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

Percentage Accurate: 90.6% → 97.5%

Time: 10.5s

Alternatives: 7

Speedup: 0.4×

• 49.6% 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: 90.6% 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: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq 10^{-44}:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(y, t \cdot -4, \left(y \cdot z\right) \cdot \left(4 \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y 4.0) 1e-44)
   (- (* x x) (fma y (* t -4.0) (* (* y z) (* 4.0 z))))
   (fma x x (* (- (* z z) t) (* y -4.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * 4.0) <= 1e-44) {
		tmp = (x * x) - fma(y, (t * -4.0), ((y * z) * (4.0 * z)));
	} else {
		tmp = fma(x, x, (((z * z) - t) * (y * -4.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * 4.0) <= 1e-44)
		tmp = Float64(Float64(x * x) - fma(y, Float64(t * -4.0), Float64(Float64(y * z) * Float64(4.0 * z))));
	else
		tmp = fma(x, x, Float64(Float64(Float64(z * z) - t) * Float64(y * -4.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * 4.0), $MachinePrecision], 1e-44], N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * N[(4.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision] * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y 4) < 9.99999999999999953e-45

   1. Initial program 91.1%

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

      1. *-un-lft-identity 91.1%

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

      2. add-cube-cbrt 90.7%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(1 \cdot \left(z \cdot z\right) - \color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right) \]

      3. prod-diff 90.7%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(\mathsf{fma}\left(1, z \cdot z, -\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)} \]

      4. associate-*l* 90.7%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\mathsf{fma}\left(1, z \cdot z, -\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) \]

      5. add-cube-cbrt 91.1%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\mathsf{fma}\left(1, z \cdot z, -\color{blue}{t}\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) \]

      6. fma-neg 91.1%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\color{blue}{\left(1 \cdot \left(z \cdot z\right) - t\right)} + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) \]

      7. *-un-lft-identity 91.1%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\left(\color{blue}{z \cdot z} - t\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) \]

      8. pow2 91.1%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\left(\color{blue}{{z}^{2}} - t\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) \]

      9. pow2 91.1%

         \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\left({z}^{2} - t\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \color{blue}{{\left(\sqrt[3]{t}\right)}^{2}}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) \]

      10. associate-*l* 91.1%

          \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\left({z}^{2} - t\right) + \mathsf{fma}\left(-\sqrt[3]{t}, {\left(\sqrt[3]{t}\right)}^{2}, \color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)\right) \]

      11. add-cube-cbrt 90.7%

          \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \left(\left({z}^{2} - t\right) + \mathsf{fma}\left(-\sqrt[3]{t}, {\left(\sqrt[3]{t}\right)}^{2}, \color{blue}{t}\right)\right) \]

   4. Applied egg-rr 90.7%

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(\left({z}^{2} - t\right) + \mathsf{fma}\left(-\sqrt[3]{t}, {\left(\sqrt[3]{t}\right)}^{2}, t\right)\right)} \]

   5. Taylor expanded in z around 0 88.2%

      \[\leadsto x \cdot x - \color{blue}{\left(-4 \cdot \left({1}^{0.3333333333333333} \cdot \left(t \cdot y\right)\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. pow-base-1 88.2%

         \[\leadsto x \cdot x - \left(-4 \cdot \left(\color{blue}{1} \cdot \left(t \cdot y\right)\right) + 4 \cdot \left(y \cdot {z}^{2}\right)\right) \]

      2. associate-*r* 88.2%

         \[\leadsto x \cdot x - \left(\color{blue}{\left(-4 \cdot 1\right) \cdot \left(t \cdot y\right)} + 4 \cdot \left(y \cdot {z}^{2}\right)\right) \]

      3. metadata-eval 88.2%

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

      4. *-commutative 88.2%

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

      5. *-commutative 88.2%

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

      6. associate-*r* 88.2%

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

      7. fma-def 91.1%

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

      8. associate-*r* 90.5%

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

      9. *-commutative 90.5%

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

      10. associate-*l* 91.1%

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

   7. Simplified 91.1%

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

      1. add-sqr-sqrt 44.1%

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

      2. pow2 44.1%

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

      3. sqrt-prod 23.8%

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

      4. *-commutative 23.8%

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

      5. sqrt-prod 23.8%

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

      6. unpow2 23.8%

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

      7. sqrt-prod 13.8%

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

      8. add-sqr-sqrt 27.7%

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

      9. metadata-eval 27.7%

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

   9. Applied egg-rr 27.7%

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

       1. pow-prod-down 23.8%

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

       2. pow2 23.8%

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

       3. add-sqr-sqrt 91.1%

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

       4. *-commutative 91.1%

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

       5. unpow-prod-down 91.1%

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

       6. metadata-eval 91.1%

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

       7. associate-*l* 90.5%

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

       8. unpow2 90.5%

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

       9. associate-*l* 97.1%

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

       10. *-commutative 97.1%

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

       11. associate-*l* 97.6%

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

       12. associate-*r* 97.6%

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

   11. Applied egg-rr 97.6%

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

   if 9.99999999999999953e-45 < (*.f64 y 4)

