Average Error: 6.1 → 3.3
Time: 18.0s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 1.34947659414785 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(4 \cdot t - 4 \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 1.34947659414785 \cdot 10^{305}:\\
\;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(4 \cdot t - 4 \cdot \left(z \cdot z\right)\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 r821661 = x;
        double r821662 = r821661 * r821661;
        double r821663 = y;
        double r821664 = 4.0;
        double r821665 = r821663 * r821664;
        double r821666 = z;
        double r821667 = r821666 * r821666;
        double r821668 = t;
        double r821669 = r821667 - r821668;
        double r821670 = r821665 * r821669;
        double r821671 = r821662 - r821670;
        return r821671;
}
↓
double f(double x, double y, double z, double t) {
        double r821672 = z;
        double r821673 = r821672 * r821672;
        double r821674 = 1.34947659414785e+305;
        bool r821675 = r821673 <= r821674;
        double r821676 = x;
        double r821677 = y;
        double r821678 = 4.0;
        double r821679 = t;
        double r821680 = r821678 * r821679;
        double r821681 = r821678 * r821673;
        double r821682 = r821680 - r821681;
        double r821683 = r821677 * r821682;
        double r821684 = fma(r821676, r821676, r821683);
        double r821685 = r821676 * r821676;
        double r821686 = r821677 * r821678;
        double r821687 = sqrt(r821679);
        double r821688 = r821672 + r821687;
        double r821689 = r821686 * r821688;
        double r821690 = r821672 - r821687;
        double r821691 = r821689 * r821690;
        double r821692 = r821685 - r821691;
        double r821693 = r821675 ? r821684 : r821692;
        return r821693;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.1
Target6.1
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) < 1.34947659414785e+305

    1. Initial program 0.1

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

    2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left({x}^{2} + 4 \cdot \left(t \cdot y\right)\right) - 4 \cdot \left({z}^{2} \cdot y\right)}\]

    3. Simplified0.1

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

    if 1.34947659414785e+305 < (* z z)

    1. Initial program 62.8

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

    3. Applied add-sqr-sqrt63.3

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

    4. Applied difference-of-squares63.3

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

    5. Applied associate-*r*33.5

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 1.34947659414785 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(4 \cdot t - 4 \cdot \left(z \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))))