Average Error: 6.2 → 0.1
Time: 8.3s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot 0\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot 0\right)
double f(double x, double y, double z, double t) {
        double r672472 = x;
        double r672473 = r672472 * r672472;
        double r672474 = y;
        double r672475 = 4.0;
        double r672476 = r672474 * r672475;
        double r672477 = z;
        double r672478 = r672477 * r672477;
        double r672479 = t;
        double r672480 = r672478 - r672479;
        double r672481 = r672476 * r672480;
        double r672482 = r672473 - r672481;
        return r672482;
}
↓
double f(double x, double y, double z, double t) {
        double r672483 = x;
        double r672484 = y;
        double r672485 = 4.0;
        double r672486 = r672484 * r672485;
        double r672487 = t;
        double r672488 = r672486 * r672487;
        double r672489 = z;
        double r672490 = -r672489;
        double r672491 = r672486 * r672490;
        double r672492 = r672491 * r672489;
        double r672493 = r672488 + r672492;
        double r672494 = 0.0;
        double r672495 = r672486 * r672494;
        double r672496 = r672493 + r672495;
        double r672497 = fma(r672483, r672483, r672496);
        return r672497;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.2
Target6.1
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 6.2

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

  2. Simplified6.2

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

  4. Applied add-sqr-sqrt35.1

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

  5. Applied prod-diff35.1

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

  6. Applied distribute-lft-in35.1

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

  7. Simplified6.2

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

  9. Applied sub-neg6.2

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

  10. Applied distribute-lft-in6.2

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

  12. Applied distribute-lft-neg-in6.2

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

  13. Applied associate-*r*6.2

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

  14. Taylor expanded around 0 0.1

    \[\leadsto \mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot \color{blue}{0}\right)\]

  15. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot 0\right)\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4) (- (* z z) t))))