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

Percentage Accurate: 90.5% → 97.1%

Time: 10.9s

Alternatives: 12

Speedup: 0.9×

• 48.6% 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: 90.5% 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: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(z\_m \cdot z\_m - t\right)\right)\right)\\ \mathbf{elif}\;z\_m \leq 2.8 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z\_m, z\_m, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{-0.25}{z\_m}}{y \cdot z\_m}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3.6e+138)
   (fma x x (* -4.0 (* y (- (* z_m z_m) t))))
   (if (<= z_m 2.8e+197)
     (fma (* (* -4.0 y) z_m) z_m (* x x))
     (/ 1.0 (/ (/ -0.25 z_m) (* y z_m))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.6e+138) {
		tmp = fma(x, x, (-4.0 * (y * ((z_m * z_m) - t))));
	} else if (z_m <= 2.8e+197) {
		tmp = fma(((-4.0 * y) * z_m), z_m, (x * x));
	} else {
		tmp = 1.0 / ((-0.25 / z_m) / (y * z_m));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.6e+138)
		tmp = fma(x, x, Float64(-4.0 * Float64(y * Float64(Float64(z_m * z_m) - t))));
	elseif (z_m <= 2.8e+197)
		tmp = fma(Float64(Float64(-4.0 * y) * z_m), z_m, Float64(x * x));
	else
		tmp = Float64(1.0 / Float64(Float64(-0.25 / z_m) / Float64(y * z_m)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 3.6e+138], N[(x * x + N[(-4.0 * N[(y * N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.8e+197], N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-0.25 / z$95$m), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.6000000000000001e138

   1. Initial program 92.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.6

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

   4. Applied rewrites 95.6%

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

   if 3.6000000000000001e138 < z < 2.7999999999999999e197

   1. Initial program 67.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 100.0%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 2.7999999999999999e197 < z

   1. Initial program 91.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.8

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

   4. Applied rewrites 95.8%

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

   6. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \mathsf{fma}\left(z, z, t\right)\right)\right)}}} \]

   7. Taylor expanded in z around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{4}}{y \cdot {z}^{2}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{4}}{y \cdot {z}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{4}}{\color{blue}{{z}^{2} \cdot y}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{4}}{\color{blue}{{z}^{2} \cdot y}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{4}}{\color{blue}{\left(z \cdot z\right)} \cdot y}} \]

      5. lower-*.f64 95.8

         \[\leadsto \frac{1}{\frac{-0.25}{\color{blue}{\left(z \cdot z\right)} \cdot y}} \]

   9. Applied rewrites 95.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.25}{\left(z \cdot z\right) \cdot y}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.8%

       \[\leadsto \frac{1}{\frac{\frac{-0.25}{z}}{\color{blue}{z \cdot y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{-0.25}{z}}{y \cdot z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;z\_m \cdot z\_m \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(-4, y \cdot \left(z\_m \cdot z\_m\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\_m\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 1e-6)
   (fma x x (* 4.0 (* y t)))
   (if (<= (* z_m z_m) 5e+261)
     (fma -4.0 (* y (* z_m z_m)) (* x x))
     (* (* (* -4.0 y) z_m) z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 1e-6) {
		tmp = fma(x, x, (4.0 * (y * t)));
	} else if ((z_m * z_m) <= 5e+261) {
		tmp = fma(-4.0, (y * (z_m * z_m)), (x * x));
	} else {
		tmp = ((-4.0 * y) * z_m) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e-6)
		tmp = fma(x, x, Float64(4.0 * Float64(y * t)));
	elseif (Float64(z_m * z_m) <= 5e+261)
		tmp = fma(-4.0, Float64(y * Float64(z_m * z_m)), Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-4.0 * y) * z_m) * z_m);
	end
	return 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], 1e-6], N[(x * x + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+261], N[(-4.0 * N[(y * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 9.99999999999999955e-7

   1. Initial program 98.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 9.99999999999999955e-7 < (*.f64 z z) < 5.0000000000000001e261

   1. Initial program 99.9%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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-sign-sub-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if 5.0000000000000001e261 < (*.f64 z z)

   1. Initial program 74.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 86.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(-4, y \cdot \left(z \cdot z\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;z\_m \cdot z\_m \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot \left(t - z\_m \cdot z\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\_m\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 1e-12)
   (fma x x (* 4.0 (* y t)))
   (if (<= (* z_m z_m) 5e+261)
     (* (* 4.0 y) (- t (* z_m z_m)))
     (* (* (* -4.0 y) z_m) z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 1e-12) {
		tmp = fma(x, x, (4.0 * (y * t)));
	} else if ((z_m * z_m) <= 5e+261) {
		tmp = (4.0 * y) * (t - (z_m * z_m));
	} else {
		tmp = ((-4.0 * y) * z_m) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e-12)
		tmp = fma(x, x, Float64(4.0 * Float64(y * t)));
	elseif (Float64(z_m * z_m) <= 5e+261)
		tmp = Float64(Float64(4.0 * y) * Float64(t - Float64(z_m * z_m)));
	else
		tmp = Float64(Float64(Float64(-4.0 * y) * z_m) * z_m);
	end
	return 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], 1e-12], N[(x * x + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+261], N[(N[(4.0 * y), $MachinePrecision] * N[(t - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 9.9999999999999998e-13

