Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B

Percentage Accurate: 90.9% → 95.7%

Time: 9.4s

Alternatives: 8

Speedup: 0.8×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * 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 = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * 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 = (x * x) - ((y * 4.0d0) * ((z * z) - t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}

Python

def code(x, y, z, t):
	return (x * x) - ((y * 4.0) * ((z * z) - t))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - ((y * 4.0) * ((z * z) - t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z\_m, z\_m, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 1e+213)
   (fma (* (* y -4.0) z_m) z_m (fma (* (- t) y) -4.0 (* x x)))
   (* (* (* y z_m) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 1e+213) {
		tmp = fma(((y * -4.0) * z_m), z_m, fma((-t * y), -4.0, (x * x)));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 1e+213)
		tmp = fma(Float64(Float64(y * -4.0) * z_m), z_m, fma(Float64(Float64(-t) * y), -4.0, Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 1e+213], N[(N[(N[(y * -4.0), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m + N[(N[((-t) * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.99999999999999984e212

   1. Initial program 92.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right) + x \cdot x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)}\right)\right) + x \cdot x \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} + x \cdot x \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z - t\right)} + x \cdot x \]

      7. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      9. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z\right)} + \left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right) \cdot z} + \left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z}, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{y \cdot 4}\right)\right) \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{4 \cdot y}\right)\right) \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right)} \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right)} \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{-4} \cdot y\right) \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

   4. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)} \]

   if 9.99999999999999984e212 < z

   1. Initial program 65.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      5. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

      6. lower-*.f64 77.3

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

   5. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.9%

      \[\leadsto \left(\left(z \cdot y\right) \cdot z\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot -4\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\right) \cdot z\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.4 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot -4, y, \mathsf{fma}\left(\left(z\_m \cdot z\_m\right) \cdot y, -4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3.4e+138)
   (fma (* (- t) -4.0) y (fma (* (* z_m z_m) y) -4.0 (* x x)))
   (* (* (* y z_m) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.4e+138) {
		tmp = fma((-t * -4.0), y, fma(((z_m * z_m) * y), -4.0, (x * x)));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.4e+138)
		tmp = fma(Float64(Float64(-t) * -4.0), y, fma(Float64(Float64(z_m * z_m) * y), -4.0, Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 3.4e+138], N[(N[((-t) * -4.0), $MachinePrecision] * y + N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.40000000000000011e138

   1. Initial program 95.1%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x \cdot x + \left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\right) + x \cdot x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)}\right)\right) + x \cdot x \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z - t\right)} + x \cdot x \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z - t\right)} + x \cdot x \]

      7. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z + \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)} + x \cdot x \]

      9. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(z \cdot z\right) + \left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \color{blue}{\left(z \cdot z\right)} + \left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z\right) \cdot z} + \left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot z}, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{y \cdot 4}\right)\right) \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{4 \cdot y}\right)\right) \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right)} \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot y\right)} \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{-4} \cdot y\right) \cdot z, z, \left(\mathsf{neg}\left(y \cdot 4\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right) + x \cdot x\right) \]

   4. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-4 \cdot y\right) \cdot z\right) \cdot z + \mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-t\right) \cdot y, -4, x \cdot x\right) + \left(\left(-4 \cdot y\right) \cdot z\right) \cdot z} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(-t\right) \cdot y\right) \cdot -4 + x \cdot x\right)} + \left(\left(-4 \cdot y\right) \cdot z\right) \cdot z \]

      4. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(-t\right) \cdot y\right) \cdot -4 + \left(x \cdot x + \left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-t\right) \cdot y\right)} \cdot -4 + \left(x \cdot x + \left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(-t\right) \cdot \left(y \cdot -4\right)} + \left(x \cdot x + \left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-t\right) \cdot \color{blue}{\left(-4 \cdot y\right)} + \left(x \cdot x + \left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(-t\right) \cdot -4\right) \cdot y} + \left(x \cdot x + \left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \color{blue}{\left(\left(\left(-4 \cdot y\right) \cdot z\right) \cdot z + x \cdot x\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \left(\color{blue}{\left(\left(-4 \cdot y\right) \cdot z\right)} \cdot z + x \cdot x\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \left(\color{blue}{\left(-4 \cdot y\right) \cdot \left(z \cdot z\right)} + x \cdot x\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \left(\left(-4 \cdot y\right) \cdot \color{blue}{\left(z \cdot z\right)} + x \cdot x\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \left(\color{blue}{\left(z \cdot z\right) \cdot \left(-4 \cdot y\right)} + x \cdot x\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \left(\left(z \cdot z\right) \cdot \color{blue}{\left(-4 \cdot y\right)} + x \cdot x\right) \]

