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

Percentage Accurate: 66.6% → 99.7%

Time: 10.8s

Alternatives: 14

Speedup: 0.9×

• 49.5% 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: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f64 N/A

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

   6. frac-add N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*r* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

4. Applied rewrites 70.7%

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

5. Taylor expanded in t around inf

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. times-frac N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 2: 95.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{y}{x}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+219}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 0.0)
     (/ 1.0 (* (/ y x) (/ y x)))
     (if (<= t_1 5e+219)
       (+ (/ (* (/ x y) x) y) t_1)
       (fma (/ x (* y y)) x (* (/ z t) (/ z t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / ((y / x) * (y / x));
	} else if (t_1 <= 5e+219) {
		tmp = (((x / y) * x) / y) + t_1;
	} else {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(y / x) * Float64(y / x)));
	elseif (t_1 <= 5e+219)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) / y) + t_1);
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+219], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 0.0

   1. Initial program 71.2%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 71.2

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{{\left(\frac{z}{t}\right)}^{2} + {\left(\frac{x}{y}\right)}^{2}}}} \]

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.7

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

   7. Applied rewrites 94.7%

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

   if 0.0 < (/.f64 (*.f64 z z) (*.f64 t t)) < 5e219

   1. Initial program 71.8%

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

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 93.8

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

   4. Applied rewrites 93.8%

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

   if 5e219 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.5%

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

   3. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 97.4%

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

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-172}:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{y}{x}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;t\_1 + \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-172)
     (/ 1.0 (* (/ y x) (/ y x)))
     (if (<= t_1 2e+297) (+ t_1 (/ (* x x) (* y y))) (/ (/ z t) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-172) {
		tmp = 1.0 / ((y / x) * (y / x));
	} else if (t_1 <= 2e+297) {
		tmp = t_1 + ((x * x) / (y * y));
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) / (t * t)
    if (t_1 <= 2d-172) then
        tmp = 1.0d0 / ((y / x) * (y / x))
    else if (t_1 <= 2d+297) then
        tmp = t_1 + ((x * x) / (y * y))
    else
        tmp = (z / t) / (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-172) {
		tmp = 1.0 / ((y / x) * (y / x));
	} else if (t_1 <= 2e+297) {
		tmp = t_1 + ((x * x) / (y * y));
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 2e-172:
		tmp = 1.0 / ((y / x) * (y / x))
	elif t_1 <= 2e+297:
		tmp = t_1 + ((x * x) / (y * y))
	else:
		tmp = (z / t) / (t / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e-172)
		tmp = Float64(1.0 / Float64(Float64(y / x) * Float64(y / x)));
	elseif (t_1 <= 2e+297)
		tmp = Float64(t_1 + Float64(Float64(x * x) / Float64(y * y)));
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 2e-172)
		tmp = 1.0 / ((y / x) * (y / x));
	elseif (t_1 <= 2e+297)
		tmp = t_1 + ((x * x) / (y * y));
	else
		tmp = (z / t) / (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-172], N[(1.0 / N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+297], N[(t$95$1 + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.0000000000000001e-172

   1. Initial program 69.6%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 69.6

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{{\left(\frac{z}{t}\right)}^{2} + {\left(\frac{x}{y}\right)}^{2}}}} \]

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if 2.0000000000000001e-172 < (/.f64 (*.f64 z z) (*.f64 t t)) < 2e297

   1. Initial program 82.1%

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

   if 2e297 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 54.5%

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

   3. Taylor expanded in t 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. 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 87.5

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 87.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-172}:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{y}{x}}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+150}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (/ z t) (/ z t))))
   (if (<= t_1 2e+150) t_2 (if (<= t_1 INFINITY) (* (/ x (* y y)) x) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) * (z / t);
	double tmp;
	if (t_1 <= 2e+150) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x / (y * y)) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) * (z / t);
	double tmp;
	if (t_1 <= 2e+150) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x / (y * y)) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = (z / t) * (z / t)
	tmp = 0
	if t_1 <= 2e+150:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (x / (y * y)) * x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(Float64(z / t) * Float64(z / t))
	tmp = 0.0
	if (t_1 <= 2e+150)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x / Float64(y * y)) * x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = (z / t) * (z / t);
	tmp = 0.0;
	if (t_1 <= 2e+150)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (x / (y * y)) * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+150], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999996e150 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 56.8%

