else
        tmp = ((x / y_m) / y_m) * x
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	double tmp;
	if (((x * x) / (y_m * y_m)) <= 6e+122) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = ((x / y_m) / y_m) * x;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	tmp = 0
	if ((x * x) / (y_m * y_m)) <= 6e+122:
		tmp = (z / t) * (z / t)
	else:
		tmp = ((x / y_m) / y_m) * x
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y_m * y_m)) <= 6e+122)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(Float64(x / y_m) / y_m) * x);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z, t)
	tmp = 0.0;
	if (((x * x) / (y_m * y_m)) <= 6e+122)
		tmp = (z / t) * (z / t);
	else
		tmp = ((x / y_m) / y_m) * x;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], 6e+122], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 5.99999999999999972e122

   1. Initial program 72.6%

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

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 75.8

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

   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. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.2

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

   7. Applied rewrites 87.2%

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

   if 5.99999999999999972e122 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 58.2%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{x}{y} \cdot {\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + \frac{z \cdot z}{t \cdot t} \]

      3. sqrt-pow1 N/A

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + \frac{z \cdot z}{t \cdot t} \]

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 18.3

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

   6. Applied rewrites 18.3%

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

   7. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)}\right)}{{y}^{2}} \]

      7. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({x}^{2}\right)}{{y}^{2}}\right)} \]

      8. distribute-neg-frac N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{{x}^{2}}{{y}^{2}}\right)\right)}\right) \]

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 75.7

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

   9. Applied rewrites 75.7%

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

Alternative 12: 53.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{z}{t \cdot t} \cdot z \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z t) :precision binary64 (* (/ z (* t t)) z))

C

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

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z, double t) {
	return (z / (t * t)) * z;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z, t):
	return (z / (t * t)) * z

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_, t_] := N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 60.4

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

5. Applied rewrites 60.4%

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

7. Applied rewrites 53.4%

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

8. Final simplification 53.4%

   \[\leadsto \frac{z}{t \cdot t} \cdot z \]
9. 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 2024337 
(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))))