   1. Initial program 95.2%

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

      1. fma-neg 100.0%

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

      2. distribute-lft-neg-in 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 4 \leq 10^{-44}:\\ \;\;\;\;x \cdot x - \mathsf{fma}\left(y, t \cdot -4, \left(y \cdot z\right) \cdot \left(4 \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, t \cdot \left(y \cdot \left(--4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* x x) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (fma x x (* t (* y (- -4.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(x, x, (t * (y * -(-4.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(x, x, Float64(t * Float64(y * Float64(-(-4.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(x * x + N[(t * N[(y * (--4.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 95.8%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

      1. fma-neg 44.4%

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

      2. distribute-lft-neg-in 44.4%

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

      3. *-commutative 44.4%

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

      4. distribute-rgt-neg-in 44.4%

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

      5. metadata-eval 44.4%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in z around 0 55.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(-1 \cdot t\right)} \cdot \left(y \cdot -4\right)\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 55.6%

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

   7. Simplified 55.6%

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

4. Final simplification 94.4%

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

Alternative 3: 92.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. fma-neg 94.0%

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

   2. distribute-lft-neg-in 94.0%

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

   3. *-commutative 94.0%

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

   4. distribute-rgt-neg-in 94.0%

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

   5. metadata-eval 94.0%

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

3. Simplified 94.0%

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

5. Final simplification 94.0%

   \[\leadsto \mathsf{fma}\left(x, x, \left(z \cdot z - t\right) \cdot \left(y \cdot -4\right)\right) \]
6. Add Preprocessing

Alternative 4: 93.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* x x) 2e+305) (+ (* x x) (* (* y 4.0) (- t (* z z)))) (pow x 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+305) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = pow(x, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * x) <= 2d+305) then
        tmp = (x * x) + ((y * 4.0d0) * (t - (z * z)))
    else
        tmp = x ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x * x) <= 2e+305) {
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	} else {
		tmp = Math.pow(x, 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x * x) <= 2e+305:
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)))
	else:
		tmp = math.pow(x, 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x * x) <= 2e+305)
		tmp = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))));
	else
		tmp = x ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x * x) <= 2e+305)
		tmp = (x * x) + ((y * 4.0) * (t - (z * z)));
	else
		tmp = x ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+305], N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[x, 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.9999999999999999e305

   1. Initial program 94.2%

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

   if 1.9999999999999999e305 < (*.f64 x x)

   1. Initial program 88.3%

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

   3. Taylor expanded in x around inf 94.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot x + \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - y \cdot \left(t \cdot -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* x x) (* (* y 4.0) (- t (* z z))))))
   (if (<= t_1 INFINITY) t_1 (- (* x x) (* y (* t -4.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (x * x) - (y * (t * -4.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = (x * x) - (y * (t * -4.0))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) + Float64(Float64(y * 4.0) * Float64(t - Float64(z * z))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x * x) - Float64(y * Float64(t * -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) + ((y * 4.0) * (t - (z * z)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = (x * x) - (y * (t * -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + N[(N[(y * 4.0), $MachinePrecision] * N[(t - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(x * x), $MachinePrecision] - N[(y * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t))) < +inf.0

   1. Initial program 95.8%

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

   if +inf.0 < (-.f64 (*.f64 x x) (*.f64 (*.f64 y 4) (-.f64 (*.f64 z z) t)))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0 44.4%

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

      1. *-commutative 44.4%

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

      2. *-commutative 44.4%

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

      3. associate-*l* 44.4%

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

   5. Simplified 44.4%

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

4. Final simplification 94.0%

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

Alternative 6: 67.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * x) - (y * (t * (-4.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 67.9%

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

   1. *-commutative 67.9%

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

   2. *-commutative 67.9%

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

   3. associate-*l* 67.9%

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

5. Simplified 67.9%

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

6. Final simplification 67.9%

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

Alternative 7: 31.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in t around inf 30.2%

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

   1. *-commutative 30.2%

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

5. Simplified 30.2%

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

6. Final simplification 30.2%

   \[\leadsto 4 \cdot \left(y \cdot t\right) \]
7. Add Preprocessing

Developer target: 90.6% 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 2024031 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

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