Average Error: 5.8 → 3.3
Time: 15.7s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 1.18911681894289682 \cdot 10^{303}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 1.18911681894289682 \cdot 10^{303}:\\
\;\;\;\;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 r800927 = x;
        double r800928 = r800927 * r800927;
        double r800929 = y;
        double r800930 = 4.0;
        double r800931 = r800929 * r800930;
        double r800932 = z;
        double r800933 = r800932 * r800932;
        double r800934 = t;
        double r800935 = r800933 - r800934;
        double r800936 = r800931 * r800935;
        double r800937 = r800928 - r800936;
        return r800937;
}
↓
double f(double x, double y, double z, double t) {
        double r800938 = z;
        double r800939 = r800938 * r800938;
        double r800940 = 1.1891168189428968e+303;
        bool r800941 = r800939 <= r800940;
        double r800942 = x;
        double r800943 = r800942 * r800942;
        double r800944 = y;
        double r800945 = 4.0;
        double r800946 = r800944 * r800945;
        double r800947 = t;
        double r800948 = r800939 - r800947;
        double r800949 = r800946 * r800948;
        double r800950 = r800943 - r800949;
        double r800951 = sqrt(r800947);
        double r800952 = r800938 + r800951;
        double r800953 = r800946 * r800952;
        double r800954 = r800938 - r800951;
        double r800955 = r800953 * r800954;
        double r800956 = r800943 - r800955;
        double r800957 = r800941 ? r800950 : r800956;
        return r800957;
}

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

Original5.8
Target5.7
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) < 1.1891168189428968e+303

    1. Initial program 0.1

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

    if 1.1891168189428968e+303 < (* z z)

    1. Initial program 61.8

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

    3. Applied add-sqr-sqrt63.0

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

    4. Applied difference-of-squares63.0

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

    5. Applied associate-*r*34.6

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 1.18911681894289682 \cdot 10^{303}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))