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

Percentage Accurate: 65.9% → 99.6%

Time: 7.8s

Alternatives: 12

Speedup: 0.8×

• 48.1% 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: 65.9% 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: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 87.6

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. pow2 N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 2: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left(\frac{x}{y} \cdot x\right) \cdot t}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t} \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 2e-154)
     (* (/ z t) (/ z t))
     (if (<= t_1 5e+255)
       (+ (* x (/ x (* y y))) (/ (* z z) (* t t)))
       (if (<= t_1 INFINITY)
         (/ (* (* (/ x y) x) t) (* t y))
         (/ (* (/ z t) z) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 2e-154) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= 5e+255) {
		tmp = (x * (x / (y * y))) + ((z * z) / (t * t));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((x / y) * x) * t) / (t * y);
	} else {
		tmp = ((z / t) * z) / t;
	}
	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-154) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= 5e+255) {
		tmp = (x * (x / (y * y))) + ((z * z) / (t * t));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (((x / y) * x) * t) / (t * y);
	} else {
		tmp = ((z / t) * z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if t_1 <= 2e-154:
		tmp = (z / t) * (z / t)
	elif t_1 <= 5e+255:
		tmp = (x * (x / (y * y))) + ((z * z) / (t * t))
	elif t_1 <= math.inf:
		tmp = (((x / y) * x) * t) / (t * y)
	else:
		tmp = ((z / t) * z) / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 2e-154)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	elseif (t_1 <= 5e+255)
		tmp = Float64(Float64(x * Float64(x / Float64(y * y))) + Float64(Float64(z * z) / Float64(t * t)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) * t) / Float64(t * y));
	else
		tmp = Float64(Float64(Float64(z / t) * z) / t);
	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-154)
		tmp = (z / t) * (z / t);
	elseif (t_1 <= 5e+255)
		tmp = (x * (x / (y * y))) + ((z * z) / (t * t));
	elseif (t_1 <= Inf)
		tmp = (((x / y) * x) * t) / (t * y);
	else
		tmp = ((z / t) * z) / t;
	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[LessEqual[t$95$1, 2e-154], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+255], N[(N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] / N[(t * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e-154

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 95.3%

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

   if 1.9999999999999999e-154 < (/.f64 (*.f64 x x) (*.f64 y y)) < 5.0000000000000002e255

   1. Initial program 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqr-neg-rev N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 82.0

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

   4. Applied rewrites 82.0%

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

   if 5.0000000000000002e255 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 82.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.5

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

   7. Applied rewrites 34.5%

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

   9. Applied rewrites 29.6%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. associate-/l* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 94.8

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

   12. Applied rewrites 94.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 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 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 75.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;\frac{\left(\frac{x}{y} \cdot x\right) \cdot t}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t} \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;t\_1 + \frac{z \cdot z}{t \cdot t}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left(\frac{x}{y} \cdot x\right) \cdot t}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t} \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 2e-154)
     (* (/ z t) (/ z t))
     (if (<= t_1 5e+255)
       (+ t_1 (/ (* z z) (* t t)))
       (if (<= t_1 INFINITY)
         (/ (* (* (/ x y) x) t) (* t y))
         (/ (* (/ z t) z) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 2e-154) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= 5e+255) {
		tmp = t_1 + ((z * z) / (t * t));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((x / y) * x) * t) / (t * y);
	} else {
		tmp = ((z / t) * z) / t;
	}
	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-154) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= 5e+255) {
		tmp = t_1 + ((z * z) / (t * t));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (((x / y) * x) * t) / (t * y);
	} else {
		tmp = ((z / t) * z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if t_1 <= 2e-154:
		tmp = (z / t) * (z / t)
	elif t_1 <= 5e+255:
		tmp = t_1 + ((z * z) / (t * t))
	elif t_1 <= math.inf:
		tmp = (((x / y) * x) * t) / (t * y)
	else:
		tmp = ((z / t) * z) / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 2e-154)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	elseif (t_1 <= 5e+255)
		tmp = Float64(t_1 + Float64(Float64(z * z) / Float64(t * t)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) * t) / Float64(t * y));
	else
		tmp = Float64(Float64(Float64(z / t) * z) / t);
	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-154)
		tmp = (z / t) * (z / t);
	elseif (t_1 <= 5e+255)
		tmp = t_1 + ((z * z) / (t * t));
	elseif (t_1 <= Inf)
		tmp = (((x / y) * x) * t) / (t * y);
	else
		tmp = ((z / t) * z) / t;
	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[LessEqual[t$95$1, 2e-154], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+255], N[(t$95$1 + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] / N[(t * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e-154

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 95.3%

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

   if 1.9999999999999999e-154 < (/.f64 (*.f64 x x) (*.f64 y y)) < 5.0000000000000002e255

   1. Initial program 81.8%

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

   if 5.0000000000000002e255 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 82.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.5

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

   7. Applied rewrites 34.5%

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

   9. Applied rewrites 29.6%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. associate-/l* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 94.8

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

   12. Applied rewrites 94.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 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 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 75.7%

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

Alternative 4: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1 + z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t} \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 2e-154)
     (* (/ z t) (/ z t))
     (if (<= t_1 INFINITY) (+ t_1 (* z (/ z (* t t)))) (/ (* (/ z t) z) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 2e-154) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1 + (z * (z / (t * t)));
	} else {
		tmp = ((z / t) * z) / t;
	}
	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-154) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 + (z * (z / (t * t)));
	} else {
		tmp = ((z / t) * z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if t_1 <= 2e-154:
		tmp = (z / t) * (z / t)
	elif t_1 <= math.inf:
		tmp = t_1 + (z * (z / (t * t)))
	else:
		tmp = ((z / t) * z) / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 2e-154)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	elseif (t_1 <= Inf)
		tmp = Float64(t_1 + Float64(z * Float64(z / Float64(t * t))));
	else
		tmp = Float64(Float64(Float64(z / t) * z) / t);
	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-154)
		tmp = (z / t) * (z / t);
	elseif (t_1 <= Inf)
		tmp = t_1 + (z * (z / (t * t)));
	else
		tmp = ((z / t) * z) / t;
	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[LessEqual[t$95$1, 2e-154], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$1 + N[(z * N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.9999999999999999e-154

