Average Error: 2.2 → 0.6
Time: 3.7s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.6772471652537959 \cdot 10^{138} \lor \neg \left(b \le 0.18959099764245008\right):\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.6772471652537959 \cdot 10^{138} \lor \neg \left(b \le 0.18959099764245008\right):\\
\;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r582966 = x;
        double r582967 = y;
        double r582968 = z;
        double r582969 = r582967 * r582968;
        double r582970 = r582966 + r582969;
        double r582971 = t;
        double r582972 = a;
        double r582973 = r582971 * r582972;
        double r582974 = r582970 + r582973;
        double r582975 = r582972 * r582968;
        double r582976 = b;
        double r582977 = r582975 * r582976;
        double r582978 = r582974 + r582977;
        return r582978;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r582979 = b;
        double r582980 = -2.677247165253796e+138;
        bool r582981 = r582979 <= r582980;
        double r582982 = 0.18959099764245008;
        bool r582983 = r582979 <= r582982;
        double r582984 = !r582983;
        bool r582985 = r582981 || r582984;
        double r582986 = x;
        double r582987 = y;
        double r582988 = z;
        double r582989 = r582987 * r582988;
        double r582990 = r582986 + r582989;
        double r582991 = t;
        double r582992 = a;
        double r582993 = r582991 * r582992;
        double r582994 = r582990 + r582993;
        double r582995 = r582992 * r582988;
        double r582996 = r582995 * r582979;
        double r582997 = r582994 + r582996;
        double r582998 = fma(r582992, r582979, r582987);
        double r582999 = fma(r582992, r582991, r582986);
        double r583000 = fma(r582998, r582988, r582999);
        double r583001 = r582985 ? r582997 : r583000;
        return r583001;
}

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

Original2.2
Target0.3
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -2.677247165253796e+138 or 0.18959099764245008 < b

    1. Initial program 0.8

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

    if -2.677247165253796e+138 < b < 0.18959099764245008

    1. Initial program 2.9

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

    2. Simplified0.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.6772471652537959 \cdot 10^{138} \lor \neg \left(b \le 0.18959099764245008\right):\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))