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

Percentage Accurate: 66.7% → 99.7%

Time: 10.0s

Alternatives: 14

Speedup: 0.7×

• 48.3% 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: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{y}, {\left(\frac{y}{x}\right)}^{-1}, {\left(\frac{z}{t}\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ x y) (pow (/ y x) -1.0) (pow (/ z t) 2.0)))

C

double code(double x, double y, double z, double t) {
	return fma((x / y), pow((y / x), -1.0), pow((z / t), 2.0));
}

Julia

function code(x, y, z, t)
	return fma(Float64(x / y), (Float64(y / x) ^ -1.0), (Float64(z / t) ^ 2.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x / y), $MachinePrecision] * N[Power[N[(y / x), $MachinePrecision], -1.0], $MachinePrecision] + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.0%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 82.1

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{x}{y}, {\left(\frac{y}{x}\right)}^{-1}, {\left(\frac{z}{t}\right)}^{2}\right) \]
8. Add Preprocessing

Alternative 2: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;{\left(\frac{\frac{y}{x}}{\frac{x}{y}}\right)}^{-1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x + t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 0.0)
     (pow (/ (/ y x) (/ x y)) -1.0)
     (if (<= t_1 5e+259)
       (+ (* (/ x (* y y)) x) t_1)
       (if (<= t_1 INFINITY)
         (* (/ z (* t t)) z)
         (pow (* (/ y x) (/ y x)) -1.0))))))

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 = pow(((y / x) / (x / y)), -1.0);
	} else if (t_1 <= 5e+259) {
		tmp = ((x / (y * y)) * x) + t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = pow(((y / x) * (y / x)), -1.0);
	}
	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 <= 0.0) {
		tmp = Math.pow(((y / x) / (x / y)), -1.0);
	} else if (t_1 <= 5e+259) {
		tmp = ((x / (y * y)) * x) + t_1;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = Math.pow(((y / x) * (y / x)), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.pow(((y / x) / (x / y)), -1.0)
	elif t_1 <= 5e+259:
		tmp = ((x / (y * y)) * x) + t_1
	elif t_1 <= math.inf:
		tmp = (z / (t * t)) * z
	else:
		tmp = math.pow(((y / x) * (y / x)), -1.0)
	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(y / x) / Float64(x / y)) ^ -1.0;
	elseif (t_1 <= 5e+259)
		tmp = Float64(Float64(Float64(x / Float64(y * y)) * x) + t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	else
		tmp = Float64(Float64(y / x) * Float64(y / x)) ^ -1.0;
	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 <= 0.0)
		tmp = ((y / x) / (x / y)) ^ -1.0;
	elseif (t_1 <= 5e+259)
		tmp = ((x / (y * y)) * x) + t_1;
	elseif (t_1 <= Inf)
		tmp = (z / (t * t)) * z;
	else
		tmp = ((y / x) * (y / x)) ^ -1.0;
	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, 0.0], N[Power[N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$1, 5e+259], N[(N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[Power[N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 77.8%

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

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-add N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 78.6

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 96.3

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

   7. Applied rewrites 96.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.3%

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

   if 0.0 < (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000033e259

   1. Initial program 85.4%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 91.0%

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

   if 5.00000000000000033e259 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 79.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 95.6%

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

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-add N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 48.0

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 54.5

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

   7. Applied rewrites 54.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 0:\\ \;\;\;\;{\left(\frac{\frac{y}{x}}{\frac{x}{y}}\right)}^{-1}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\frac{x}{y \cdot y} \cdot x + \frac{z \cdot z}{t \cdot t}\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+259)
     (fma (/ x y) (/ x y) t_1)
     (if (<= t_1 INFINITY)
       (* (/ z (* t t)) z)
       (pow (* (/ y x) (/ y x)) -1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+259) {
		tmp = fma((x / y), (x / y), t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = pow(((y / x) * (y / x)), -1.0);
	}
	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+259)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	else
		tmp = Float64(Float64(y / x) * Float64(y / x)) ^ -1.0;
	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+259], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[Power[N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.00000000000000033e259

   1. Initial program 80.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.3

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 97.3

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

   6. Applied rewrites 97.3%

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

   if 5.00000000000000033e259 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 79.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 95.6%

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

   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

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-add N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 48.0

