Average Error: 6.1 → 3.9
Time: 4.7s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 1.2800532743869097 \cdot 10^{275}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 1.2800532743869097 \cdot 10^{275}:\\
\;\;\;\;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 r652213 = x;
        double r652214 = r652213 * r652213;
        double r652215 = y;
        double r652216 = 4.0;
        double r652217 = r652215 * r652216;
        double r652218 = z;
        double r652219 = r652218 * r652218;
        double r652220 = t;
        double r652221 = r652219 - r652220;
        double r652222 = r652217 * r652221;
        double r652223 = r652214 - r652222;
        return r652223;
}
↓
double f(double x, double y, double z, double t) {
        double r652224 = z;
        double r652225 = r652224 * r652224;
        double r652226 = 1.2800532743869097e+275;
        bool r652227 = r652225 <= r652226;
        double r652228 = x;
        double r652229 = r652228 * r652228;
        double r652230 = y;
        double r652231 = 4.0;
        double r652232 = r652230 * r652231;
        double r652233 = t;
        double r652234 = r652225 - r652233;
        double r652235 = r652232 * r652234;
        double r652236 = r652229 - r652235;
        double r652237 = sqrt(r652233);
        double r652238 = r652224 + r652237;
        double r652239 = r652232 * r652238;
        double r652240 = r652224 - r652237;
        double r652241 = r652239 * r652240;
        double r652242 = r652229 - r652241;
        double r652243 = r652227 ? r652236 : r652242;
        return r652243;
}

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
Herbie3.9
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 1.2800532743869097e+275

    1. Initial program 0.1

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

    if 1.2800532743869097e+275 < (* z z)

    1. Initial program 52.0

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

    3. Applied add-sqr-sqrt58.4

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

    4. Applied difference-of-squares58.4

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

    5. Applied associate-*r*32.9

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 1.2800532743869097 \cdot 10^{275}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))