      15. *-commutative N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \left(\left(z \cdot z\right) \cdot \color{blue}{\left(y \cdot -4\right)} + x \cdot x\right) \]

      16. associate-*l* N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \left(\color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} + x \cdot x\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \left(\color{blue}{\left(\left(z \cdot z\right) \cdot y\right)} \cdot -4 + x \cdot x\right) \]

      18. lift-fma.f64 N/A

          \[\leadsto \left(\left(-t\right) \cdot -4\right) \cdot y + \color{blue}{\mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right)} \]

      19. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-t\right) \cdot -4, y, \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right)\right)} \]

   6. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-t\right) \cdot -4, y, \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right)\right)} \]

   if 3.40000000000000011e138 < z

   1. Initial program 65.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      5. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

      6. lower-*.f64 75.2

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.5%

      \[\leadsto \left(\left(z \cdot y\right) \cdot z\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\left(-t\right) \cdot -4, y, \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\right) \cdot z\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5 \cdot 10^{-68}:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{elif}\;z\_m \cdot z\_m \leq 10^{+123}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 5e-68)
   (* (* 4.0 t) y)
   (if (<= (* z_m z_m) 1e+123) (* x x) (* (* (* y z_m) z_m) -4.0))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 5e-68) {
		tmp = (4.0 * t) * y;
	} else if ((z_m * z_m) <= 1e+123) {
		tmp = x * x;
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	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 ((z_m * z_m) <= 5d-68) then
        tmp = (4.0d0 * t) * y
    else if ((z_m * z_m) <= 1d+123) then
        tmp = x * x
    else
        tmp = ((y * z_m) * z_m) * (-4.0d0)
    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 ((z_m * z_m) <= 5e-68) {
		tmp = (4.0 * t) * y;
	} else if ((z_m * z_m) <= 1e+123) {
		tmp = x * x;
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if (z_m * z_m) <= 5e-68:
		tmp = (4.0 * t) * y
	elif (z_m * z_m) <= 1e+123:
		tmp = x * x
	else:
		tmp = ((y * z_m) * z_m) * -4.0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5e-68)
		tmp = Float64(Float64(4.0 * t) * y);
	elseif (Float64(z_m * z_m) <= 1e+123)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if ((z_m * z_m) <= 5e-68)
		tmp = (4.0 * t) * y;
	elseif ((z_m * z_m) <= 1e+123)
		tmp = x * x;
	else
		tmp = ((y * z_m) * z_m) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e-68], N[(N[(4.0 * t), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+123], N[(x * x), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 4.99999999999999971e-68

   1. Initial program 100.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]

      3. lower-*.f64 56.7

         \[\leadsto \color{blue}{\left(t \cdot y\right)} \cdot 4 \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.7%

      \[\leadsto \left(t \cdot 4\right) \cdot \color{blue}{y} \]

   if 4.99999999999999971e-68 < (*.f64 z z) < 9.99999999999999978e122

   1. Initial program 97.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right) \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right) \cdot -4} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({z}^{2} - t\right) \cdot y\right)} \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({z}^{2} - t\right) \cdot y\right)} \cdot -4 \]

      5. lower--.f64 N/A

         \[\leadsto \left(\color{blue}{\left({z}^{2} - t\right)} \cdot y\right) \cdot -4 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right) \cdot -4 \]

      7. lower-*.f64 43.2

         \[\leadsto \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right) \cdot -4 \]

   5. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4} \]

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.0

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

   8. Applied rewrites 61.0%

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

   if 9.99999999999999978e122 < (*.f64 z z)

   1. Initial program 80.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      5. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

      6. lower-*.f64 74.9

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

   5. Applied rewrites 74.9%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.8%

      \[\leadsto \left(\left(z \cdot y\right) \cdot z\right) \cdot -4 \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-68}:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{elif}\;z \cdot z \leq 10^{+123}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\right) \cdot z\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+268}:\\ \;\;\;\;x \cdot x - \left(z\_m \cdot z\_m - t\right) \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 2e+268)
   (- (* x x) (* (- (* z_m z_m) t) (* 4.0 y)))
   (* (* (* y z_m) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 2e+268) {
		tmp = (x * x) - (((z_m * z_m) - t) * (4.0 * y));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	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 ((z_m * z_m) <= 2d+268) then
        tmp = (x * x) - (((z_m * z_m) - t) * (4.0d0 * y))
    else
        tmp = ((y * z_m) * z_m) * (-4.0d0)
    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 ((z_m * z_m) <= 2e+268) {
		tmp = (x * x) - (((z_m * z_m) - t) * (4.0 * y));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if (z_m * z_m) <= 2e+268:
		tmp = (x * x) - (((z_m * z_m) - t) * (4.0 * y))
	else:
		tmp = ((y * z_m) * z_m) * -4.0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 2e+268)
		tmp = Float64(Float64(x * x) - Float64(Float64(Float64(z_m * z_m) - t) * Float64(4.0 * y)));
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if ((z_m * z_m) <= 2e+268)
		tmp = (x * x) - (((z_m * z_m) - t) * (4.0 * y));
	else
		tmp = ((y * z_m) * z_m) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 2e+268], N[(N[(x * x), $MachinePrecision] - N[(N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision] * N[(4.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e268

