Average Error: 6.0 → 3.0
Time: 25.3s
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 r26529143 = x;
        double r26529144 = r26529143 * r26529143;
        double r26529145 = y;
        double r26529146 = 4.0;
        double r26529147 = r26529145 * r26529146;
        double r26529148 = z;
        double r26529149 = r26529148 * r26529148;
        double r26529150 = t;
        double r26529151 = r26529149 - r26529150;
        double r26529152 = r26529147 * r26529151;
        double r26529153 = r26529144 - r26529152;
        return r26529153;
}
↓
double f(double x, double y, double z, double t) {
        double r26529154 = z;
        double r26529155 = r26529154 * r26529154;
        double r26529156 = 2.0384901101651157e+302;
        bool r26529157 = r26529155 <= r26529156;
        double r26529158 = t;
        double r26529159 = r26529155 - r26529158;
        double r26529160 = -r26529159;
        double r26529161 = y;
        double r26529162 = 4.0;
        double r26529163 = r26529161 * r26529162;
        double r26529164 = r26529163 * r26529159;
        double r26529165 = fma(r26529160, r26529163, r26529164);
        double r26529166 = x;
        double r26529167 = -r26529164;
        double r26529168 = fma(r26529166, r26529166, r26529167);
        double r26529169 = r26529165 + r26529168;
        double r26529170 = r26529166 * r26529166;
        double r26529171 = sqrt(r26529158);
        double r26529172 = r26529171 + r26529154;
        double r26529173 = r26529163 * r26529172;
        double r26529174 = r26529154 - r26529171;
        double r26529175 = r26529173 * r26529174;
        double r26529176 = r26529170 - r26529175;
        double r26529177 = r26529157 ? r26529169 : r26529176;
        return r26529177;
}

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))))