Average Error: 6.9 → 2.2
Time: 3.4s
Precision: 64
\[\left(x \cdot y - z \cdot y\right) \cdot t\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -5.2825142284684672 \cdot 10^{125} \lor \neg \left(x \cdot y - z \cdot y \le 3.0182044239998063 \cdot 10^{117}\right):\\ \;\;\;\;1 \cdot \left(\left(t \cdot y\right) \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, -z \cdot y\right) \cdot t\\ \end{array}\]
\left(x \cdot y - z \cdot y\right) \cdot t
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y - z \cdot y \le -5.2825142284684672 \cdot 10^{125} \lor \neg \left(x \cdot y - z \cdot y \le 3.0182044239998063 \cdot 10^{117}\right):\\
\;\;\;\;1 \cdot \left(\left(t \cdot y\right) \cdot \left(x - z\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r510826 = x;
        double r510827 = y;
        double r510828 = r510826 * r510827;
        double r510829 = z;
        double r510830 = r510829 * r510827;
        double r510831 = r510828 - r510830;
        double r510832 = t;
        double r510833 = r510831 * r510832;
        return r510833;
}
↓
double f(double x, double y, double z, double t) {
        double r510834 = x;
        double r510835 = y;
        double r510836 = r510834 * r510835;
        double r510837 = z;
        double r510838 = r510837 * r510835;
        double r510839 = r510836 - r510838;
        double r510840 = -5.282514228468467e+125;
        bool r510841 = r510839 <= r510840;
        double r510842 = 3.0182044239998063e+117;
        bool r510843 = r510839 <= r510842;
        double r510844 = !r510843;
        bool r510845 = r510841 || r510844;
        double r510846 = 1.0;
        double r510847 = t;
        double r510848 = r510847 * r510835;
        double r510849 = r510834 - r510837;
        double r510850 = r510848 * r510849;
        double r510851 = r510846 * r510850;
        double r510852 = -r510838;
        double r510853 = fma(r510834, r510835, r510852);
        double r510854 = r510853 * r510847;
        double r510855 = r510845 ? r510851 : r510854;
        return r510855;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.9
Target3.2
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;t \lt -9.2318795828867769 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t \lt 2.5430670515648771 \cdot 10^{83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* z y)) < -5.282514228468467e+125 or 3.0182044239998063e+117 < (- (* x y) (* z y))

    1. Initial program 17.9

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

    3. Applied *-un-lft-identity17.9

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

    4. Applied associate-*l*17.9

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

    5. Simplified2.7

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

    if -5.282514228468467e+125 < (- (* x y) (* z y)) < 3.0182044239998063e+117

    1. Initial program 1.9

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

    3. Applied fma-neg1.9

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -5.2825142284684672 \cdot 10^{125} \lor \neg \left(x \cdot y - z \cdot y \le 3.0182044239998063 \cdot 10^{117}\right):\\ \;\;\;\;1 \cdot \left(\left(t \cdot y\right) \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, -z \cdot y\right) \cdot t\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

  (* (- (* x y) (* z y)) t))