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

Percentage Accurate: 65.7% → 98.1%

Time: 9.1s

Alternatives: 11

Speedup: 0.8×

• 48.2% 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: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 1.35 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{-x}{y}}{y \cdot \frac{-1}{x}} + \frac{\frac{z\_m}{t} \cdot z\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z\_m}{t}}{t}, z\_m, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 1.35e+180)
   (+ (/ (/ (- x) y) (* y (/ -1.0 x))) (/ (* (/ z_m t) z_m) t))
   (fma (/ (/ z_m t) t) z_m (* (/ (/ x y) y) x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1.35e+180) {
		tmp = ((-x / y) / (y * (-1.0 / x))) + (((z_m / t) * z_m) / t);
	} else {
		tmp = fma(((z_m / t) / t), z_m, (((x / y) / y) * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1.35e+180)
		tmp = Float64(Float64(Float64(Float64(-x) / y) / Float64(y * Float64(-1.0 / x))) + Float64(Float64(Float64(z_m / t) * z_m) / t));
	else
		tmp = fma(Float64(Float64(z_m / t) / t), z_m, Float64(Float64(Float64(x / y) / y) * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 1.35e+180], N[(N[(N[((-x) / y), $MachinePrecision] / N[(y * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z$95$m / t), $MachinePrecision] * z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision] * z$95$m + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.35000000000000008e180

   1. Initial program 68.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-pow.f64 N/A

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

      17. inv-pow N/A

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

      18. lower-pow.f64 97.0

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

   6. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{{y}^{-1}}{{x}^{-1}}} + \frac{\frac{z}{t} \cdot z}{t} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-times N/A

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

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

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

      7. lift-pow.f64 N/A

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

      8. unpow-1 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. unpow-1 N/A

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

      19. distribute-neg-frac N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 97.0

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

   8. Applied rewrites 97.0%

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

   if 1.35000000000000008e180 < z

   1. Initial program 68.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \frac{z\_m \cdot z\_m}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z\_m}{t}}{t}, z\_m, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* z_m z_m) (* t t))))
   (if (<= t_1 2e+304)
     (+ (* (/ x y) (/ x y)) t_1)
     (fma (/ (/ z_m t) t) z_m (* (/ x (* y y)) x)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (z_m * z_m) / (t * t);
	double tmp;
	if (t_1 <= 2e+304) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = fma(((z_m / t) / t), z_m, ((x / (y * y)) * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(z_m * z_m) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 2e+304)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	else
		tmp = fma(Float64(Float64(z_m / t) / t), z_m, Float64(Float64(x / Float64(y * y)) * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+304], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision] * z$95$m + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999999e304

   1. Initial program 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   if 1.9999999999999999e304 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 63.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 94.3%

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

Alternative 3: 91.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot z\_m}{t \cdot t} \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m}{t \cdot t}, z\_m, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z\_m}{t}}{t}, z\_m, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* z_m z_m) (* t t)) 2e+175)
   (fma (/ z_m (* t t)) z_m (* (/ (/ x y) y) x))
   (fma (/ (/ z_m t) t) z_m (* (/ x (* y y)) x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((z_m * z_m) / (t * t)) <= 2e+175) {
		tmp = fma((z_m / (t * t)), z_m, (((x / y) / y) * x));
	} else {
		tmp = fma(((z_m / t) / t), z_m, ((x / (y * y)) * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(z_m * z_m) / Float64(t * t)) <= 2e+175)
		tmp = fma(Float64(z_m / Float64(t * t)), z_m, Float64(Float64(Float64(x / y) / y) * x));
	else
		tmp = fma(Float64(Float64(z_m / t) / t), z_m, Float64(Float64(x / Float64(y * y)) * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(z$95$m * z$95$m), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision], 2e+175], N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision] * z$95$m + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1.9999999999999999e175

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} + \frac{{z}^{2}}{{t}^{2}}} \]
   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 94.3

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

   5. Applied rewrites 94.3%

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

   7. Applied rewrites 94.4%

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

   if 1.9999999999999999e175 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 95.4

