Average Error: 6.1 → 3.3
Time: 4.1s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 5.2999359490719286 \cdot 10^{294}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(1 \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 5.2999359490719286 \cdot 10^{294}:\\
\;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(1 \cdot \left(z \cdot z - t\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r726826 = x;
        double r726827 = r726826 * r726826;
        double r726828 = y;
        double r726829 = 4.0;
        double r726830 = r726828 * r726829;
        double r726831 = z;
        double r726832 = r726831 * r726831;
        double r726833 = t;
        double r726834 = r726832 - r726833;
        double r726835 = r726830 * r726834;
        double r726836 = r726827 - r726835;
        return r726836;
}
↓
double f(double x, double y, double z, double t) {
        double r726837 = z;
        double r726838 = r726837 * r726837;
        double r726839 = 5.299935949071929e+294;
        bool r726840 = r726838 <= r726839;
        double r726841 = x;
        double r726842 = r726841 * r726841;
        double r726843 = y;
        double r726844 = 4.0;
        double r726845 = r726843 * r726844;
        double r726846 = 1.0;
        double r726847 = t;
        double r726848 = r726838 - r726847;
        double r726849 = r726846 * r726848;
        double r726850 = r726845 * r726849;
        double r726851 = r726842 - r726850;
        double r726852 = sqrt(r726847);
        double r726853 = r726837 + r726852;
        double r726854 = r726845 * r726853;
        double r726855 = r726837 - r726852;
        double r726856 = r726854 * r726855;
        double r726857 = r726842 - r726856;
        double r726858 = r726840 ? r726851 : r726857;
        return r726858;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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) < 5.299935949071929e+294

    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 *-un-lft-identity0.1

      \[\leadsto x \cdot x - \left(y \cdot 4\right) \cdot \color{blue}{\left(1 \cdot \left(z \cdot z - t\right)\right)}\]

    if 5.299935949071929e+294 < (* z z)

    1. Initial program 58.7

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

    3. Applied add-sqr-sqrt61.6

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

    4. Applied difference-of-squares61.6

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

    5. Applied associate-*r*31.0

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 5.2999359490719286 \cdot 10^{294}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(1 \cdot \left(z \cdot z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))))