   1. Initial program 98.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 96.3

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

   7. Applied rewrites 96.3%

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

   if 9.9999999999999998e-13 < (*.f64 z z) < 5.0000000000000001e261

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
   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. sub-neg N/A

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

      6. unpow2 N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\left(z \cdot z + \color{blue}{-1 \cdot t}\right) \cdot y\right) \cdot -4 \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(z, z, -1 \cdot t\right)} \cdot y\right) \cdot -4 \]

      9. mul-1-neg N/A

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

      10. lower-neg.f64 74.2

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 74.2%

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

   if 5.0000000000000001e261 < (*.f64 z z)

   1. Initial program 74.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 86.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;z\_m \cdot z\_m \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\left(\mathsf{fma}\left(z\_m, z\_m, -t\right) \cdot y\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\_m\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 1e-12)
   (fma x x (* 4.0 (* y t)))
   (if (<= (* z_m z_m) 5e+261)
     (* (* (fma z_m z_m (- t)) y) -4.0)
     (* (* (* -4.0 y) z_m) z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 1e-12) {
		tmp = fma(x, x, (4.0 * (y * t)));
	} else if ((z_m * z_m) <= 5e+261) {
		tmp = (fma(z_m, z_m, -t) * y) * -4.0;
	} else {
		tmp = ((-4.0 * y) * z_m) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e-12)
		tmp = fma(x, x, Float64(4.0 * Float64(y * t)));
	elseif (Float64(z_m * z_m) <= 5e+261)
		tmp = Float64(Float64(fma(z_m, z_m, Float64(-t)) * y) * -4.0);
	else
		tmp = Float64(Float64(Float64(-4.0 * y) * z_m) * z_m);
	end
	return 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], 1e-12], N[(x * x + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+261], N[(N[(N[(z$95$m * z$95$m + (-t)), $MachinePrecision] * y), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 9.9999999999999998e-13

   1. Initial program 98.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 96.3

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

   7. Applied rewrites 96.3%

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

   if 9.9999999999999998e-13 < (*.f64 z z) < 5.0000000000000001e261

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-4 \cdot \left(y \cdot \left({z}^{2} - t\right)\right)} \]
   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. sub-neg N/A

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

      6. unpow2 N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\left(z \cdot z + \color{blue}{-1 \cdot t}\right) \cdot y\right) \cdot -4 \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(z, z, -1 \cdot t\right)} \cdot y\right) \cdot -4 \]

      9. mul-1-neg N/A

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

      10. lower-neg.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if 5.0000000000000001e261 < (*.f64 z z)

   1. Initial program 74.0%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 86.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+261}:\\ \;\;\;\;\left(\mathsf{fma}\left(z, z, -t\right) \cdot y\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \left(4 \cdot y\right) \cdot t\\ \mathbf{if}\;z\_m \leq 4 \cdot 10^{-234}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z\_m \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z\_m \leq 0.0015:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\_m\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (* (* 4.0 y) t)))
   (if (<= z_m 4e-234)
     t_1
     (if (<= z_m 2.1e-26)
       (* x x)
       (if (<= z_m 0.0015) t_1 (* (* (* -4.0 y) z_m) z_m))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (4.0 * y) * t;
	double tmp;
	if (z_m <= 4e-234) {
		tmp = t_1;
	} else if (z_m <= 2.1e-26) {
		tmp = x * x;
	} else if (z_m <= 0.0015) {
		tmp = t_1;
	} else {
		tmp = ((-4.0 * y) * 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) :: t_1
    real(8) :: tmp
    t_1 = (4.0d0 * y) * t
    if (z_m <= 4d-234) then
        tmp = t_1
    else if (z_m <= 2.1d-26) then
        tmp = x * x
    else if (z_m <= 0.0015d0) then
        tmp = t_1
    else
        tmp = (((-4.0d0) * y) * 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 t_1 = (4.0 * y) * t;
	double tmp;
	if (z_m <= 4e-234) {
		tmp = t_1;
	} else if (z_m <= 2.1e-26) {
		tmp = x * x;
	} else if (z_m <= 0.0015) {
		tmp = t_1;
	} else {
		tmp = ((-4.0 * y) * z_m) * z_m;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (4.0 * y) * t
	tmp = 0
	if z_m <= 4e-234:
		tmp = t_1
	elif z_m <= 2.1e-26:
		tmp = x * x
	elif z_m <= 0.0015:
		tmp = t_1
	else:
		tmp = ((-4.0 * y) * z_m) * z_m
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(4.0 * y) * t)
	tmp = 0.0
	if (z_m <= 4e-234)
		tmp = t_1;
	elseif (z_m <= 2.1e-26)
		tmp = Float64(x * x);
	elseif (z_m <= 0.0015)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-4.0 * y) * z_m) * z_m);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (4.0 * y) * t;
	tmp = 0.0;
	if (z_m <= 4e-234)
		tmp = t_1;
	elseif (z_m <= 2.1e-26)
		tmp = x * x;
	elseif (z_m <= 0.0015)
		tmp = t_1;
	else
		tmp = ((-4.0 * y) * z_m) * z_m;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z$95$m, 4e-234], t$95$1, If[LessEqual[z$95$m, 2.1e-26], N[(x * x), $MachinePrecision], If[LessEqual[z$95$m, 0.0015], t$95$1, N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.9999999999999998e-234 or 2.10000000000000008e-26 < z < 0.0015

