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

Percentage Accurate: 66.8% → 96.6%

Time: 6.8s

Alternatives: 12

Speedup: 0.8×

• 48.6% 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: 66.8% 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: 96.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 80.1

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. pow2 N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-/.f64 97.1

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

4. Applied rewrites 97.1%

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

Alternative 2: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{y}}{y}, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{\frac{z}{t} \cdot z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 0.0)
     (* (/ x y) (/ x y))
     (if (<= t_1 5e+261)
       (fma (/ (/ x y) y) x t_1)
       (fma (/ x (* y y)) x (/ (* (/ z t) z) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= 5e+261) {
		tmp = fma(((x / y) / y), x, t_1);
	} else {
		tmp = fma((x / (y * y)), x, (((z / t) * z) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= 5e+261)
		tmp = fma(Float64(Float64(x / y) / y), x, t_1);
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(Float64(z / t) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+261], N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 66.6%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.4

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

   4. Applied rewrites 94.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 26.4

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

   6. Applied rewrites 26.4%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 94.4

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

   9. Applied rewrites 94.4%

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

   if 0.0 < (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000001e261

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.9

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 96.9

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

   6. Applied rewrites 96.9%

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

   if 5.0000000000000001e261 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 60.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 67.2

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 96.0

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

   6. Applied rewrites 96.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 3: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-79)
     (* (/ x y) (/ x y))
     (if (<= t_1 2e+212) (fma (/ x (* y y)) x t_1) (* (/ z t) (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 2e-79) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= 2e+212) {
		tmp = fma((x / (y * y)), x, t_1);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e-79)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= 2e+212)
		tmp = fma(Float64(x / Float64(y * y)), x, t_1);
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-79], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+212], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2e-79

   1. Initial program 68.3%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 25.7

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

   6. Applied rewrites 25.7%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 92.9

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

   9. Applied rewrites 92.9%

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

   if 2e-79 < (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999998e212

   1. Initial program 82.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 98.1

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-neg-rev N/A

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

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

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

      14. distribute-rgt-neg-in N/A

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

      15. lift-*.f64 N/A

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

      16. frac-2neg-rev N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 98.1

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

   6. Applied rewrites 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 90.0

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

   8. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. times-frac N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 90.0

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

   10. Applied rewrites 90.0%

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

   if 1.9999999999999998e212 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 60.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 68.0

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 84.1

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

   7. Applied rewrites 84.1%

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

Alternative 4: 94.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_1 + ((z / t) * (z / t));
	} else {
		tmp = fma(((z / t) / t), z, (((x / y) / y) * x));
	}
	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+105)
		tmp = Float64(t_1 + Float64(Float64(z / t) * Float64(z / t)));
	else
		tmp = fma(Float64(Float64(z / t) / t), z, Float64(Float64(Float64(x / y) / y) * x));
	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+105], N[(t$95$1 + N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision] * z + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.9999999999999994e104

   1. Initial program 75.4%

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

      1. lift-/.f64 N/A

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

      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. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 9.9999999999999994e104 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 58.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 95.4%

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

Alternative 5: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-79} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (or (<= t_1 2e-79) (not (<= t_1 INFINITY)))
     (* (/ x y) (/ x y))
     (* (/ z (* t t)) z))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if ((t_1 <= 2e-79) || !(t_1 <= ((double) INFINITY))) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / (t * t)) * z;
	}
	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 <= 2e-79) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = (x / y) * (x / y);
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if (t_1 <= 2e-79) or not (t_1 <= math.inf):
		tmp = (x / y) * (x / y)
	else:
		tmp = (z / (t * t)) * z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if ((t_1 <= 2e-79) || !(t_1 <= Inf))
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if ((t_1 <= 2e-79) || ~((t_1 <= Inf)))
		tmp = (x / y) * (x / y);
	else
		tmp = (z / (t * t)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 2e-79], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2e-79 or +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 53.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 74.2

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

   4. Applied rewrites 74.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 20.0

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

   6. Applied rewrites 20.0%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 83.8

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

   9. Applied rewrites 83.8%

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

   if 2e-79 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 86.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-79} \lor \neg \left(\frac{z \cdot z}{t \cdot t} \leq \infty\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 2e-79)
     (* (/ x y) (/ x y))
     (if (<= t_1 INFINITY) (* (/ z (* t t)) z) (* (/ (/ 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 <= 2e-79) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = ((x / y) / y) * x;
	}
	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 <= 2e-79) {
		tmp = (x / y) * (x / y);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = ((x / y) / y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 2e-79:
		tmp = (x / y) * (x / y)
	elif t_1 <= math.inf:
		tmp = (z / (t * t)) * z
	else:
		tmp = ((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 <= 2e-79)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	else
		tmp = Float64(Float64(Float64(x / y) / y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 2e-79)
		tmp = (x / y) * (x / y);
	elseif (t_1 <= Inf)
		tmp = (z / (t * t)) * z;
	else
		tmp = ((x / y) / y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-79], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2e-79

