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

Average Error: 5.8 → 3.0

Time: 15.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) = -\infty:\\ \;\;\;\;x \cdot x - \sqrt{\left(\left(\sqrt{t} + z\right) \cdot \left(y \cdot 4\right)\right) \cdot \left(z - \sqrt{t}\right)} \cdot \sqrt{\left(\left(\sqrt{t} + z\right) \cdot \left(y \cdot 4\right)\right) \cdot \left(z - \sqrt{t}\right)}\\ \mathbf{elif}\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \le 3.667644209517229293046663353825080797984 \cdot 10^{304}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(\sqrt{t} + z\right) \cdot \left(y \cdot 4\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r39165699 = x;
        double r39165700 = r39165699 * r39165699;
        double r39165701 = y;
        double r39165702 = 4.0;
        double r39165703 = r39165701 * r39165702;
        double r39165704 = z;
        double r39165705 = r39165704 * r39165704;
        double r39165706 = t;
        double r39165707 = r39165705 - r39165706;
        double r39165708 = r39165703 * r39165707;
        double r39165709 = r39165700 - r39165708;
        return r39165709;
}

↓

double f(double x, double y, double z, double t) {
        double r39165710 = x;
        double r39165711 = r39165710 * r39165710;
        double r39165712 = y;
        double r39165713 = 4.0;
        double r39165714 = r39165712 * r39165713;
        double r39165715 = z;
        double r39165716 = r39165715 * r39165715;
        double r39165717 = t;
        double r39165718 = r39165716 - r39165717;
        double r39165719 = r39165714 * r39165718;
        double r39165720 = r39165711 - r39165719;
        double r39165721 = -inf.0;
        bool r39165722 = r39165720 <= r39165721;
        double r39165723 = sqrt(r39165717);
        double r39165724 = r39165723 + r39165715;
        double r39165725 = r39165724 * r39165714;
        double r39165726 = r39165715 - r39165723;
        double r39165727 = r39165725 * r39165726;
        double r39165728 = sqrt(r39165727);
        double r39165729 = r39165728 * r39165728;
        double r39165730 = r39165711 - r39165729;
        double r39165731 = 3.6676442095172293e+304;
        bool r39165732 = r39165720 <= r39165731;
        double r39165733 = r39165711 - r39165727;
        double r39165734 = r39165732 ? r39165720 : r39165733;
        double r39165735 = r39165722 ? r39165730 : r39165734;
        return r39165735;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.8
Target	5.8
Herbie	3.0

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

Derivation

1. Split input into 3 regimes

2. if (- (* x x) (* (* y 4.0) (- (* z z) t))) < -inf.0

   1. Initial program 64.0

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

   3. Applied add-sqr-sqrt 64.0

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

   4. Applied difference-of-squares 64.0

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

   5. Applied associate-*r* 31.2

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

   7. Applied add-sqr-sqrt 31.3

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

   if -inf.0 < (- (* x x) (* (* y 4.0) (- (* z z) t))) < 3.6676442095172293e+304

   1. Initial program 0.1

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

   if 3.6676442095172293e+304 < (- (* x x) (* (* y 4.0) (- (* z z) t)))

   1. Initial program 57.0

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

   3. Applied add-sqr-sqrt 60.1

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

   4. Applied difference-of-squares 60.1

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

   5. Applied associate-*r* 31.4

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) = -\infty:\\ \;\;\;\;x \cdot x - \sqrt{\left(\left(\sqrt{t} + z\right) \cdot \left(y \cdot 4\right)\right) \cdot \left(z - \sqrt{t}\right)} \cdot \sqrt{\left(\left(\sqrt{t} + z\right) \cdot \left(y \cdot 4\right)\right) \cdot \left(z - \sqrt{t}\right)}\\ \mathbf{elif}\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \le 3.667644209517229293046663353825080797984 \cdot 10^{304}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(\sqrt{t} + z\right) \cdot \left(y \cdot 4\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))