   1. Initial program 89.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 33.9

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

   5. Applied rewrites 33.9%

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

   if 3.9999999999999998e-234 < z < 2.10000000000000008e-26

   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 y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if 0.0015 < z

   1. Initial program 90.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 76.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-234}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 0.0015:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_1 := \left(4 \cdot y\right) \cdot t\\ \mathbf{if}\;z\_m \leq 4 \cdot 10^{-234}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z\_m \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z\_m \leq 0.0015:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(z\_m \cdot z\_m\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (let* ((t_1 (* (* 4.0 y) t)))
   (if (<= z_m 4e-234)
     t_1
     (if (<= z_m 2.1e-26)
       (* x x)
       (if (<= z_m 0.0015) t_1 (* (* y (* z_m z_m)) -4.0))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double t_1 = (4.0 * y) * t;
	double tmp;
	if (z_m <= 4e-234) {
		tmp = t_1;
	} else if (z_m <= 2.1e-26) {
		tmp = x * x;
	} else if (z_m <= 0.0015) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (4.0d0 * y) * t
    if (z_m <= 4d-234) then
        tmp = t_1
    else if (z_m <= 2.1d-26) then
        tmp = x * x
    else if (z_m <= 0.0015d0) then
        tmp = t_1
    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 t_1 = (4.0 * y) * t;
	double tmp;
	if (z_m <= 4e-234) {
		tmp = t_1;
	} else if (z_m <= 2.1e-26) {
		tmp = x * x;
	} else if (z_m <= 0.0015) {
		tmp = t_1;
	} else {
		tmp = (y * (z_m * z_m)) * -4.0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	t_1 = (4.0 * y) * t
	tmp = 0
	if z_m <= 4e-234:
		tmp = t_1
	elif z_m <= 2.1e-26:
		tmp = x * x
	elif z_m <= 0.0015:
		tmp = t_1
	else:
		tmp = (y * (z_m * z_m)) * -4.0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	t_1 = Float64(Float64(4.0 * y) * t)
	tmp = 0.0
	if (z_m <= 4e-234)
		tmp = t_1;
	elseif (z_m <= 2.1e-26)
		tmp = Float64(x * x);
	elseif (z_m <= 0.0015)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * Float64(z_m * z_m)) * -4.0);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m, t)
	t_1 = (4.0 * y) * t;
	tmp = 0.0;
	if (z_m <= 4e-234)
		tmp = t_1;
	elseif (z_m <= 2.1e-26)
		tmp = x * x;
	elseif (z_m <= 0.0015)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z$95$m, 4e-234], t$95$1, If[LessEqual[z$95$m, 2.1e-26], N[(x * x), $MachinePrecision], If[LessEqual[z$95$m, 0.0015], t$95$1, N[(N[(y * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.9999999999999998e-234 or 2.10000000000000008e-26 < z < 0.0015

   1. Initial program 89.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 33.9

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

   5. Applied rewrites 33.9%

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

   if 3.9999999999999998e-234 < z < 2.10000000000000008e-26

   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 y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if 0.0015 < z

