Average Error: 2.1 → 0.2
Time: 3.4s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.5804298148702942 \cdot 10^{51} \lor \neg \left(z \le 8.799567967124541 \cdot 10^{-53}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.5804298148702942 \cdot 10^{51} \lor \neg \left(z \le 8.799567967124541 \cdot 10^{-53}\right):\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r579969 = x;
        double r579970 = y;
        double r579971 = z;
        double r579972 = r579970 * r579971;
        double r579973 = r579969 + r579972;
        double r579974 = t;
        double r579975 = a;
        double r579976 = r579974 * r579975;
        double r579977 = r579973 + r579976;
        double r579978 = r579975 * r579971;
        double r579979 = b;
        double r579980 = r579978 * r579979;
        double r579981 = r579977 + r579980;
        return r579981;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r579982 = z;
        double r579983 = -2.5804298148702942e+51;
        bool r579984 = r579982 <= r579983;
        double r579985 = 8.799567967124541e-53;
        bool r579986 = r579982 <= r579985;
        double r579987 = !r579986;
        bool r579988 = r579984 || r579987;
        double r579989 = a;
        double r579990 = b;
        double r579991 = y;
        double r579992 = fma(r579989, r579990, r579991);
        double r579993 = t;
        double r579994 = x;
        double r579995 = fma(r579989, r579993, r579994);
        double r579996 = fma(r579992, r579982, r579995);
        double r579997 = 1.0;
        double r579998 = r579982 * r579990;
        double r579999 = r579993 + r579998;
        double r580000 = fma(r579982, r579991, r579994);
        double r580001 = fma(r579989, r579999, r580000);
        double r580002 = r579997 * r580001;
        double r580003 = r579988 ? r579996 : r580002;
        return r580003;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original2.1
Target0.4
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.5804298148702942e+51 or 8.799567967124541e-53 < z

    1. Initial program 4.7

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]

    2. Simplified0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)}\]

    if -2.5804298148702942e+51 < z < 8.799567967124541e-53

    1. Initial program 0.4

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

    3. Applied associate-+l+0.4

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

    4. Simplified0.4

      \[\leadsto \left(x + y \cdot z\right) + \color{blue}{\mathsf{fma}\left(t, a, \left(a \cdot z\right) \cdot b\right)}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity0.4

      \[\leadsto \left(x + y \cdot z\right) + \color{blue}{1 \cdot \mathsf{fma}\left(t, a, \left(a \cdot z\right) \cdot b\right)}\]

    7. Applied *-un-lft-identity0.4

      \[\leadsto \color{blue}{1 \cdot \left(x + y \cdot z\right)} + 1 \cdot \mathsf{fma}\left(t, a, \left(a \cdot z\right) \cdot b\right)\]

    8. Applied distribute-lft-out0.4

      \[\leadsto \color{blue}{1 \cdot \left(\left(x + y \cdot z\right) + \mathsf{fma}\left(t, a, \left(a \cdot z\right) \cdot b\right)\right)}\]

    9. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.5804298148702942 \cdot 10^{51} \lor \neg \left(z \le 8.799567967124541 \cdot 10^{-53}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(a, t + z \cdot b, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))