Average Error: 6.0 → 3.0
Time: 21.3s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.339683571742010173916707057681142627271 \cdot 10^{154} \lor \neg \left(z \le 1.343821229368198900070514089418053427071 \cdot 10^{154}\right):\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.339683571742010173916707057681142627271 \cdot 10^{154} \lor \neg \left(z \le 1.343821229368198900070514089418053427071 \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}:\\
\;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r475828 = x;
        double r475829 = r475828 * r475828;
        double r475830 = y;
        double r475831 = 4.0;
        double r475832 = r475830 * r475831;
        double r475833 = z;
        double r475834 = r475833 * r475833;
        double r475835 = t;
        double r475836 = r475834 - r475835;
        double r475837 = r475832 * r475836;
        double r475838 = r475829 - r475837;
        return r475838;
}
↓
double f(double x, double y, double z, double t) {
        double r475839 = z;
        double r475840 = -1.3396835717420102e+154;
        bool r475841 = r475839 <= r475840;
        double r475842 = 1.343821229368199e+154;
        bool r475843 = r475839 <= r475842;
        double r475844 = !r475843;
        bool r475845 = r475841 || r475844;
        double r475846 = x;
        double r475847 = r475846 * r475846;
        double r475848 = y;
        double r475849 = 4.0;
        double r475850 = r475848 * r475849;
        double r475851 = t;
        double r475852 = sqrt(r475851);
        double r475853 = r475839 + r475852;
        double r475854 = r475850 * r475853;
        double r475855 = r475839 - r475852;
        double r475856 = r475854 * r475855;
        double r475857 = r475847 - r475856;
        double r475858 = -r475851;
        double r475859 = fma(r475839, r475839, r475858);
        double r475860 = r475850 * r475859;
        double r475861 = r475847 - r475860;
        double r475862 = r475845 ? r475857 : r475861;
        return r475862;
}

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 < -1.3396835717420102e+154 or 1.343821229368199e+154 < z

    1. Initial program 64.0

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

    3. Applied add-sqr-sqrt64.0

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

    4. Applied difference-of-squares64.0

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

    5. Applied associate-*r*31.8

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

    if -1.3396835717420102e+154 < z < 1.343821229368199e+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

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.339683571742010173916707057681142627271 \cdot 10^{154} \lor \neg \left(z \le 1.343821229368198900070514089418053427071 \cdot 10^{154}\right):\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \mathsf{fma}\left(z, z, -t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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))))