   1. Initial program 68.3%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 25.7

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

   6. Applied rewrites 25.7%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 92.9

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

   9. Applied rewrites 92.9%

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

   if 2e-79 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 86.3%

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

   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 51.7

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

   9. Applied rewrites 51.7%

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

   10. Taylor expanded in x around -inf

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. mul-1-neg N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. unpow2 N/A

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

       9. associate-/l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-/r* N/A

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

       14. lower-/.f64 N/A

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

       15. lower-/.f64 51.7

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

   12. Applied rewrites 51.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{\frac{z}{t} \cdot z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+261)
     (+ (* (/ x y) (/ x y)) t_1)
     (fma (/ x (* y y)) x (/ (* (/ z t) z) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+261) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = fma((x / (y * y)), x, (((z / t) * z) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+261)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(Float64(z / t) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+261], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000001e261

   1. Initial program 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   if 5.0000000000000001e261 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 60.5%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 67.2

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 96.0

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

   6. Applied rewrites 96.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 8: 89.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, \frac{\frac{z}{t} \cdot z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e-79)
   (* (/ 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)) <= 2e-79) {
		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)) <= 2e-79)
		tmp = Float64(Float64(x / y) * Float64(x / y));
	else
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(Float64(Float64(z / t) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e-79], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 2e-79

   1. Initial program 68.3%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 25.7

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

   6. Applied rewrites 25.7%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 92.9

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

   9. Applied rewrites 92.9%

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

   if 2e-79 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 74.7

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 96.9

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

   6. Applied rewrites 96.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 91.4

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

   8. Applied rewrites 91.4%

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

Alternative 9: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.55 \cdot 10^{-228}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, z \cdot \frac{z}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 2.55e-228)
   (* (/ z t) (/ z t))
   (if (<= t 1.35e+154)
     (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 tmp;
	if (t <= 2.55e-228) {
		tmp = (z / t) * (z / t);
	} else if (t <= 1.35e+154) {
		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)
	tmp = 0.0
	if (t <= 2.55e-228)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	elseif (t <= 1.35e+154)
		tmp = fma(Float64(x / Float64(y * y)), x, Float64(z * Float64(z / Float64(t * t))));
	else
		tmp = Float64(Float64(x / y) * Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 2.55e-228], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+154], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x + N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.5500000000000001e-228

   1. Initial program 68.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.2

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 67.4

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

   7. Applied rewrites 67.4%

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

   if 2.5500000000000001e-228 < t < 1.35000000000000003e154

   1. Initial program 68.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 85.8

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-neg-rev N/A

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

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

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

      14. distribute-rgt-neg-in N/A

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

      15. lift-*.f64 N/A

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

      16. frac-2neg-rev N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 84.4

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

   8. Applied rewrites 84.4%

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

   if 1.35000000000000003e154 < t

   1. Initial program 52.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 17.2

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

   6. Applied rewrites 17.2%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 85.8

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

   9. Applied rewrites 85.8%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.55 \cdot 10^{-228}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y \cdot y}, x, z \cdot \frac{z}{t \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.7% accurate, 0.8× speedup

Math

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

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(Float64(z / t) * z) / t))
end

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 80.1

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. pow2 N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-/.f64 97.1

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

4. Applied rewrites 97.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. times-frac N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lift-/.f64 N/A

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

   12. lower-*.f64 94.7

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

6. Applied rewrites 94.7%

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

Alternative 11: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* z z) (* t t)) 2e-79) (* (/ 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)) <= 2e-79) {
		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)) <= 2d-79) 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)) <= 2e-79) {
		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)) <= 2e-79:
		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)) <= 2e-79)
		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)) <= 2e-79)
		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], 2e-79], 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)) < 2e-79

   1. Initial program 68.3%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 25.7

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

   6. Applied rewrites 25.7%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. remove-double-neg N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 92.9

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

   9. Applied rewrites 92.9%

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

   if 2e-79 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 74.7

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 12: 53.2% 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 66.6%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 53.9%

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

8. Final simplification 53.9%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024320 
(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))))