Average Error: 6.0 → 3.1
Time: 5.0s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 1.33207297841188052 \cdot 10^{300}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 1.33207297841188052 \cdot 10^{300}:\\
\;\;\;\;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 r644598 = x;
        double r644599 = r644598 * r644598;
        double r644600 = y;
        double r644601 = 4.0;
        double r644602 = r644600 * r644601;
        double r644603 = z;
        double r644604 = r644603 * r644603;
        double r644605 = t;
        double r644606 = r644604 - r644605;
        double r644607 = r644602 * r644606;
        double r644608 = r644599 - r644607;
        return r644608;
}
↓
double f(double x, double y, double z, double t) {
        double r644609 = z;
        double r644610 = r644609 * r644609;
        double r644611 = 1.3320729784118805e+300;
        bool r644612 = r644610 <= r644611;
        double r644613 = x;
        double r644614 = r644613 * r644613;
        double r644615 = y;
        double r644616 = 4.0;
        double r644617 = r644615 * r644616;
        double r644618 = t;
        double r644619 = r644610 - r644618;
        double r644620 = r644617 * r644619;
        double r644621 = r644614 - r644620;
        double r644622 = sqrt(r644618);
        double r644623 = r644609 + r644622;
        double r644624 = r644617 * r644623;
        double r644625 = r644609 - r644622;
        double r644626 = r644624 * r644625;
        double r644627 = r644614 - r644626;
        double r644628 = r644612 ? r644621 : r644627;
        return r644628;
}

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

Original6.0
Target6.0
Herbie3.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 1.3320729784118805e+300

    1. Initial program 0.1

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

    if 1.3320729784118805e+300 < (* z z)

    1. Initial program 61.2

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

    3. Applied add-sqr-sqrt62.7

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

    4. Applied difference-of-squares62.7

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

    5. Applied associate-*r*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 simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 1.33207297841188052 \cdot 10^{300}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))