Average Error: 6.1 → 3.3
Time: 9.7s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 1.34947659414785 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(-y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 1.34947659414785 \cdot 10^{305}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r851292 = x;
        double r851293 = r851292 * r851292;
        double r851294 = y;
        double r851295 = 4.0;
        double r851296 = r851294 * r851295;
        double r851297 = z;
        double r851298 = r851297 * r851297;
        double r851299 = t;
        double r851300 = r851298 - r851299;
        double r851301 = r851296 * r851300;
        double r851302 = r851293 - r851301;
        return r851302;
}
↓
double f(double x, double y, double z, double t) {
        double r851303 = z;
        double r851304 = r851303 * r851303;
        double r851305 = 1.34947659414785e+305;
        bool r851306 = r851304 <= r851305;
        double r851307 = x;
        double r851308 = y;
        double r851309 = 4.0;
        double r851310 = r851308 * r851309;
        double r851311 = -r851310;
        double r851312 = t;
        double r851313 = r851304 - r851312;
        double r851314 = r851311 * r851313;
        double r851315 = fma(r851307, r851307, r851314);
        double r851316 = sqrt(r851312);
        double r851317 = r851303 + r851316;
        double r851318 = r851311 * r851317;
        double r851319 = r851303 - r851316;
        double r851320 = r851318 * r851319;
        double r851321 = fma(r851307, r851307, r851320);
        double r851322 = r851306 ? r851315 : r851321;
        return r851322;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.1
Target6.1
Herbie3.3
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 1.34947659414785e+305

    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

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

    if 1.34947659414785e+305 < (* z z)

    1. Initial program 62.8

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

    3. Applied fma-neg62.8

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

    4. Simplified62.8

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

    6. Applied add-sqr-sqrt63.3

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

    7. Applied difference-of-squares63.3

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

    8. Applied associate-*r*33.5

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 1.34947659414785 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(-y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(-y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))))