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

   5. Applied rewrites 95.4%

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

   7. Applied rewrites 94.7%

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

Alternative 4: 88.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10:\\ \;\;\;\;\frac{\frac{z\_m}{t}}{\frac{t}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m}{t \cdot t}, z\_m, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 10.0)
   (/ (/ z_m t) (/ t z_m))
   (fma (/ z_m (* t t)) z_m (* (/ (/ x y) y) x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 10.0) {
		tmp = (z_m / t) / (t / z_m);
	} else {
		tmp = fma((z_m / (t * t)), z_m, (((x / y) / y) * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 10.0)
		tmp = Float64(Float64(z_m / t) / Float64(t / z_m));
	else
		tmp = fma(Float64(z_m / Float64(t * t)), z_m, Float64(Float64(Float64(x / y) / y) * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 10.0], N[(N[(z$95$m / t), $MachinePrecision] / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 10

   1. Initial program 69.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 89.9%

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

   if 10 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 67.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 96.6

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 94.7%

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

Alternative 5: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 10:\\ \;\;\;\;\frac{\frac{z\_m}{t}}{\frac{t}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m}{t \cdot t}, z\_m, \frac{x}{y \cdot y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 10.0)
   (/ (/ z_m t) (/ t z_m))
   (fma (/ z_m (* t t)) z_m (* (/ x (* y y)) x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 10.0) {
		tmp = (z_m / t) / (t / z_m);
	} else {
		tmp = fma((z_m / (t * t)), z_m, ((x / (y * y)) * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 10.0)
		tmp = Float64(Float64(z_m / t) / Float64(t / z_m));
	else
		tmp = fma(Float64(z_m / Float64(t * t)), z_m, Float64(Float64(x / Float64(y * y)) * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 10.0], N[(N[(z$95$m / t), $MachinePrecision] / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 10

   1. Initial program 69.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 89.9%

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

   if 10 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 67.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 96.6

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 94.7%

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

   9. Applied rewrites 84.2%

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

Alternative 6: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{\frac{y}{x} \cdot y} + \frac{\frac{z\_m}{t} \cdot z\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z\_m}{t}}{t}, z\_m, \frac{\frac{x}{y}}{y} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= t 3.8e-226)
   (+ (/ x (* (/ y x) y)) (/ (* (/ z_m t) z_m) t))
   (fma (/ (/ z_m t) t) z_m (* (/ (/ x y) y) x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (t <= 3.8e-226) {
		tmp = (x / ((y / x) * y)) + (((z_m / t) * z_m) / t);
	} else {
		tmp = fma(((z_m / t) / t), z_m, (((x / y) / y) * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (t <= 3.8e-226)
		tmp = Float64(Float64(x / Float64(Float64(y / x) * y)) + Float64(Float64(Float64(z_m / t) * z_m) / t));
	else
		tmp = fma(Float64(Float64(z_m / t) / t), z_m, Float64(Float64(Float64(x / y) / y) * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[t, 3.8e-226], N[(N[(x / N[(N[(y / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z$95$m / t), $MachinePrecision] * z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision] * z$95$m + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999981e-226

   1. Initial program 69.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.0

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

   4. Applied rewrites 82.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 96.2

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

   6. Applied rewrites 96.2%

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

   if 3.79999999999999981e-226 < t

   1. Initial program 68.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l/ N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 7: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+241}:\\ \;\;\;\;\frac{\frac{z\_m}{t}}{\frac{t}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{t}}{t} \cdot z\_m\right) \cdot \left(-z\_m\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 2e+241)
   (/ (/ z_m t) (/ t z_m))
   (* (* (/ (/ -1.0 t) t) z_m) (- z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 2e+241) {
		tmp = (z_m / t) / (t / z_m);
	} else {
		tmp = (((-1.0 / t) / t) * z_m) * -z_m;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 2d+241) then
        tmp = (z_m / t) / (t / z_m)
    else
        tmp = ((((-1.0d0) / t) / t) * z_m) * -z_m
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 2e+241) {
		tmp = (z_m / t) / (t / z_m);
	} else {
		tmp = (((-1.0 / t) / t) * z_m) * -z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 2e+241:
		tmp = (z_m / t) / (t / z_m)
	else:
		tmp = (((-1.0 / t) / t) * z_m) * -z_m
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 2e+241)
		tmp = Float64(Float64(z_m / t) / Float64(t / z_m));
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 / t) / t) * z_m) * Float64(-z_m));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 2e+241)
		tmp = (z_m / t) / (t / z_m);
	else
		tmp = (((-1.0 / t) / t) * z_m) * -z_m;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 2e+241], N[(N[(z$95$m / t), $MachinePrecision] / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 / t), $MachinePrecision] / t), $MachinePrecision] * z$95$m), $MachinePrecision] * (-z$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 2.0000000000000001e241