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

   3. Taylor expanded in t 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. 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 80.9

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

   5. Applied rewrites 80.9%

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

   if 1.99999999999999996e150 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 77.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.7

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

   10. Applied rewrites 88.7%

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

   12. Applied rewrites 88.4%

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

Alternative 5: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \frac{z}{t \cdot t} \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 2e+253)
   (fma (/ x (* y y)) x (* (/ z t) (/ z t)))
   (fma (/ (/ x y) y) x (* (/ z (* t t)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 2e+253) {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	} else {
		tmp = fma(((x / y) / y), x, ((z / (t * t)) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 2e+253)
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	else
		tmp = fma(Float64(Float64(x / y) / y), x, Float64(Float64(z / Float64(t * t)) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 2e+253], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x + N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e253

   1. Initial program 72.4%

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

   3. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 97.7%

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

   if 1.9999999999999999e253 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 54.7%

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

   3. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 94.4

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

   5. Applied rewrites 94.4%

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

   7. Applied rewrites 91.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, \frac{z}{t \cdot t} \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-172}:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e-172)
   (/ 1.0 (* (/ y x) (/ y x)))
   (fma (/ x (* y y)) x (* (/ z t) (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e-172) {
		tmp = 1.0 / ((y / x) * (y / x));
	} else {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 2e-172)
		tmp = Float64(1.0 / Float64(Float64(y / x) * Float64(y / x)));
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e-172], N[(1.0 / N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2.0000000000000001e-172

   1. Initial program 69.6%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 69.6

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{{\left(\frac{z}{t}\right)}^{2} + {\left(\frac{x}{y}\right)}^{2}}}} \]

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if 2.0000000000000001e-172 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 61.0%

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

   3. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 98.6

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

   5. Applied rewrites 98.6%

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

   7. Applied rewrites 94.9%

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

Alternative 7: 72.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{x}{y \cdot y} \cdot x\\ \mathbf{if}\;t\_1 \leq 1.7 \cdot 10^{-129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (* (/ x (* y y)) x)))
   (if (<= t_1 1.7e-129) t_2 (if (<= t_1 INFINITY) t_1 t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = (x / (y * y)) * x;
	double tmp;
	if (t_1 <= 1.7e-129) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = (x / (y * y)) * x;
	double tmp;
	if (t_1 <= 1.7e-129) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = (x / (y * y)) * x
	tmp = 0
	if t_1 <= 1.7e-129:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	t_2 = Float64(Float64(x / Float64(y * y)) * x)
	tmp = 0.0
	if (t_1 <= 1.7e-129)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	t_2 = (x / (y * y)) * x;
	tmp = 0.0;
	if (t_1 <= 1.7e-129)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, 1.7e-129], t$95$2, If[LessEqual[t$95$1, Infinity], t$95$1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.70000000000000007e-129 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 76.0

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

   10. Applied rewrites 76.0%

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

   12. Applied rewrites 64.2%

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

   if 1.70000000000000007e-129 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 72.1%

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

   3. Taylor expanded in t 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. 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 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 77.1%

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

Alternative 8: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 0.01:\\ \;\;\;\;\frac{1}{\frac{y}{x} \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 0.01)
   (/ 1.0 (* (/ y x) (/ y x)))
   (/ (/ z t) (/ t z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 0.01) {
		tmp = 1.0 / ((y / x) * (y / x));
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) / (t * t)) <= 0.01d0) then
        tmp = 1.0d0 / ((y / x) * (y / x))
    else
        tmp = (z / t) / (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 0.01) {
		tmp = 1.0 / ((y / x) * (y / x));
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 0.01:
		tmp = 1.0 / ((y / x) * (y / x))
	else:
		tmp = (z / t) / (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 0.01)
		tmp = Float64(1.0 / Float64(Float64(y / x) * Float64(y / x)));
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 0.01)
		tmp = 1.0 / ((y / x) * (y / x));
	else
		tmp = (z / t) / (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 0.01], N[(1.0 / N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 0.0100000000000000002