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 54.5

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

   7. Applied rewrites 54.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z \cdot z}{t \cdot t}\right)\\ \mathbf{elif}\;\frac{z \cdot z}{t \cdot t} \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-225} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x \cdot x} \cdot y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (or (<= t_1 2e-225) (not (<= t_1 INFINITY)))
     (/ (/ z t) (/ t z))
     (pow (* (/ y (* x x)) y) -1.0))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if ((t_1 <= 2e-225) || !(t_1 <= ((double) INFINITY))) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = pow(((y / (x * x)) * y), -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if ((t_1 <= 2e-225) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = Math.pow(((y / (x * x)) * y), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if (t_1 <= 2e-225) or not (t_1 <= math.inf):
		tmp = (z / t) / (t / z)
	else:
		tmp = math.pow(((y / (x * x)) * y), -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if ((t_1 <= 2e-225) || !(t_1 <= Inf))
		tmp = Float64(Float64(z / t) / Float64(t / z));
	else
		tmp = Float64(Float64(y / Float64(x * x)) * y) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	tmp = 0.0;
	if ((t_1 <= 2e-225) || ~((t_1 <= Inf)))
		tmp = (z / t) / (t / z);
	else
		tmp = ((y / (x * x)) * y) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 2e-225], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e-225 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 56.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 80.2

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

   5. Applied rewrites 80.2%

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

   7. Applied rewrites 71.0%

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

   9. Applied rewrites 84.7%

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

   if 1.9999999999999999e-225 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 81.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-add N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.3%

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{\frac{x}{y} \cdot x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{-225} \lor \neg \left(\frac{x \cdot x}{y \cdot y} \leq \infty\right):\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x \cdot x} \cdot y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-225} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x \cdot x} \cdot y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (or (<= t_1 2e-225) (not (<= t_1 INFINITY)))
     (* (/ z t) (/ z t))
     (pow (* (/ y (* x x)) y) -1.0))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if ((t_1 <= 2e-225) || !(t_1 <= ((double) INFINITY))) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = pow(((y / (x * x)) * y), -1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if ((t_1 <= 2e-225) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = Math.pow(((y / (x * x)) * y), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if (t_1 <= 2e-225) or not (t_1 <= math.inf):
		tmp = (z / t) * (z / t)
	else:
		tmp = math.pow(((y / (x * x)) * y), -1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if ((t_1 <= 2e-225) || !(t_1 <= Inf))
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(y / Float64(x * x)) * y) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	tmp = 0.0;
	if ((t_1 <= 2e-225) || ~((t_1 <= Inf)))
		tmp = (z / t) * (z / t);
	else
		tmp = ((y / (x * x)) * y) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 2e-225], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e-225 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 56.4%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 76.0

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   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 84.6

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

   7. Applied rewrites 84.6%

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

   if 1.9999999999999999e-225 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 81.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-add N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.3%

      \[\leadsto \frac{1}{\frac{y}{\color{blue}{\frac{x}{y} \cdot x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{-225} \lor \neg \left(\frac{x \cdot x}{y \cdot y} \leq \infty\right):\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x \cdot x} \cdot y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{y}{x}}{\frac{x}{y}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 2e-225)
   (/ (/ z t) (/ t z))
   (pow (/ (/ y x) (/ x y)) -1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 2e-225) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = pow(((y / x) / (x / y)), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 2d-225) then
        tmp = (z / t) / (t / z)
    else
        tmp = ((y / x) / (x / y)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 2e-225) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = Math.pow(((y / x) / (x / y)), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 2e-225:
		tmp = (z / t) / (t / z)
	else:
		tmp = math.pow(((y / x) / (x / y)), -1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 2e-225)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	else
		tmp = Float64(Float64(y / x) / Float64(x / y)) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 2e-225)
		tmp = (z / t) / (t / z);
	else
		tmp = ((y / x) / (x / y)) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 2e-225], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e-225

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 94.3%

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

   if 1.9999999999999999e-225 < (/.f64 (*.f64 x x) (*.f64 y y))

   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

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-add N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 78.9

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

   7. Applied rewrites 78.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{x} \cdot \frac{y}{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{y}{x}}{\frac{x}{y}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 2e-225)
   (/ (/ z t) (/ t z))
   (pow (* (/ y x) (/ y x)) -1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 2e-225) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = pow(((y / x) * (y / x)), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 2d-225) then
        tmp = (z / t) / (t / z)
    else
        tmp = ((y / x) * (y / x)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 2e-225) {
		tmp = (z / t) / (t / z);
	} else {
		tmp = Math.pow(((y / x) * (y / x)), -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 2e-225:
		tmp = (z / t) / (t / z)
	else:
		tmp = math.pow(((y / x) * (y / x)), -1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 2e-225)
		tmp = Float64(Float64(z / t) / Float64(t / z));
	else
		tmp = Float64(Float64(y / x) * Float64(y / x)) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 2e-225)
		tmp = (z / t) / (t / z);
	else
		tmp = ((y / x) * (y / x)) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 2e-225], N[(N[(z / t), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e-225

