Average Error: 6.2 → 3.2
Time: 6.0s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 1.521145199433776357765735930876915790283 \cdot 10^{298}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 1.521145199433776357765735930876915790283 \cdot 10^{298}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r661570 = x;
        double r661571 = r661570 * r661570;
        double r661572 = y;
        double r661573 = 4.0;
        double r661574 = r661572 * r661573;
        double r661575 = z;
        double r661576 = r661575 * r661575;
        double r661577 = t;
        double r661578 = r661576 - r661577;
        double r661579 = r661574 * r661578;
        double r661580 = r661571 - r661579;
        return r661580;
}
↓
double f(double x, double y, double z, double t) {
        double r661581 = z;
        double r661582 = r661581 * r661581;
        double r661583 = 1.5211451994337764e+298;
        bool r661584 = r661582 <= r661583;
        double r661585 = x;
        double r661586 = y;
        double r661587 = 4.0;
        double r661588 = r661586 * r661587;
        double r661589 = t;
        double r661590 = r661589 - r661582;
        double r661591 = r661588 * r661590;
        double r661592 = fma(r661585, r661585, r661591);
        double r661593 = sqrt(r661589);
        double r661594 = r661593 + r661581;
        double r661595 = r661588 * r661594;
        double r661596 = r661593 - r661581;
        double r661597 = r661595 * r661596;
        double r661598 = fma(r661585, r661585, r661597);
        double r661599 = r661584 ? r661592 : r661598;
        return r661599;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.2
Target6.2
Herbie3.2
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 1.5211451994337764e+298

    1. Initial program 0.1

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

    2. Simplified0.1

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

    if 1.5211451994337764e+298 < (* z z)

    1. Initial program 60.7

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

    2. Simplified60.7

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

    4. Applied add-sqr-sqrt62.4

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

    5. Applied difference-of-squares62.4

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

    6. Applied associate-*r*31.3

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 1.521145199433776357765735930876915790283 \cdot 10^{298}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))