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

Percentage Accurate: 67.0% → 99.7%

Time: 10.8s

Alternatives: 12

Speedup: 0.8×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 81.5

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. pow2 N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 94.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+88}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 0.0)
     (* (/ x y) (/ x y))
     (if (<= t_1 1e+88)
       (+ (/ (* (/ x y) x) y) t_1)
       (fma (/ z t) (/ z t) (* (/ x (* y y)) x))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= 1e+88) {
		tmp = (((x / y) * x) / y) + t_1;
	} else {
		tmp = fma((z / t), (z / t), ((x / (y * y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= 1e+88)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) / y) + t_1);
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x / Float64(y * y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+88], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 80.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 24.2

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 22.6%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 97.7

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

   10. Applied rewrites 97.7%

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

   if 0.0 < (/.f64 (*.f64 z z) (*.f64 t t)) < 9.99999999999999959e87

   1. Initial program 67.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 92.1

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

   4. Applied rewrites 92.1%

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

   if 9.99999999999999959e87 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 55.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. +-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 84.7

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 98.2

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

   6. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 98.2

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 95.9

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

   8. Applied rewrites 95.9%

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

Alternative 3: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{t} \cdot \left(\frac{-1}{t} \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e-270)
     (* (/ x y) (/ x y))
     (if (<= t_1 5e+222)
       (fma (/ x (* y y)) x t_1)
       (* (/ (- z) t) (* (/ -1.0 t) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e-270) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= 5e+222) {
		tmp = fma((x / (y * y)), x, t_1);
	} else {
		tmp = (-z / t) * ((-1.0 / t) * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e-270)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= 5e+222)
		tmp = fma(Float64(x / Float64(y * y)), x, t_1);
	else
		tmp = Float64(Float64(Float64(-z) / t) * Float64(Float64(-1.0 / t) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-270], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+222], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[((-z) / t), $MachinePrecision] * N[(N[(-1.0 / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999998e-270

   1. Initial program 80.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 24.1

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

   5. Applied rewrites 24.1%

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 97.0

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

   10. Applied rewrites 97.0%

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

   if 4.9999999999999998e-270 < (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000023e222

   1. Initial program 69.0%

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

   3. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

   7. Applied rewrites 82.0%

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

   9. Applied rewrites 81.2%

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

   if 5.00000000000000023e222 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 53.4%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z \cdot z}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{t} \cdot \left(\frac{-1}{t} \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ t_2 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))) (t_2 (* (/ x y) (/ x y))))
   (if (<= t_1 5e+23) t_2 (if (<= t_1 INFINITY) (* (/ z (* t t)) z) 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 <= 5e+23) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / (t * t)) * z;
	} 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 <= 5e+23) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (z / (t * t)) * z;
	} 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 <= 5e+23:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (z / (t * t)) * z
	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 <= 5e+23)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	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 <= 5e+23)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (z / (t * t)) * z;
	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, 5e+23], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e23 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 63.1%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 30.1

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

   5. Applied rewrites 30.1%

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

   7. Applied rewrites 19.6%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 83.8

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

   10. Applied rewrites 83.8%

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

   if 4.9999999999999999e23 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 72.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 81.8%

      \[\leadsto \frac{\left(-z\right) \cdot -1}{t \cdot t} \cdot \color{blue}{z} \]
   8. Step-by-step derivation

   9. Applied rewrites 81.8%

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

Alternative 5: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e-270)
   (* (/ x y) (/ x y))
   (fma (/ z t) (/ z t) (* (/ x (* y y)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e-270) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = fma((z / t), (z / t), ((x / (y * y)) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e-270)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x / Float64(y * y)) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e-270], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999998e-270

   1. Initial program 80.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 24.1

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

   5. Applied rewrites 24.1%

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 97.0

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

   10. Applied rewrites 97.0%

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

   if 4.9999999999999998e-270 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 80.8