   1. Initial program 71.9%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 71.9

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{{\left(\frac{z}{t}\right)}^{2} + {\left(\frac{x}{y}\right)}^{2}}}} \]

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.3%

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

   3. Taylor expanded in t 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. 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 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

Alternative 9: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 0.01:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 0.01) (* (/ x y) (/ x y)) (/ (/ z t) (/ t z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 0.01) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) / (t * t)) <= 0.01d0) then
        tmp = (x / y) * (x / y)
    else
        tmp = (z / t) / (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 0.01) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 0.01:
		tmp = (x / y) * (x / y)
	else:
		tmp = (z / t) / (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 0.01)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 0.01)
		tmp = (x / y) * (x / y);
	else
		tmp = (z / t) / (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 0.0100000000000000002

   1. Initial program 71.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.3%

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

   3. Taylor expanded in t 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. 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 84.8

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

   5. Applied rewrites 84.8%

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

   7. Applied rewrites 84.8%

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

Alternative 10: 97.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in t around inf

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 96.9

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

5. Applied rewrites 96.9%

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

Alternative 11: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 0.01:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 0.01) (* (/ x y) (/ x y)) (* (/ z t) (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 0.01) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) / (t * t)) <= 0.01d0) then
        tmp = (x / y) * (x / y)
    else
        tmp = (z / t) * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 0.01) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 0.01:
		tmp = (x / y) * (x / y)
	else:
		tmp = (z / t) * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 0.01)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 0.01)
		tmp = (x / y) * (x / y);
	else
		tmp = (z / t) * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 0.0100000000000000002

   1. Initial program 71.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 66.3%

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

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.3%

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

   3. Taylor expanded in t 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. 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 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 12: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 0.01:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 0.01) (* (/ (/ x y) y) x) (* (/ z t) (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 0.01) {
		tmp = ((x / y) / y) * x;
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) / (t * t)) <= 0.01d0) then
        tmp = ((x / y) / y) * x
    else
        tmp = (z / t) * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 0.01) {
		tmp = ((x / y) / y) * x;
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 0.01:
		tmp = ((x / y) / y) * x
	else:
		tmp = (z / t) * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 0.01)
		tmp = Float64(Float64(Float64(x / y) / y) * x);
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 0.01)
		tmp = ((x / y) / y) * x;
	else
		tmp = (z / t) * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 0.0100000000000000002

   1. Initial program 71.9%

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

   3. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.3%

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

   3. Taylor expanded in t 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. 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 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 13: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e-133) (* (/ x (* y y)) x) (* (/ z (* t t)) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e-133) {
		tmp = (x / (y * y)) * x;
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) / (t * t)) <= 2d-133) then
        tmp = (x / (y * y)) * x
    else
        tmp = (z / (t * t)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 2e-133) {
		tmp = (x / (y * y)) * x;
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 2e-133:
		tmp = (x / (y * y)) * x
	else:
		tmp = (z / (t * t)) * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 2e-133)
		tmp = Float64(Float64(x / Float64(y * y)) * x);
	else
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 2e-133)
		tmp = (x / (y * y)) * x;
	else
		tmp = (z / (t * t)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e-133], N[(N[(x / N[(y * y), $MachinePrecision]), $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)) < 2.0000000000000001e-133

   1. Initial program 71.2%

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

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   4. Applied rewrites 67.9%

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

   5. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 87.3

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

   10. Applied rewrites 87.3%

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

   12. Applied rewrites 73.0%

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

   if 2.0000000000000001e-133 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 59.8%

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

   3. Taylor expanded in t 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. 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 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 70.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (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 = (z * z) / (t * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in t 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. 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 62.9

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

5. Applied rewrites 62.9%

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

7. Applied rewrites 50.4%

   \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} \]
8. 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 2024270 
(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))))