Average Error: 6.0 → 3.0
Time: 23.1s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 2.03849011016511565786308164633915074586 \cdot 10^{302}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z \cdot z - t\right), y \cdot 4, \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right) + \mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 2.03849011016511565786308164633915074586 \cdot 10^{302}:\\
\;\;\;\;\mathsf{fma}\left(-\left(z \cdot z - t\right), y \cdot 4, \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right) + \mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r25681175 = x;
        double r25681176 = r25681175 * r25681175;
        double r25681177 = y;
        double r25681178 = 4.0;
        double r25681179 = r25681177 * r25681178;
        double r25681180 = z;
        double r25681181 = r25681180 * r25681180;
        double r25681182 = t;
        double r25681183 = r25681181 - r25681182;
        double r25681184 = r25681179 * r25681183;
        double r25681185 = r25681176 - r25681184;
        return r25681185;
}
↓
double f(double x, double y, double z, double t) {
        double r25681186 = z;
        double r25681187 = r25681186 * r25681186;
        double r25681188 = 2.0384901101651157e+302;
        bool r25681189 = r25681187 <= r25681188;
        double r25681190 = t;
        double r25681191 = r25681187 - r25681190;
        double r25681192 = -r25681191;
        double r25681193 = y;
        double r25681194 = 4.0;
        double r25681195 = r25681193 * r25681194;
        double r25681196 = r25681195 * r25681191;
        double r25681197 = fma(r25681192, r25681195, r25681196);
        double r25681198 = x;
        double r25681199 = -r25681196;
        double r25681200 = fma(r25681198, r25681198, r25681199);
        double r25681201 = r25681197 + r25681200;
        double r25681202 = r25681198 * r25681198;
        double r25681203 = sqrt(r25681190);
        double r25681204 = r25681203 + r25681186;
        double r25681205 = r25681195 * r25681204;
        double r25681206 = r25681186 - r25681203;
        double r25681207 = r25681205 * r25681206;
        double r25681208 = r25681202 - r25681207;
        double r25681209 = r25681189 ? r25681201 : r25681208;
        return r25681209;
}

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 z) < 2.0384901101651157e+302

    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 prod-diff0.1

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

    if 2.0384901101651157e+302 < (* z z)

    1. Initial program 61.2

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

    3. Applied add-sqr-sqrt62.8

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

    4. Applied difference-of-squares62.8

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

    5. Applied associate-*r*30.3

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 2.03849011016511565786308164633915074586 \cdot 10^{302}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z \cdot z - t\right), y \cdot 4, \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right) + \mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))