Average Error: 6.1 → 3.5
Time: 5.2s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 2.19138132073588015 \cdot 10^{282}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 2.19138132073588015 \cdot 10^{282}:\\
\;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\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 r646778 = x;
        double r646779 = r646778 * r646778;
        double r646780 = y;
        double r646781 = 4.0;
        double r646782 = r646780 * r646781;
        double r646783 = z;
        double r646784 = r646783 * r646783;
        double r646785 = t;
        double r646786 = r646784 - r646785;
        double r646787 = r646782 * r646786;
        double r646788 = r646779 - r646787;
        return r646788;
}
↓
double f(double x, double y, double z, double t) {
        double r646789 = z;
        double r646790 = r646789 * r646789;
        double r646791 = 2.1913813207358801e+282;
        bool r646792 = r646790 <= r646791;
        double r646793 = x;
        double r646794 = r646793 * r646793;
        double r646795 = y;
        double r646796 = 4.0;
        double r646797 = r646795 * r646796;
        double r646798 = t;
        double r646799 = r646790 - r646798;
        double r646800 = r646797 * r646799;
        double r646801 = r646794 - r646800;
        double r646802 = sqrt(r646798);
        double r646803 = r646789 + r646802;
        double r646804 = r646797 * r646803;
        double r646805 = r646789 - r646802;
        double r646806 = r646804 * r646805;
        double r646807 = r646794 - r646806;
        double r646808 = r646792 ? r646801 : r646807;
        return r646808;
}

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
Herbie3.5
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 2.1913813207358801e+282

    1. Initial program 0.1

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]

    if 2.1913813207358801e+282 < (* z z)

    1. Initial program 56.0

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

    3. Applied add-sqr-sqrt60.0

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

    4. Applied difference-of-squares60.0

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

    5. Applied associate-*r*31.7

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

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 2.19138132073588015 \cdot 10^{282}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))))