   1. Initial program 90.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-234}:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;z \leq 0.0015:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(z \cdot z\right)\right) \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z\_m, z\_m, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* z_m z_m) 1e-6)
   (fma x x (* 4.0 (* y t)))
   (fma (* (* -4.0 y) z_m) z_m (* x x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((z_m * z_m) <= 1e-6) {
		tmp = fma(x, x, (4.0 * (y * t)));
	} else {
		tmp = fma(((-4.0 * y) * z_m), z_m, (x * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 1e-6)
		tmp = fma(x, x, Float64(4.0 * Float64(y * t)));
	else
		tmp = fma(Float64(Float64(-4.0 * y) * z_m), z_m, Float64(x * x));
	end
	return 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], 1e-6], N[(x * x + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999955e-7

   1. Initial program 98.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 9.99999999999999955e-7 < (*.f64 z z)

   1. Initial program 85.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 88.1%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.5

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

   7. Applied rewrites 89.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(z\_m \cdot z\_m - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z\_m, z\_m, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 3.6e+138)
   (fma x x (* -4.0 (* y (- (* z_m z_m) t))))
   (fma (* (* -4.0 y) z_m) z_m (* x x))))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.6e+138) {
		tmp = fma(x, x, (-4.0 * (y * ((z_m * z_m) - t))));
	} else {
		tmp = fma(((-4.0 * y) * z_m), z_m, (x * x));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.6e+138)
		tmp = fma(x, x, Float64(-4.0 * Float64(y * Float64(Float64(z_m * z_m) - t))));
	else
		tmp = fma(Float64(Float64(-4.0 * y) * z_m), z_m, Float64(x * x));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 3.6e+138], N[(x * x + N[(-4.0 * N[(y * N[(N[(z$95$m * z$95$m), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.6000000000000001e138

   1. Initial program 92.8%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.6

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

   4. Applied rewrites 95.6%

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

   if 3.6000000000000001e138 < z

   1. Initial program 83.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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 88.9%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.2

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

   7. Applied rewrites 97.2%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4 \cdot y\right) \cdot z, z, x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 9.2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\_m\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 9.2e+62) (fma x x (* 4.0 (* y t))) (* (* (* -4.0 y) z_m) z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 9.2e+62) {
		tmp = fma(x, x, (4.0 * (y * t)));
	} else {
		tmp = ((-4.0 * y) * z_m) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 9.2e+62)
		tmp = fma(x, x, Float64(4.0 * Float64(y * t)));
	else
		tmp = Float64(Float64(Float64(-4.0 * y) * z_m) * z_m);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 9.2e+62], N[(x * x + N[(4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.19999999999999936e62

   1. Initial program 92.3%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 95.2

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 73.5

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

   7. Applied rewrites 73.5%

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

   if 9.19999999999999936e62 < z

   1. Initial program 88.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 83.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 9.2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\_m\right) \cdot z\_m\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= z_m 9.2e+62) (fma (* y t) 4.0 (* x x)) (* (* (* -4.0 y) z_m) z_m)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 9.2e+62) {
		tmp = fma((y * t), 4.0, (x * x));
	} else {
		tmp = ((-4.0 * y) * z_m) * z_m;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 9.2e+62)
		tmp = fma(Float64(y * t), 4.0, Float64(x * x));
	else
		tmp = Float64(Float64(Float64(-4.0 * y) * z_m) * z_m);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_, t_] := If[LessEqual[z$95$m, 9.2e+62], N[(N[(y * t), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * y), $MachinePrecision] * z$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.19999999999999936e62

   1. Initial program 92.3%

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

   3. Taylor expanded in z around 0

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

      1. cancel-sign-sub-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 72.5

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

   5. Applied rewrites 72.5%

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

   if 9.19999999999999936e62 < z

   1. Initial program 88.6%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 83.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t, 4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-4 \cdot y\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.34:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m t)
 :precision binary64
 (if (<= (* x x) 0.34) (* (* 4.0 y) t) (* x x)))

C

z_m = fabs(z);
double code(double x, double y, double z_m, double t) {
	double tmp;
	if ((x * x) <= 0.34) {
		tmp = (4.0 * y) * t;
	} 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) <= 0.34d0) then
        tmp = (4.0d0 * y) * t
    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) <= 0.34) {
		tmp = (4.0 * y) * t;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m, t):
	tmp = 0
	if (x * x) <= 0.34:
		tmp = (4.0 * y) * t
	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) <= 0.34)
		tmp = Float64(Float64(4.0 * y) * t);
	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) <= 0.34)
		tmp = (4.0 * y) * t;
	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], 0.34], N[(N[(4.0 * y), $MachinePrecision] * t), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.340000000000000024

   1. Initial program 97.7%

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
   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

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if 0.340000000000000024 < (*.f64 x x)

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.34:\\ \;\;\;\;\left(4 \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.4% 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 91.5%

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

3. Taylor expanded in y around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 39.8

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

5. Applied rewrites 39.8%

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

Developer Target 1: 90.5% 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 2024243 
(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))))