   1. Initial program 99.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   if 1.9999999999999999e268 < (*.f64 z z)

   1. Initial program 71.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      5. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

      6. lower-*.f64 77.2

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.6%

      \[\leadsto \left(\left(z \cdot y\right) \cdot z\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+268}:\\ \;\;\;\;x \cdot x - \left(z \cdot z - t\right) \cdot \left(4 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\right) \cdot z\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \left(y \cdot z\_m\right) \cdot z\_m\\ \mathbf{if}\;z\_m \leq 9.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z\_m \leq 6.5 \cdot 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (* (* y z_m) z_m)))
   (if (<= z_m 9.5e+41)
     (fma (* 4.0 t) y (* x x))
     (if (<= z_m 6.5e+213) (fma t_1 -4.0 (* x x)) (* t_1 -4.0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (y * z_m) * z_m;
	double tmp;
	if (z_m <= 9.5e+41) {
		tmp = fma((4.0 * t), y, (x * x));
	} else if (z_m <= 6.5e+213) {
		tmp = fma(t_1, -4.0, (x * x));
	} else {
		tmp = t_1 * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(y * z_m) * z_m)
	tmp = 0.0
	if (z_m <= 9.5e+41)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	elseif (z_m <= 6.5e+213)
		tmp = fma(t_1, -4.0, Float64(x * x));
	else
		tmp = Float64(t_1 * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]}, If[LessEqual[z$95$m, 9.5e+41], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 6.5e+213], N[(t$95$1 * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 9.4999999999999996e41

   1. Initial program 94.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

      1. cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + \color{blue}{4} \cdot \left(t \cdot y\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right) + {x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} + {x}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y}, 4, {x}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, \color{blue}{x \cdot x}\right) \]

      8. lower-*.f64 72.3

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.8%

      \[\leadsto \mathsf{fma}\left(t \cdot 4, \color{blue}{y}, x \cdot x\right) \]

   if 9.4999999999999996e41 < z < 6.49999999999999982e213

   1. Initial program 81.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{{x}^{2} - 4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(y \cdot {z}^{2}\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + \color{blue}{-4} \cdot \left(y \cdot {z}^{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right) + {x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} + {x}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot {z}^{2}, -4, {x}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{2} \cdot y}, -4, {x}^{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{z}^{2} \cdot y}, -4, {x}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot z\right)} \cdot y, -4, {x}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot z\right)} \cdot y, -4, {x}^{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, \color{blue}{x \cdot x}\right) \]

      11. lower-*.f64 81.8

          \[\leadsto \mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot z\right) \cdot y, -4, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.7%

      \[\leadsto \mathsf{fma}\left(\left(z \cdot y\right) \cdot z, -4, x \cdot x\right) \]

   if 6.49999999999999982e213 < z

   1. Initial program 65.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      5. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

      6. lower-*.f64 77.3

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

   5. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.9%

      \[\leadsto \left(\left(z \cdot y\right) \cdot z\right) \cdot -4 \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+213}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot z\right) \cdot z, -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\right) \cdot z\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\_m\right) \cdot z\_m\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3.5e+61) (fma (* 4.0 t) y (* x x)) (* (* (* y z_m) z_m) -4.0)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.5e+61) {
		tmp = fma((4.0 * t), y, (x * x));
	} else {
		tmp = ((y * z_m) * z_m) * -4.0;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.5e+61)
		tmp = fma(Float64(4.0 * t), y, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(y * z_m) * z_m) * -4.0);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 3.5e+61], N[(N[(4.0 * t), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.50000000000000018e61

   1. Initial program 94.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{{x}^{2} - -4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

      1. cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{{x}^{2} + \left(\mathsf{neg}\left(-4\right)\right) \cdot \left(t \cdot y\right)} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{2} + \color{blue}{4} \cdot \left(t \cdot y\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right) + {x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} + {x}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, 4, {x}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot y}, 4, {x}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, \color{blue}{x \cdot x}\right) \]

