Average Error: 1.9 → 0.2
Time: 17.3s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.157691058688136 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \mathbf{elif}\;z \le 2.415354527944333 \cdot 10^{-97}:\\ \;\;\;\;a \cdot \left(b \cdot z\right) + \left(\left(z \cdot y + x\right) + t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array}\]
\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 -5.157691058688136 \cdot 10^{+22}:\\
\;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\

\mathbf{elif}\;z \le 2.415354527944333 \cdot 10^{-97}:\\
\;\;\;\;a \cdot \left(b \cdot z\right) + \left(\left(z \cdot y + x\right) + t \cdot a\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r28036920 = x;
        double r28036921 = y;
        double r28036922 = z;
        double r28036923 = r28036921 * r28036922;
        double r28036924 = r28036920 + r28036923;
        double r28036925 = t;
        double r28036926 = a;
        double r28036927 = r28036925 * r28036926;
        double r28036928 = r28036924 + r28036927;
        double r28036929 = r28036926 * r28036922;
        double r28036930 = b;
        double r28036931 = r28036929 * r28036930;
        double r28036932 = r28036928 + r28036931;
        return r28036932;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r28036933 = z;
        double r28036934 = -5.157691058688136e+22;
        bool r28036935 = r28036933 <= r28036934;
        double r28036936 = t;
        double r28036937 = a;
        double r28036938 = b;
        double r28036939 = y;
        double r28036940 = fma(r28036937, r28036938, r28036939);
        double r28036941 = x;
        double r28036942 = fma(r28036933, r28036940, r28036941);
        double r28036943 = fma(r28036936, r28036937, r28036942);
        double r28036944 = 2.415354527944333e-97;
        bool r28036945 = r28036933 <= r28036944;
        double r28036946 = r28036938 * r28036933;
        double r28036947 = r28036937 * r28036946;
        double r28036948 = r28036933 * r28036939;
        double r28036949 = r28036948 + r28036941;
        double r28036950 = r28036936 * r28036937;
        double r28036951 = r28036949 + r28036950;
        double r28036952 = r28036947 + r28036951;
        double r28036953 = r28036945 ? r28036952 : r28036943;
        double r28036954 = r28036935 ? r28036943 : r28036953;
        return r28036954;
}

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

Original1.9
Target0.4
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -1.1820553527347888 \cdot 10^{+19}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.157691058688136e+22 or 2.415354527944333e-97 < z

    1. Initial program 3.6

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

    2. Simplified0.4

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

    if -5.157691058688136e+22 < z < 2.415354527944333e-97

    1. Initial program 0.5

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

    3. Applied associate-*l*0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.157691058688136 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \mathbf{elif}\;z \le 2.415354527944333 \cdot 10^{-97}:\\ \;\;\;\;a \cdot \left(b \cdot z\right) + \left(\left(z \cdot y + x\right) + t \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"

  :herbie-target
  (if (< z -1.1820553527347888e+19) (+ (* 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)))