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

Percentage Accurate: 66.4% → 96.3%

Time: 11.5s

Alternatives: 17

Speedup: 0.8×

• 48.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{\frac{z \cdot z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e+218)
   (+ (* (/ x y) (/ x y)) (/ (/ (* z z) t) t))
   (+ (* x (/ x (* y y))) (/ (/ z t) (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+218) {
		tmp = ((x / y) * (x / y)) + (((z * z) / t) / t);
	} else {
		tmp = (x * (x / (y * y))) + ((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)) <= 5d+218) then
        tmp = ((x / y) * (x / y)) + (((z * z) / t) / t)
    else
        tmp = (x * (x / (y * y))) + ((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)) <= 5e+218) {
		tmp = ((x / y) * (x / y)) + (((z * z) / t) / t);
	} else {
		tmp = (x * (x / (y * y))) + ((z / t) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 5e+218:
		tmp = ((x / y) * (x / y)) + (((z * z) / t) / t)
	else:
		tmp = (x * (x / (y * y))) + ((z / t) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e+218)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + Float64(Float64(Float64(z * z) / t) / t));
	else
		tmp = Float64(Float64(x * Float64(x / Float64(y * y))) + 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)) <= 5e+218)
		tmp = ((x / y) * (x / y)) + (((z * z) / t) / t);
	else
		tmp = (x * (x / (y * y))) + ((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], 5e+218], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.99999999999999983e218

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.1

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

   4. Applied rewrites 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.6

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

   6. Applied rewrites 98.6%

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

   if 4.99999999999999983e218 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 61.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 90.9

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

   4. Applied rewrites 90.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 96.3

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

   6. Applied rewrites 96.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{\frac{z \cdot z}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\frac{z}{t}}{t}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 0.0)
     (/ (/ z t) (/ t z))
     (if (<= t_1 2e+283) (fma z (/ (/ z t) t) t_1) (* (/ x y) (/ x y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (z / t) / (t / z);
	} else if (t_1 <= 2e+283) {
		tmp = fma(z, ((z / t) / t), t_1);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	elseif (t_1 <= 2e+283)
		tmp = fma(z, Float64(Float64(z / t) / t), t_1);
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+283], N[(z * N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 0.0

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.3

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.9

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

   7. Applied rewrites 70.9%

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

   9. Applied rewrites 95.6%

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

   if 0.0 < (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e283

   1. Initial program 79.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 94.5

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

   6. Applied rewrites 94.5%

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

   if 1.99999999999999991e283 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 84.9%

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

Alternative 3: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 10^{-305}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t \cdot t}, z, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 1e-305)
     (/ (/ z t) (/ t z))
     (if (<= t_1 2e+283) (fma (/ z (* t t)) z t_1) (* (/ x y) (/ x y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 1e-305) {
		tmp = (z / t) / (t / z);
	} else if (t_1 <= 2e+283) {
		tmp = fma((z / (t * t)), z, t_1);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 1e-305)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	elseif (t_1 <= 2e+283)
		tmp = fma(Float64(z / Float64(t * t)), z, t_1);
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-305], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+283], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z + t$95$1), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.99999999999999996e-306

   1. Initial program 68.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 76.7

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.2

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 95.4%

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

   if 9.99999999999999996e-306 < (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e283

   1. Initial program 80.3%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.6

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

   4. Applied rewrites 87.6%

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

   if 1.99999999999999991e283 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 84.9%

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

Alternative 4: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-274}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+274}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z \cdot z}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 5e-274)
     (/ (/ z t) (/ t z))
     (if (<= t_1 2e+274)
       (fma (/ x (* y y)) x (/ (* z z) (* t t)))
       (* (/ x y) (/ x y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 5e-274) {
		tmp = (z / t) / (t / z);
	} else if (t_1 <= 2e+274) {
		tmp = fma((x / (y * y)), x, ((z * z) / (t * t)));
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 5e-274)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	elseif (t_1 <= 2e+274)
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z * z) / Float64(t * t)));
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-274], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+274], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 5e-274

   1. Initial program 68.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 70.2

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 94.6%

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

   if 5e-274 < (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999984e274

   1. Initial program 82.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.9

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

   4. Applied rewrites 82.9%

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

   if 1.99999999999999984e274 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 61.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

   7. Applied rewrites 85.0%

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

Alternative 5: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{\frac{z \cdot z}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 2e+283)
     (fma (/ z t) (/ z t) t_1)
     (+ (* (/ x y) (/ x y)) (/ (/ (* z z) t) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 2e+283) {
		tmp = fma((z / t), (z / t), t_1);
	} else {
		tmp = ((x / y) * (x / y)) + (((z * z) / t) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 2e+283)
		tmp = fma(Float64(z / t), Float64(z / t), t_1);
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + Float64(Float64(Float64(z * z) / t) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+283], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e283

