Average Error: 5.8 → 3.3
Time: 15.5s
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 r640058 = x;
        double r640059 = r640058 * r640058;
        double r640060 = y;
        double r640061 = 4.0;
        double r640062 = r640060 * r640061;
        double r640063 = z;
        double r640064 = r640063 * r640063;
        double r640065 = t;
        double r640066 = r640064 - r640065;
        double r640067 = r640062 * r640066;
        double r640068 = r640059 - r640067;
        return r640068;
}
↓
double f(double x, double y, double z, double t) {
        double r640069 = z;
        double r640070 = r640069 * r640069;
        double r640071 = 1.1891168189428968e+303;
        bool r640072 = r640070 <= r640071;
        double r640073 = x;
        double r640074 = r640073 * r640073;
        double r640075 = y;
        double r640076 = 4.0;
        double r640077 = r640075 * r640076;
        double r640078 = t;
        double r640079 = r640070 - r640078;
        double r640080 = r640077 * r640079;
        double r640081 = r640074 - r640080;
        double r640082 = sqrt(r640078);
        double r640083 = r640069 + r640082;
        double r640084 = r640077 * r640083;
        double r640085 = r640069 - r640082;
        double r640086 = r640084 * r640085;
        double r640087 = r640074 - r640086;
        double r640088 = r640072 ? r640081 : r640087;
        return r640088;
}

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))))