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

Average Error: 6.1 → 3.1

Time: 26.9s

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 r85558475 = x;
        double r85558476 = r85558475 * r85558475;
        double r85558477 = y;
        double r85558478 = 4.0;
        double r85558479 = r85558477 * r85558478;
        double r85558480 = z;
        double r85558481 = r85558480 * r85558480;
        double r85558482 = t;
        double r85558483 = r85558481 - r85558482;
        double r85558484 = r85558479 * r85558483;
        double r85558485 = r85558476 - r85558484;
        return r85558485;
}

↓

double f(double x, double y, double z, double t) {
        double r85558486 = z;
        double r85558487 = r85558486 * r85558486;
        double r85558488 = 6.01449937593609e+305;
        bool r85558489 = r85558487 <= r85558488;
        double r85558490 = x;
        double r85558491 = r85558490 * r85558490;
        double r85558492 = y;
        double r85558493 = 4.0;
        double r85558494 = r85558492 * r85558493;
        double r85558495 = t;
        double r85558496 = r85558487 - r85558495;
        double r85558497 = r85558494 * r85558496;
        double r85558498 = r85558491 - r85558497;
        double r85558499 = sqrt(r85558495);
        double r85558500 = r85558486 + r85558499;
        double r85558501 = r85558494 * r85558500;
        double r85558502 = r85558486 - r85558499;
        double r85558503 = r85558501 * r85558502;
        double r85558504 = r85558491 - r85558503;
        double r85558505 = r85558489 ? r85558498 : r85558504;
        return r85558505;
}

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 +o rules:numerics
(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))))