Average Error: 1.9 → 0.2
Time: 18.3s
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 r28421920 = x;
        double r28421921 = y;
        double r28421922 = z;
        double r28421923 = r28421921 * r28421922;
        double r28421924 = r28421920 + r28421923;
        double r28421925 = t;
        double r28421926 = a;
        double r28421927 = r28421925 * r28421926;
        double r28421928 = r28421924 + r28421927;
        double r28421929 = r28421926 * r28421922;
        double r28421930 = b;
        double r28421931 = r28421929 * r28421930;
        double r28421932 = r28421928 + r28421931;
        return r28421932;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r28421933 = z;
        double r28421934 = -5.157691058688136e+22;
        bool r28421935 = r28421933 <= r28421934;
        double r28421936 = t;
        double r28421937 = a;
        double r28421938 = b;
        double r28421939 = y;
        double r28421940 = fma(r28421937, r28421938, r28421939);
        double r28421941 = x;
        double r28421942 = fma(r28421933, r28421940, r28421941);
        double r28421943 = fma(r28421936, r28421937, r28421942);
        double r28421944 = 2.415354527944333e-97;
        bool r28421945 = r28421933 <= r28421944;
        double r28421946 = r28421938 * r28421933;
        double r28421947 = r28421937 * r28421946;
        double r28421948 = r28421933 * r28421939;
        double r28421949 = r28421948 + r28421941;
        double r28421950 = r28421936 * r28421937;
        double r28421951 = r28421949 + r28421950;
        double r28421952 = r28421947 + r28421951;
        double r28421953 = r28421945 ? r28421952 : r28421943;
        double r28421954 = r28421935 ? r28421943 : r28421953;
        return r28421954;
}

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)))