   1. Initial program 73.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if 1.99999999999999991e283 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.8

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

   6. Applied rewrites 95.8%

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

Alternative 6: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq 10^{+105}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (* (/ x y) (/ x y))))
   (if (<= t_1 1e+105) t_2 (if (<= t_1 INFINITY) (* z (/ z (* t t))) 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) * (x / y);
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (z / (t * t));
	} 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) * (x / y);
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z * (z / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = (x / y) * (x / y)
	tmp = 0
	if t_1 <= 1e+105:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = z * (z / (t * t))
	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 / y) * Float64(x / y))
	tmp = 0.0
	if (t_1 <= 1e+105)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	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) * (x / y);
	tmp = 0.0;
	if (t_1 <= 1e+105)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = z * (z / (t * t));
	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 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+105], t$95$2, If[LessEqual[t$95$1, Infinity], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999994e104 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 56.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 76.5%

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

   if 9.9999999999999994e104 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

Alternative 7: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := x \cdot \frac{\frac{x}{y}}{y}\\ \mathbf{if}\;t\_1 \leq 10^{+105}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (* x (/ (/ x y) y))))
   (if (<= t_1 1e+105) t_2 (if (<= t_1 INFINITY) (* z (/ z (* t t))) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = x * ((x / y) / y);
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (z / (t * t));
	} 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 * ((x / y) / y);
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z * (z / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = x * ((x / y) / y)
	tmp = 0
	if t_1 <= 1e+105:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = z * (z / (t * t))
	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(x * Float64(Float64(x / y) / y))
	tmp = 0.0
	if (t_1 <= 1e+105)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	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 * ((x / y) / y);
	tmp = 0.0;
	if (t_1 <= 1e+105)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = z * (z / (t * t));
	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[(x * N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+105], t$95$2, If[LessEqual[t$95$1, Infinity], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999994e104 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 56.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 71.4%

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

   if 9.9999999999999994e104 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

Alternative 8: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{+105}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e+105)
     (* x (/ x (* y y)))
     (if (<= t_1 INFINITY) (* z (/ z (* t t))) (* x (* x (/ 1.0 (* y y))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = x * (x / (y * y));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (z / (t * t));
	} else {
		tmp = x * (x * (1.0 / (y * y)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = x * (x / (y * y));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z * (z / (t * t));
	} else {
		tmp = x * (x * (1.0 / (y * y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 1e+105:
		tmp = x * (x / (y * y))
	elif t_1 <= math.inf:
		tmp = z * (z / (t * t))
	else:
		tmp = x * (x * (1.0 / (y * y)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e+105)
		tmp = Float64(x * Float64(x / Float64(y * y)));
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	else
		tmp = Float64(x * Float64(x * Float64(1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 1e+105)
		tmp = x * (x / (y * y));
	elseif (t_1 <= Inf)
		tmp = z * (z / (t * t));
	else
		tmp = x * (x * (1.0 / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+105], N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999994e104

   1. Initial program 74.7%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 9.9999999999999994e104 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 45.2

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

   5. Applied rewrites 45.2%

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

   7. Applied rewrites 45.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{+105}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{z}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 2e+283)
     (fma (/ z t) (/ z t) t_1)
     (+ (* (/ x y) (/ x y)) (* z (/ z (* t t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 2e+283) {
		tmp = fma((z / t), (z / t), t_1);
	} else {
		tmp = ((x / y) * (x / y)) + (z * (z / (t * t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 2e+283)
		tmp = fma(Float64(z / t), Float64(z / t), t_1);
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + Float64(z * Float64(z / Float64(t * t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+283], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999991e283

   1. Initial program 73.1%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if 1.99999999999999991e283 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.7

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

   4. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.8

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

   6. Applied rewrites 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 95.1

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

   8. Applied rewrites 95.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + z \cdot \frac{z}{t \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+218)
     (fma (/ x y) (/ x y) t_1)
     (fma (/ z t) (/ z t) (/ (* x x) (* y y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+218) {
		tmp = fma((x / y), (x / y), t_1);
	} else {
		tmp = fma((z / t), (z / t), ((x * x) / (y * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+218)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x * x) / Float64(y * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+218], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.99999999999999983e218

   1. Initial program 74.3%

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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 4.99999999999999983e218 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 61.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 90.9