   1. Initial program 71.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 82.2%

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

   if 2.0000000000000001e241 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 66.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 42.9%

      \[\leadsto \frac{\frac{-1}{t}}{-\frac{t}{z}} \cdot z \]
   8. Step-by-step derivation

   9. Applied rewrites 44.3%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 2 \cdot 10^{+241}:\\ \;\;\;\;\frac{\frac{z}{t}}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{-1}{t}}{t} \cdot z\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 3.3 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{z\_m}{t}}{\frac{t}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t \cdot t} \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 3.3e+264)
   (/ (/ z_m t) (/ t z_m))
   (* (/ z_m (* t t)) z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 3.3e+264) {
		tmp = (z_m / t) / (t / z_m);
	} else {
		tmp = (z_m / (t * t)) * z_m;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 3.3d+264) then
        tmp = (z_m / t) / (t / z_m)
    else
        tmp = (z_m / (t * t)) * z_m
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 3.3e+264) {
		tmp = (z_m / t) / (t / z_m);
	} else {
		tmp = (z_m / (t * t)) * z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 3.3e+264:
		tmp = (z_m / t) / (t / z_m)
	else:
		tmp = (z_m / (t * t)) * z_m
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 3.3e+264)
		tmp = Float64(Float64(z_m / t) / Float64(t / z_m));
	else
		tmp = Float64(Float64(z_m / Float64(t * t)) * z_m);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 3.3e+264)
		tmp = (z_m / t) / (t / z_m);
	else
		tmp = (z_m / (t * t)) * z_m;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 3.3e+264], N[(N[(z$95$m / t), $MachinePrecision] / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 3.3e264

   1. Initial program 71.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 77.0

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 82.2%

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

   if 3.3e264 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 66.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 43.5%

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

Alternative 9: 93.8% accurate, 0.8× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (fma (/ (/ z_m t) t) z_m (* (/ (/ x y) y) x)))

C

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

Julia

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := N[(N[(N[(z$95$m / t), $MachinePrecision] / t), $MachinePrecision] * z$95$m + N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} + \frac{{z}^{2}}{{t}^{2}}} \]
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 94.9

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

5. Applied rewrites 94.9%

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

Alternative 10: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 3.3 \cdot 10^{+264}:\\ \;\;\;\;\frac{z\_m}{t} \cdot \frac{z\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{t \cdot t} \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (/ (* x x) (* y y)) 3.3e+264)
   (* (/ z_m t) (/ z_m t))
   (* (/ z_m (* t t)) z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 3.3e+264) {
		tmp = (z_m / t) * (z_m / t);
	} else {
		tmp = (z_m / (t * t)) * z_m;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * x) / (y * y)) <= 3.3d+264) then
        tmp = (z_m / t) * (z_m / t)
    else
        tmp = (z_m / (t * t)) * z_m
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	double tmp;
	if (((x * x) / (y * y)) <= 3.3e+264) {
		tmp = (z_m / t) * (z_m / t);
	} else {
		tmp = (z_m / (t * t)) * z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if ((x * x) / (y * y)) <= 3.3e+264:
		tmp = (z_m / t) * (z_m / t)
	else:
		tmp = (z_m / (t * t)) * z_m
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x * x) / Float64(y * y)) <= 3.3e+264)
		tmp = Float64(Float64(z_m / t) * Float64(z_m / t));
	else
		tmp = Float64(Float64(z_m / Float64(t * t)) * z_m);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if (((x * x) / (y * y)) <= 3.3e+264)
		tmp = (z_m / t) * (z_m / t);
	else
		tmp = (z_m / (t * t)) * z_m;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision], 3.3e+264], N[(N[(z$95$m / t), $MachinePrecision] * N[(z$95$m / t), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 3.3e264

   1. Initial program 71.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 82.2

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

   8. Applied rewrites 82.2%

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

   if 3.3e264 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 66.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   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 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 43.5%

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

Alternative 11: 53.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \frac{z\_m}{t \cdot t} \cdot z\_m \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t) :precision binary64 (* (/ z_m (* t t)) z_m))

C

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

Fortran

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

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m, double t) {
	return (z_m / (t * t)) * z_m;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	return (z_m / (t * t)) * z_m

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := N[(N[(z$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 60.4

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

5. Applied rewrites 60.4%

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

7. Applied rewrites 55.8%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024324 
(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))))