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 97.3

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

   6. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.3

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 91.7

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

   8. Applied rewrites 91.7%

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

Alternative 6: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{z}{t} \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e-270)
   (* (/ x y) (/ x y))
   (fma (/ x (* y y)) x (* (/ z t) (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e-270) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = fma((x / (y * y)), x, ((z / t) * (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e-270)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(z / t) * Float64(z / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e-270], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999998e-270

   1. Initial program 80.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 24.1

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

   5. Applied rewrites 24.1%

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 97.0

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

   10. Applied rewrites 97.0%

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

   if 4.9999999999999998e-270 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 57.8%

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

   3. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 91.7%

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

Alternative 7: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{t} \cdot \left(\frac{-1}{t} \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e+23)
   (* (/ x y) (/ x y))
   (* (/ (- z) t) (* (/ -1.0 t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+23) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (-z / t) * ((-1.0 / 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+23) then
        tmp = (x / y) * (x / y)
    else
        tmp = (-z / t) * (((-1.0d0) / 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+23) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (-z / t) * ((-1.0 / t) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 5e+23:
		tmp = (x / y) * (x / y)
	else:
		tmp = (-z / t) * ((-1.0 / t) * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e+23)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(Float64(Float64(-z) / t) * Float64(Float64(-1.0 / t) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 5e+23)
		tmp = (x / y) * (x / y);
	else
		tmp = (-z / t) * ((-1.0 / 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+23], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[((-z) / t), $MachinePrecision] * N[(N[(-1.0 / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e23

   1. Initial program 78.1%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.5

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

   5. Applied rewrites 25.5%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if 4.9999999999999999e23 < (/.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 t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{t} \cdot \left(\frac{-1}{t} \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e+23) (* (/ x y) (/ x y)) (/ (/ z t) (/ t z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+23) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) / (t * t)) <= 5d+23) then
        tmp = (x / y) * (x / y)
    else
        tmp = (z / t) / (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+23) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) / (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 5e+23:
		tmp = (x / y) * (x / y)
	else:
		tmp = (z / t) / (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e+23)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(Float64(z / t) / Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 5e+23)
		tmp = (x / y) * (x / y);
	else
		tmp = (z / t) / (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e+23], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e23

   1. Initial program 78.1%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.5

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

   5. Applied rewrites 25.5%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if 4.9999999999999999e23 < (/.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 t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.1%

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

Alternative 9: 97.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 81.5

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. pow2 N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/r/ N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 97.1

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

6. Applied rewrites 97.1%

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

Alternative 10: 96.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

3. Taylor expanded in t around inf

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 96.8

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

5. Applied rewrites 96.8%

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

Alternative 11: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 5e+23) (* (/ x y) (/ x y)) (* (/ z t) (/ z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+23) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * z) / (t * t)) <= 5d+23) then
        tmp = (x / y) * (x / y)
    else
        tmp = (z / t) * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) / (t * t)) <= 5e+23) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * z) / (t * t)) <= 5e+23:
		tmp = (x / y) * (x / y)
	else:
		tmp = (z / t) * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) / Float64(t * t)) <= 5e+23)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * z) / (t * t)) <= 5e+23)
		tmp = (x / y) * (x / y);
	else
		tmp = (z / t) * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 5e+23], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 4.9999999999999999e23

   1. Initial program 78.1%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 25.5

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

   5. Applied rewrites 25.5%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in t around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 91.9

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

   10. Applied rewrites 91.9%

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

   if 4.9999999999999999e23 < (/.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 t around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 12: 52.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 52.2

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

5. Applied rewrites 52.2%

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

7. Applied rewrites 45.9%

   \[\leadsto \frac{\left(-z\right) \cdot -1}{t \cdot t} \cdot \color{blue}{z} \]
8. Step-by-step derivation

9. Applied rewrites 45.9%

   \[\leadsto \frac{z}{t \cdot t} \cdot z \]
10. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024268 
(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))))