Average Error: 6.2 → 3.3
Time: 5.5s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 8.492390771886926299590046168507855606141 \cdot 10^{288}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 8.492390771886926299590046168507855606141 \cdot 10^{288}:\\
\;\;\;\;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 r712910 = x;
        double r712911 = r712910 * r712910;
        double r712912 = y;
        double r712913 = 4.0;
        double r712914 = r712912 * r712913;
        double r712915 = z;
        double r712916 = r712915 * r712915;
        double r712917 = t;
        double r712918 = r712916 - r712917;
        double r712919 = r712914 * r712918;
        double r712920 = r712911 - r712919;
        return r712920;
}
↓
double f(double x, double y, double z, double t) {
        double r712921 = z;
        double r712922 = r712921 * r712921;
        double r712923 = 8.492390771886926e+288;
        bool r712924 = r712922 <= r712923;
        double r712925 = x;
        double r712926 = r712925 * r712925;
        double r712927 = y;
        double r712928 = 4.0;
        double r712929 = r712927 * r712928;
        double r712930 = t;
        double r712931 = r712922 - r712930;
        double r712932 = r712929 * r712931;
        double r712933 = r712926 - r712932;
        double r712934 = sqrt(r712930);
        double r712935 = r712921 + r712934;
        double r712936 = r712929 * r712935;
        double r712937 = r712921 - r712934;
        double r712938 = r712936 * r712937;
        double r712939 = r712926 - r712938;
        double r712940 = r712924 ? r712933 : r712939;
        return r712940;
}

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

Original6.2
Target6.2
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) < 8.492390771886926e+288

    1. Initial program 0.1

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

    if 8.492390771886926e+288 < (* z z)

    1. Initial program 56.1

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

    3. Applied add-sqr-sqrt59.7

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

    4. Applied difference-of-squares59.7

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

    5. Applied associate-*r*29.4

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 8.492390771886926299590046168507855606141 \cdot 10^{288}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))))