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

Average Error: 6.1 → 3.1

Time: 29.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 6.014499375936089855166656281444919682607 \cdot 10^{305}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r181883653 = x;
        double r181883654 = r181883653 * r181883653;
        double r181883655 = y;
        double r181883656 = 4.0;
        double r181883657 = r181883655 * r181883656;
        double r181883658 = z;
        double r181883659 = r181883658 * r181883658;
        double r181883660 = t;
        double r181883661 = r181883659 - r181883660;
        double r181883662 = r181883657 * r181883661;
        double r181883663 = r181883654 - r181883662;
        return r181883663;
}

↓

double f(double x, double y, double z, double t) {
        double r181883664 = z;
        double r181883665 = r181883664 * r181883664;
        double r181883666 = 6.01449937593609e+305;
        bool r181883667 = r181883665 <= r181883666;
        double r181883668 = x;
        double r181883669 = r181883668 * r181883668;
        double r181883670 = y;
        double r181883671 = 4.0;
        double r181883672 = r181883670 * r181883671;
        double r181883673 = t;
        double r181883674 = r181883665 - r181883673;
        double r181883675 = r181883672 * r181883674;
        double r181883676 = r181883669 - r181883675;
        double r181883677 = sqrt(r181883673);
        double r181883678 = r181883664 + r181883677;
        double r181883679 = r181883672 * r181883678;
        double r181883680 = r181883664 - r181883677;
        double r181883681 = r181883679 * r181883680;
        double r181883682 = r181883669 - r181883681;
        double r181883683 = r181883667 ? r181883676 : r181883682;
        return r181883683;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.1
Target	6.1
Herbie	3.1

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

Derivation

1. Split input into 2 regimes

2. if (* z z) < 6.01449937593609e+305

   1. Initial program 0.1

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

   if 6.01449937593609e+305 < (* z z)

   1. Initial program 62.8

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 63.2

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

   4. Applied difference-of-squares 63.2

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

   5. Applied associate-*r* 31.6

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 6.014499375936089855166656281444919682607 \cdot 10^{305}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"

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

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