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

   4. Applied rewrites 90.9%

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

Alternative 11: 89.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 94.0

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

   4. Applied rewrites 94.0%

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

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

   1. Initial program 0.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-/.f64 68.4

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

   6. Applied rewrites 68.4%

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

Alternative 12: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := x \cdot \frac{x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 10^{+105}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (* x (/ x (* y y)))))
   (if (<= t_1 1e+105) t_2 (if (<= t_1 INFINITY) (* z (/ z (* t t))) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double t_2 = x * (x / (y * y));
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = z * (z / (t * t));
	} 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 * (x / (y * y));
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = z * (z / (t * t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	t_2 = x * (x / (y * y))
	tmp = 0
	if t_1 <= 1e+105:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = z * (z / (t * t))
	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(x * Float64(x / Float64(y * y)))
	tmp = 0.0
	if (t_1 <= 1e+105)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(z * Float64(z / Float64(t * t)));
	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 * (x / (y * y));
	tmp = 0.0;
	if (t_1 <= 1e+105)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = z * (z / (t * t));
	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[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+105], t$95$2, If[LessEqual[t$95$1, Infinity], N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999994e104 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 56.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if 9.9999999999999994e104 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

Alternative 13: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5000000000000:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 5000000000000.0)
   (/ (/ z t) (/ t z))
   (* (/ x y) (* x (/ 1.0 y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 5000000000000.0) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = (x / y) * (x * (1.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 5000000000000.0d0) then
        tmp = (z / t) / (t / z)
    else
        tmp = (x / y) * (x * (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 5000000000000.0) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = (x / y) * (x * (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 5000000000000.0:
		tmp = (z / t) / (t / z)
	else:
		tmp = (x / y) * (x * (1.0 / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 5000000000000.0)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	else
		tmp = Float64(Float64(x / y) * Float64(x * Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 5000000000000.0)
		tmp = (z / t) / (t / z);
	else
		tmp = (x / y) * (x * (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 5000000000000.0], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 5e12

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

   9. Applied rewrites 88.6%

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

   if 5e12 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 65.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 79.8%

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

   9. Applied rewrites 79.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5000000000000:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5000000000000:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 5000000000000.0)
   (* (/ z t) (/ z t))
   (* (/ x y) (* x (/ 1.0 y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 5000000000000.0) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (x / y) * (x * (1.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 5000000000000.0d0) then
        tmp = (z / t) * (z / t)
    else
        tmp = (x / y) * (x * (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 5000000000000.0) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (x / y) * (x * (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 5000000000000.0:
		tmp = (z / t) * (z / t)
	else:
		tmp = (x / y) * (x * (1.0 / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 5000000000000.0)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(x / y) * Float64(x * Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 5000000000000.0)
		tmp = (z / t) * (z / t);
	else
		tmp = (x / y) * (x * (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 5000000000000.0], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 5e12

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

   9. Applied rewrites 88.5%

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

   if 5e12 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 65.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 79.8%

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

   9. Applied rewrites 79.8%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5000000000000:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(x \cdot \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5000000000000:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 5000000000000.0)
   (* (/ z t) (/ z t))
   (* (/ x y) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 5000000000000.0) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 5000000000000.0d0) then
        tmp = (z / t) * (z / t)
    else
        tmp = (x / y) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 5000000000000.0) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (x / y) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 5000000000000.0:
		tmp = (z / t) * (z / t)
	else:
		tmp = (x / y) * (x / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 5000000000000.0)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 5000000000000.0)
		tmp = (z / t) * (z / t);
	else
		tmp = (x / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 5000000000000.0], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 5e12

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

   9. Applied rewrites 88.5%

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

   if 5e12 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 65.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 79.8%

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

Alternative 16: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{+105}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{\frac{z}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 1e+105) (* (/ x y) (/ x y)) (* z (/ (/ z t) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 1e+105) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = z * ((z / t) / 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)) <= 1d+105) then
        tmp = (x / y) * (x / y)
    else
        tmp = z * ((z / t) / 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)) <= 1e+105) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = z * ((z / t) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 1e+105:
		tmp = (x / y) * (x / y)
	else:
		tmp = z * ((z / t) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 1e+105)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(z * Float64(Float64(z / t) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 1e+105)
		tmp = (x / y) * (x / y);
	else
		tmp = z * ((z / t) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 1e+105], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 9.9999999999999994e104

   1. Initial program 74.7%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 84.5%

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

   if 9.9999999999999994e104 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 62.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.6

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

   9. Applied rewrites 76.9%

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

Alternative 17: 52.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 54.1

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

5. Applied rewrites 54.1%

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

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024234 
(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))))