   1. Initial program 72.7%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 79.7%

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

   9. Applied rewrites 94.3%

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

   if 1.9999999999999999e-225 < (/.f64 (*.f64 x x) (*.f64 y y))

   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

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-add N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 78.9

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

   7. Applied rewrites 78.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 82.1

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 9: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y} + \frac{\frac{z}{t} \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+259)
     (fma (/ x y) (/ x y) t_1)
     (if (<= t_1 INFINITY)
       (* (/ z (* t t)) z)
       (+ (/ (* x x) (* y y)) (/ (* (/ 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+259) {
		tmp = fma((x / y), (x / y), t_1);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / (t * t)) * z;
	} else {
		tmp = ((x * x) / (y * y)) + (((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+259)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	else
		tmp = Float64(Float64(Float64(x * x) / Float64(y * y)) + 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+259], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + 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)) < 5.00000000000000033e259

   1. Initial program 80.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.3

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 97.3

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

   6. Applied rewrites 97.3%

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

   if 5.00000000000000033e259 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 79.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 95.6%

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

   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

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

      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. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 69.0

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

   4. Applied rewrites 69.0%

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

Alternative 10: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+290)
     (fma (/ x y) (/ x y) t_1)
     (fma (/ (/ 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 <= 5e+290) {
		tmp = fma((x / y), (x / y), t_1);
	} else {
		tmp = fma(((z / t) / t), z, (((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 <= 5e+290)
		tmp = fma(Float64(x / y), Float64(x / y), t_1);
	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[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+290], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + t$95$1), $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 z z) (*.f64 t t)) < 4.9999999999999998e290

   1. Initial program 80.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 97.4

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 97.4

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

   6. Applied rewrites 97.4%

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

   if 4.9999999999999998e290 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 58.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 95.0

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

   5. Applied rewrites 95.0%

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

Alternative 11: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} + \frac{\frac{z}{t} \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 2e+265)
   (+ (/ (/ x y) (/ y x)) (/ (* (/ 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 tmp;
	if ((z * z) <= 2e+265) {
		tmp = ((x / y) / (y / x)) + (((z / t) * z) / t);
	} else {
		tmp = fma(((z / t) / t), z, (((x / y) / y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 2e+265)
		tmp = Float64(Float64(Float64(x / y) / Float64(y / x)) + Float64(Float64(Float64(z / t) * 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_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+265], N[(N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $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 z z) < 2.00000000000000013e265

   1. Initial program 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 79.6

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

   4. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 98.7

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

   6. Applied rewrites 98.7%

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

   if 2.00000000000000013e265 < (*.f64 z z)

   1. Initial program 62.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 12: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\frac{z}{t} \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 6.5e+166)
   (fma (/ x y) (/ x y) (/ (* (/ 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 tmp;
	if (z <= 6.5e+166) {
		tmp = fma((x / y), (x / y), (((z / t) * z) / t));
	} else {
		tmp = fma(((z / t) / t), z, (((x / y) / y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 6.5e+166)
		tmp = fma(Float64(x / y), Float64(x / y), Float64(Float64(Float64(z / t) * 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_] := If[LessEqual[z, 6.5e+166], N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $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 z < 6.5000000000000005e166

   1. Initial program 71.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 85.1

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 97.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.3

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

   6. Applied rewrites 97.3%

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

   if 6.5000000000000005e166 < z

   1. Initial program 56.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 13: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+172}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 6e+172) (* (/ z t) (/ z t)) (* (/ z (* t t)) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 6e+172) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 6d+172) then
        tmp = (z / t) * (z / t)
    else
        tmp = (z / (t * t)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 6e+172) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 6e+172:
		tmp = (z / t) * (z / t)
	else:
		tmp = (z / (t * t)) * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 6e+172)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 6e+172)
		tmp = (z / t) * (z / t);
	else
		tmp = (z / (t * t)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 6e+172], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.9999999999999998e172

   1. Initial program 69.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.9

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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 56.3

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

   7. Applied rewrites 56.3%

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

   if 5.9999999999999998e172 < x

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      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 46.0

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

   5. Applied rewrites 46.0%

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

   7. Applied rewrites 57.4%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+172}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.4% 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 70.0%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   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 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 52.4%

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

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024318 
(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))))