Average Error: 5.9 → 3.1
Time: 5.9s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 7.154430218809535669448245048773269532254 \cdot 10^{294}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right) + \left(y \cdot 4\right) \cdot \mathsf{fma}\left(-z, z, z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 7.154430218809535669448245048773269532254 \cdot 10^{294}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \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)\\

\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 r678431 = x;
        double r678432 = r678431 * r678431;
        double r678433 = y;
        double r678434 = 4.0;
        double r678435 = r678433 * r678434;
        double r678436 = z;
        double r678437 = r678436 * r678436;
        double r678438 = t;
        double r678439 = r678437 - r678438;
        double r678440 = r678435 * r678439;
        double r678441 = r678432 - r678440;
        return r678441;
}
↓
double f(double x, double y, double z, double t) {
        double r678442 = z;
        double r678443 = r678442 * r678442;
        double r678444 = 7.154430218809536e+294;
        bool r678445 = r678443 <= r678444;
        double r678446 = x;
        double r678447 = y;
        double r678448 = 4.0;
        double r678449 = r678447 * r678448;
        double r678450 = t;
        double r678451 = r678450 - r678443;
        double r678452 = r678449 * r678451;
        double r678453 = -r678442;
        double r678454 = fma(r678453, r678442, r678443);
        double r678455 = r678449 * r678454;
        double r678456 = r678452 + r678455;
        double r678457 = fma(r678446, r678446, r678456);
        double r678458 = sqrt(r678450);
        double r678459 = r678458 + r678442;
        double r678460 = r678449 * r678459;
        double r678461 = r678458 - r678442;
        double r678462 = r678460 * r678461;
        double r678463 = fma(r678446, r678446, r678462);
        double r678464 = r678445 ? r678457 : r678463;
        return r678464;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.9
Target5.9
Herbie3.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 7.154430218809536e+294

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-sqr-sqrt32.3

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

    5. Applied prod-diff32.3

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

    6. Applied distribute-lft-in32.3

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

    7. Simplified0.1

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

    if 7.154430218809536e+294 < (* z z)

    1. Initial program 59.6

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

    2. Simplified59.6

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

    4. Applied add-sqr-sqrt61.4

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

    5. Applied difference-of-squares61.4

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

    6. Applied associate-*r*30.7

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 7.154430218809535669448245048773269532254 \cdot 10^{294}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right) + \left(y \cdot 4\right) \cdot \mathsf{fma}\left(-z, z, z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))))