Average Error: 6.3 → 3.4
Time: 17.5s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 4.903521817442067659834887694586779442201 \cdot 10^{301}:\\ \;\;\;\;\left(x \cdot x - y \cdot \left(\left(z \cdot z\right) \cdot 4\right)\right) - y \cdot \left(4 \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 4.903521817442067659834887694586779442201 \cdot 10^{301}:\\
\;\;\;\;\left(x \cdot x - y \cdot \left(\left(z \cdot z\right) \cdot 4\right)\right) - y \cdot \left(4 \cdot \left(-t\right)\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 r417595 = x;
        double r417596 = r417595 * r417595;
        double r417597 = y;
        double r417598 = 4.0;
        double r417599 = r417597 * r417598;
        double r417600 = z;
        double r417601 = r417600 * r417600;
        double r417602 = t;
        double r417603 = r417601 - r417602;
        double r417604 = r417599 * r417603;
        double r417605 = r417596 - r417604;
        return r417605;
}
↓
double f(double x, double y, double z, double t) {
        double r417606 = z;
        double r417607 = r417606 * r417606;
        double r417608 = 4.903521817442068e+301;
        bool r417609 = r417607 <= r417608;
        double r417610 = x;
        double r417611 = r417610 * r417610;
        double r417612 = y;
        double r417613 = 4.0;
        double r417614 = r417607 * r417613;
        double r417615 = r417612 * r417614;
        double r417616 = r417611 - r417615;
        double r417617 = t;
        double r417618 = -r417617;
        double r417619 = r417613 * r417618;
        double r417620 = r417612 * r417619;
        double r417621 = r417616 - r417620;
        double r417622 = r417612 * r417613;
        double r417623 = sqrt(r417617);
        double r417624 = r417606 + r417623;
        double r417625 = r417622 * r417624;
        double r417626 = r417606 - r417623;
        double r417627 = r417625 * r417626;
        double r417628 = r417611 - r417627;
        double r417629 = r417609 ? r417621 : r417628;
        return r417629;
}

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

Original6.3
Target6.3
Herbie3.4
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 4.903521817442068e+301

    1. Initial program 0.1

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

    3. Applied associate-*l*0.1

      \[\leadsto x \cdot x - \color{blue}{y \cdot \left(4 \cdot \left(z \cdot z - t\right)\right)}\]
    4. Using strategy rm

    5. Applied sub-neg0.1

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

    6. Applied distribute-lft-in0.1

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

    7. Applied distribute-lft-in0.1

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

    8. Applied associate--r+0.1

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

    9. Simplified0.1

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

    if 4.903521817442068e+301 < (* z z)

    1. Initial program 61.6

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
    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*32.5

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 4.903521817442067659834887694586779442201 \cdot 10^{301}:\\ \;\;\;\;\left(x \cdot x - y \cdot \left(\left(z \cdot z\right) \cdot 4\right)\right) - y \cdot \left(4 \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))