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

Percentage Accurate: 91.1% → 97.4%

Time: 9.8s

Alternatives: 8

Speedup: 0.8×

• 49.5% 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: 91.1% 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: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq 10^{+290}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, -t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* z z) t) 1e+290)
   (fma x x (* y (* (fma z z (- t)) -4.0)))
   (fma x x (* z (* y (* z -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * z) - t) <= 1e+290) {
		tmp = fma(x, x, (y * (fma(z, z, -t) * -4.0)));
	} else {
		tmp = fma(x, x, (z * (y * (z * -4.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(z * z) - t) <= 1e+290)
		tmp = fma(x, x, Float64(y * Float64(fma(z, z, Float64(-t)) * -4.0)));
	else
		tmp = fma(x, x, Float64(z * Float64(y * Float64(z * -4.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision], 1e+290], N[(x * x + N[(y * N[(N[(z * z + (-t)), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(z * N[(y * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 z z) t) < 1.00000000000000006e290

   1. Initial program 97.7%

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

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. metadata-eval 98.3

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

   4. Applied rewrites 98.3%

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

   if 1.00000000000000006e290 < (-.f64 (*.f64 z z) t)

   1. Initial program 54.9%

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

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. metadata-eval 65.1

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

   4. Applied rewrites 65.1%

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

   6. Applied rewrites 83.4%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 94.8

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

   9. Applied rewrites 94.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z - t \leq 10^{+290}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(\mathsf{fma}\left(z, z, -t\right) \cdot -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, z \cdot \left(y \cdot \left(z \cdot -4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot z\right) \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z z) 1e+64)
   (fma y (* t 4.0) (* x x))
   (if (<= (* z z) 2e+301)
     (fma y (* (* z z) -4.0) (* x x))
     (* z (* z (* y -4.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * z) <= 1e+64) {
		tmp = fma(y, (t * 4.0), (x * x));
	} else if ((z * z) <= 2e+301) {
		tmp = fma(y, ((z * z) * -4.0), (x * x));
	} else {
		tmp = z * (z * (y * -4.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * z) <= 1e+64)
		tmp = fma(y, Float64(t * 4.0), Float64(x * x));
	elseif (Float64(z * z) <= 2e+301)
		tmp = fma(y, Float64(Float64(z * z) * -4.0), Float64(x * x));
	else
		tmp = Float64(z * Float64(z * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+64], N[(y * N[(t * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+301], N[(y * N[(N[(z * z), $MachinePrecision] * -4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.00000000000000002e64

   1. Initial program 98.4%

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

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if 1.00000000000000002e64 < (*.f64 z z) < 2.00000000000000011e301

   1. Initial program 96.3%

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if 2.00000000000000011e301 < (*.f64 z z)

   1. Initial program 71.5%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 88.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, t \cdot 4, x \cdot x\right)\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot z\right) \cdot -4, x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 91.2% 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 2024223 
(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))))