Average Error: 6.0 → 3.0
Time: 29.9s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.74920595474597988684148347386440472749 \cdot 10^{149} \lor \neg \left(z \le 1.205875807953862938552183354802704820011 \cdot 10^{154}\right):\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.74920595474597988684148347386440472749 \cdot 10^{149} \lor \neg \left(z \le 1.205875807953862938552183354802704820011 \cdot 10^{154}\right):\\
\;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r429303 = x;
        double r429304 = r429303 * r429303;
        double r429305 = y;
        double r429306 = 4.0;
        double r429307 = r429305 * r429306;
        double r429308 = z;
        double r429309 = r429308 * r429308;
        double r429310 = t;
        double r429311 = r429309 - r429310;
        double r429312 = r429307 * r429311;
        double r429313 = r429304 - r429312;
        return r429313;
}
↓
double f(double x, double y, double z, double t) {
        double r429314 = z;
        double r429315 = -1.74920595474598e+149;
        bool r429316 = r429314 <= r429315;
        double r429317 = 1.205875807953863e+154;
        bool r429318 = r429314 <= r429317;
        double r429319 = !r429318;
        bool r429320 = r429316 || r429319;
        double r429321 = x;
        double r429322 = r429321 * r429321;
        double r429323 = y;
        double r429324 = 4.0;
        double r429325 = r429323 * r429324;
        double r429326 = t;
        double r429327 = sqrt(r429326);
        double r429328 = r429314 + r429327;
        double r429329 = r429325 * r429328;
        double r429330 = r429314 - r429327;
        double r429331 = r429329 * r429330;
        double r429332 = r429322 - r429331;
        double r429333 = r429314 * r429314;
        double r429334 = r429326 - r429333;
        double r429335 = r429323 * r429334;
        double r429336 = r429324 * r429335;
        double r429337 = fma(r429321, r429321, r429336);
        double r429338 = r429320 ? r429332 : r429337;
        return r429338;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.0
Target6.0
Herbie3.0
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.74920595474598e+149 or 1.205875807953863e+154 < z

    1. Initial program 61.6

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

    3. Applied add-sqr-sqrt62.6

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

    4. Applied difference-of-squares62.6

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

    5. Applied associate-*r*30.5

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

    if -1.74920595474598e+149 < z < 1.205875807953863e+154

    1. Initial program 0.1

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

    3. Applied fma-neg0.1

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

    4. Simplified0.1

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.74920595474597988684148347386440472749 \cdot 10^{149} \lor \neg \left(z \le 1.205875807953862938552183354802704820011 \cdot 10^{154}\right):\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))