Average Error: 3.6 → 0.3
Time: 2.3s
Precision: 64
\[x \cdot \left(1 - y \cdot z\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot z = -\infty \lor \neg \left(y \cdot z \le 2.669129644743586370097185904379114062632 \cdot 10^{138}\right):\\ \;\;\;\;x \cdot 1 + \left(x \cdot y\right) \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right) + x \cdot \mathsf{fma}\left(-z, y, z \cdot y\right)\\ \end{array}\]
x \cdot \left(1 - y \cdot z\right)
↓
\begin{array}{l}
\mathbf{if}\;y \cdot z = -\infty \lor \neg \left(y \cdot z \le 2.669129644743586370097185904379114062632 \cdot 10^{138}\right):\\
\;\;\;\;x \cdot 1 + \left(x \cdot y\right) \cdot \left(-z\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(1 - y \cdot z\right) + x \cdot \mathsf{fma}\left(-z, y, z \cdot y\right)\\

\end{array}
double f(double x, double y, double z) {
        double r284994 = x;
        double r284995 = 1.0;
        double r284996 = y;
        double r284997 = z;
        double r284998 = r284996 * r284997;
        double r284999 = r284995 - r284998;
        double r285000 = r284994 * r284999;
        return r285000;
}
↓
double f(double x, double y, double z) {
        double r285001 = y;
        double r285002 = z;
        double r285003 = r285001 * r285002;
        double r285004 = -inf.0;
        bool r285005 = r285003 <= r285004;
        double r285006 = 2.6691296447435864e+138;
        bool r285007 = r285003 <= r285006;
        double r285008 = !r285007;
        bool r285009 = r285005 || r285008;
        double r285010 = x;
        double r285011 = 1.0;
        double r285012 = r285010 * r285011;
        double r285013 = r285010 * r285001;
        double r285014 = -r285002;
        double r285015 = r285013 * r285014;
        double r285016 = r285012 + r285015;
        double r285017 = r285011 - r285003;
        double r285018 = r285010 * r285017;
        double r285019 = r285002 * r285001;
        double r285020 = fma(r285014, r285001, r285019);
        double r285021 = r285010 * r285020;
        double r285022 = r285018 + r285021;
        double r285023 = r285009 ? r285016 : r285022;
        return r285023;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if (* y z) < -inf.0 or 2.6691296447435864e+138 < (* y z)

    1. Initial program 27.8

      \[x \cdot \left(1 - y \cdot z\right)\]
    2. Using strategy rm

    3. Applied sub-neg27.8

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(-y \cdot z\right)\right)}\]

    4. Applied distribute-lft-in27.8

      \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-y \cdot z\right)}\]
    5. Using strategy rm

    6. Applied distribute-rgt-neg-in27.8

      \[\leadsto x \cdot 1 + x \cdot \color{blue}{\left(y \cdot \left(-z\right)\right)}\]

    7. Applied associate-*r*2.0

      \[\leadsto x \cdot 1 + \color{blue}{\left(x \cdot y\right) \cdot \left(-z\right)}\]

    if -inf.0 < (* y z) < 2.6691296447435864e+138

    1. Initial program 0.1

      \[x \cdot \left(1 - y \cdot z\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.1

      \[\leadsto x \cdot \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}} - y \cdot z\right)\]

    4. Applied prod-diff0.1

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -z \cdot y\right) + \mathsf{fma}\left(-z, y, z \cdot y\right)\right)}\]

    5. Applied distribute-lft-in0.1

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -z \cdot y\right) + x \cdot \mathsf{fma}\left(-z, y, z \cdot y\right)}\]

    6. Simplified0.1

      \[\leadsto \color{blue}{x \cdot \left(1 - y \cdot z\right)} + x \cdot \mathsf{fma}\left(-z, y, z \cdot y\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z = -\infty \lor \neg \left(y \cdot z \le 2.669129644743586370097185904379114062632 \cdot 10^{138}\right):\\ \;\;\;\;x \cdot 1 + \left(x \cdot y\right) \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right) + x \cdot \mathsf{fma}\left(-z, y, z \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1 (* y z))))