      8. lower-*.f64 72.7

         \[\leadsto \mathsf{fma}\left(t \cdot y, 4, \color{blue}{x \cdot x}\right) \]

   5. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot y, 4, x \cdot x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.2%

      \[\leadsto \mathsf{fma}\left(t \cdot 4, \color{blue}{y}, x \cdot x\right) \]

   if 3.50000000000000018e61 < z

   1. Initial program 73.2%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot {z}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot {z}^{2}\right) \cdot -4} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({z}^{2} \cdot y\right)} \cdot -4 \]

      5. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

      6. lower-*.f64 77.0

         \[\leadsto \left(\color{blue}{\left(z \cdot z\right)} \cdot y\right) \cdot -4 \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(\left(z \cdot z\right) \cdot y\right) \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.0%

      \[\leadsto \left(\left(z \cdot y\right) \cdot z\right) \cdot -4 \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot t, y, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z\right) \cdot z\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 82000000000000:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* x x) 82000000000000.0) (* (* 4.0 t) y) (* x x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((x * x) <= 82000000000000.0) {
		tmp = (4.0 * t) * y;
	} else {
		tmp = x * x;
	}
	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) <= 82000000000000.0d0) then
        tmp = (4.0d0 * t) * y
    else
        tmp = x * x
    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) <= 82000000000000.0) {
		tmp = (4.0 * t) * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if (x * x) <= 82000000000000.0:
		tmp = (4.0 * t) * y
	else:
		tmp = x * x
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * x) <= 82000000000000.0)
		tmp = Float64(Float64(4.0 * t) * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	tmp = 0.0;
	if ((x * x) <= 82000000000000.0)
		tmp = (4.0 * t) * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 82000000000000.0], N[(N[(4.0 * t), $MachinePrecision] * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 8.2e13

   1. Initial program 91.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{4 \cdot \left(t \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]

      3. lower-*.f64 39.8

         \[\leadsto \color{blue}{\left(t \cdot y\right)} \cdot 4 \]

   5. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 4} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.8%

      \[\leadsto \left(t \cdot 4\right) \cdot \color{blue}{y} \]

   if 8.2e13 < (*.f64 x x)

   1. Initial program 88.4%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right) \cdot -4} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right) \cdot -4} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({z}^{2} - t\right) \cdot y\right)} \cdot -4 \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({z}^{2} - t\right) \cdot y\right)} \cdot -4 \]

      5. lower--.f64 N/A

         \[\leadsto \left(\color{blue}{\left({z}^{2} - t\right)} \cdot y\right) \cdot -4 \]

      6. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right) \cdot -4 \]

      7. lower-*.f64 43.0

         \[\leadsto \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right) \cdot -4 \]

   5. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4} \]

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.4

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

   8. Applied rewrites 67.4%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 82000000000000:\\ \;\;\;\;\left(4 \cdot t\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.8% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

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 = x * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

   \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right) \cdot -4} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot \left({z}^{2} - t\right)\right) \cdot -4} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left({z}^{2} - t\right) \cdot y\right)} \cdot -4 \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left({z}^{2} - t\right) \cdot y\right)} \cdot -4 \]

   5. lower--.f64 N/A

      \[\leadsto \left(\color{blue}{\left({z}^{2} - t\right)} \cdot y\right) \cdot -4 \]

   6. unpow2 N/A

      \[\leadsto \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right) \cdot -4 \]

   7. lower-*.f64 65.7

      \[\leadsto \left(\left(\color{blue}{z \cdot z} - t\right) \cdot y\right) \cdot -4 \]

5. Applied rewrites 65.7%

   \[\leadsto \color{blue}{\left(\left(z \cdot z - t\right) \cdot y\right) \cdot -4} \]

6. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 36.2

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

8. Applied rewrites 36.2%

   \[\leadsto \color{blue}{x \cdot x} \]
9. Add Preprocessing

Developer Target 1: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x x) (* 4.0 (* y (- (* z z) t)))))

C

double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * 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 = (x * x) - (4.0d0 * (y * ((z * z) - t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * x) - (4.0 * (y * ((z * z) - t)));
}

Python

def code(x, y, z, t):
	return (x * x) - (4.0 * (y * ((z * z) - t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(4.0 * Float64(y * Float64(Float64(z * z) - t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * x) - (4.0 * (y * ((z * z) - t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(4.0 * N[(y * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) (* 4 (* y (- (* z z) t)))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))