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

Average Error: 6.0 → 0.2

Time: 14.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r385772 = x;
        double r385773 = r385772 * r385772;
        double r385774 = y;
        double r385775 = 4.0;
        double r385776 = r385774 * r385775;
        double r385777 = z;
        double r385778 = r385777 * r385777;
        double r385779 = t;
        double r385780 = r385778 - r385779;
        double r385781 = r385776 * r385780;
        double r385782 = r385773 - r385781;
        return r385782;
}

↓

double f(double x, double y, double z, double t) {
        double r385783 = x;
        double r385784 = y;
        double r385785 = -r385784;
        double r385786 = z;
        double r385787 = r385785 * r385786;
        double r385788 = 4.0;
        double r385789 = cbrt(r385788);
        double r385790 = r385786 * r385789;
        double r385791 = r385787 * r385790;
        double r385792 = cbrt(r385789);
        double r385793 = r385789 * r385792;
        double r385794 = r385792 * r385792;
        double r385795 = r385793 * r385794;
        double r385796 = r385791 * r385795;
        double r385797 = t;
        double r385798 = r385797 * r385784;
        double r385799 = r385798 * r385788;
        double r385800 = r385796 + r385799;
        double r385801 = fma(r385783, r385783, r385800);
        return r385801;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.0
Target	6.0
Herbie	0.2

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

Derivation

1. Initial program 6.0

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

2. Simplified 6.0

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

3. Taylor expanded around inf 6.0

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

4. Simplified 6.0

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

6. Applied fma-udef 6.0

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

7. Applied distribute-lft-in 6.0

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

8. Simplified 0.1

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

9. Simplified 0.1

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

11. Applied add-cube-cbrt 0.4

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

12. Applied associate-*l* 0.4

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

13. Simplified 0.4

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

15. Applied add-cube-cbrt 0.2

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

16. Applied add-cube-cbrt 0.2

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

17. Applied swap-sqr 0.2

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

18. Simplified 0.2

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

19. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019174 +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))))