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 95.3%

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

   if 1.9999999999999999e-154 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 82.0%

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

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. frac-2neg N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 88.8

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

   4. Applied rewrites 88.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 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 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 75.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{x \cdot x}{y \cdot y} \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y \cdot y} + z \cdot \frac{z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t} \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left(\frac{x}{y} \cdot x\right) \cdot t}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t} \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 2e+220)
     (* (/ z t) (/ z t))
     (if (<= t_1 INFINITY)
       (/ (* (* (/ x y) x) t) (* t y))
       (/ (* (/ z t) z) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= 2e+220) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (((x / y) * x) * t) / (t * y);
	} else {
		tmp = ((z / t) * z) / t;
	}
	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+220) {
		tmp = (z / t) * (z / t);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (((x / y) * x) * t) / (t * y);
	} else {
		tmp = ((z / t) * z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if t_1 <= 2e+220:
		tmp = (z / t) * (z / t)
	elif t_1 <= math.inf:
		tmp = (((x / y) * x) * t) / (t * y)
	else:
		tmp = ((z / t) * z) / t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= 2e+220)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) * t) / Float64(t * y));
	else
		tmp = Float64(Float64(Float64(z / t) * z) / t);
	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+220)
		tmp = (z / t) * (z / t);
	elseif (t_1 <= Inf)
		tmp = (((x / y) * x) * t) / (t * y);
	else
		tmp = ((z / t) * z) / t;
	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[LessEqual[t$95$1, 2e+220], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * t), $MachinePrecision] / N[(t * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 2e220

   1. Initial program 74.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 82.4%

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

   if 2e220 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 82.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l/ N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 34.8

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

   7. Applied rewrites 34.8%

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

   9. Applied rewrites 30.1%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. associate-/l* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 93.9

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

   12. Applied rewrites 93.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 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 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 75.7%

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

Alternative 6: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, x \cdot \frac{x}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+255)
     (+ (* (/ x y) (/ x y)) t_1)
     (fma (/ z t) (/ z t) (* x (/ x (* y y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+255) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = fma((z / t), (z / t), (x * (x / (y * y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+255)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(x * Float64(x / Float64(y * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+255], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000002e255

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   if 5.0000000000000002e255 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 59.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 87.7

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-neg-rev N/A

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

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

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

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

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

      15. lift-*.f64 N/A

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

      16. frac-2neg-rev N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 96.2

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

   6. Applied rewrites 96.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+255}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, x \cdot \frac{x}{y \cdot y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.1

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 95.1

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

   6. Applied rewrites 95.1%

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

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

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 79.8

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

   4. Applied rewrites 79.8%

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

Alternative 8: 87.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.5%

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

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.1

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 95.1

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

   6. Applied rewrites 95.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 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 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 75.7%

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

Alternative 9: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, x \cdot \frac{x}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* t t) 0.0)
   (fma (/ z t) (/ z t) (* x (/ x (* y y))))
   (fma (/ (/ z t) t) z (* (/ (/ x y) y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t * t) <= 0.0) {
		tmp = fma((z / t), (z / t), (x * (x / (y * y))));
	} 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(t * t) <= 0.0)
		tmp = fma(Float64(z / t), Float64(z / t), Float64(x * Float64(x / Float64(y * y))));
	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[(t * t), $MachinePrecision], 0.0], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t), $MachinePrecision] / t), $MachinePrecision] * z + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 86.4

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-neg-rev N/A

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

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

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

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

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

      15. lift-*.f64 N/A

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

      16. frac-2neg-rev N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

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

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 95.5

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

   6. Applied rewrites 95.5%

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

   if 0.0 < (*.f64 t t)

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} + \frac{{z}^{2}}{{t}^{2}}} \]
   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 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot t \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, x \cdot \frac{x}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{t}}{t}, z, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 1.08 \cdot 10^{+275}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z \cdot z}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 1.08e+275)
   (* (/ z t) (/ z t))
   (/ (/ (* z z) t) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1.08e+275) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = ((z * z) / t) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 1.08d+275) then
        tmp = (z / t) * (z / t)
    else
        tmp = ((z * z) / t) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 1.08e+275) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = ((z * z) / t) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 1.08e+275:
		tmp = (z / t) * (z / t)
	else:
		tmp = ((z * z) / t) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 1.08e+275)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(Float64(z * z) / t) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 1.08e+275)
		tmp = (z / t) * (z / t);
	else
		tmp = ((z * z) / t) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 1.08e+275], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 1.0800000000000001e275

   1. Initial program 74.8%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 81.6%

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

   if 1.0800000000000001e275 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 67.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 33.4

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

   5. Applied rewrites 33.4%

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

   7. Applied rewrites 37.2%

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

   9. Applied rewrites 40.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 1.08 \cdot 10^{+275}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z \cdot z}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 57.1

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

5. Applied rewrites 57.1%

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

7. Applied rewrites 61.1%

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

Alternative 12: 52.3% 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 71.4%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
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 57.1

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

5. Applied rewrites 57.1%

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

7. Applied rewrites 53.3%

   \[\leadsto \frac{z}{t \cdot t} \cdot z \]
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 2024326 
(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))))