Average Error: 2.1 → 0.3
Time: 13.8s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -44.4689510467186224 \lor \neg \left(a \le 5.91085096997945351 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\sqrt[3]{\left(a \cdot z\right) \cdot b} \cdot \left(\sqrt[3]{a \cdot z} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;a \le -44.4689510467186224 \lor \neg \left(a \le 5.91085096997945351 \cdot 10^{-37}\right):\\
\;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\sqrt[3]{\left(a \cdot z\right) \cdot b} \cdot \left(\sqrt[3]{a \cdot z} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r390933 = x;
        double r390934 = y;
        double r390935 = z;
        double r390936 = r390934 * r390935;
        double r390937 = r390933 + r390936;
        double r390938 = t;
        double r390939 = a;
        double r390940 = r390938 * r390939;
        double r390941 = r390937 + r390940;
        double r390942 = r390939 * r390935;
        double r390943 = b;
        double r390944 = r390942 * r390943;
        double r390945 = r390941 + r390944;
        return r390945;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r390946 = a;
        double r390947 = -44.46895104671862;
        bool r390948 = r390946 <= r390947;
        double r390949 = 5.9108509699794535e-37;
        bool r390950 = r390946 <= r390949;
        double r390951 = !r390950;
        bool r390952 = r390948 || r390951;
        double r390953 = z;
        double r390954 = y;
        double r390955 = b;
        double r390956 = t;
        double r390957 = fma(r390953, r390955, r390956);
        double r390958 = x;
        double r390959 = fma(r390957, r390946, r390958);
        double r390960 = fma(r390953, r390954, r390959);
        double r390961 = r390954 * r390953;
        double r390962 = r390958 + r390961;
        double r390963 = r390956 * r390946;
        double r390964 = r390962 + r390963;
        double r390965 = r390946 * r390953;
        double r390966 = r390965 * r390955;
        double r390967 = cbrt(r390966);
        double r390968 = cbrt(r390965);
        double r390969 = cbrt(r390955);
        double r390970 = r390968 * r390969;
        double r390971 = r390967 * r390970;
        double r390972 = r390971 * r390967;
        double r390973 = r390964 + r390972;
        double r390974 = r390952 ? r390960 : r390973;
        return r390974;
}

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

Derivation

  1. Split input into 2 regimes

  2. if a < -44.46895104671862 or 5.9108509699794535e-37 < a

    1. Initial program 4.6

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

    2. Simplified0.1

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

    if -44.46895104671862 < a < 5.9108509699794535e-37

    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 add-cube-cbrt0.5

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{\left(\sqrt[3]{\left(a \cdot z\right) \cdot b} \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}\right) \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}}\]
    4. Using strategy rm

    5. Applied cbrt-prod0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -44.4689510467186224 \lor \neg \left(a \le 5.91085096997945351 \cdot 10^{-37}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, \mathsf{fma}\left(\mathsf{fma}\left(z, b, t\right), a, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\sqrt[3]{\left(a \cdot z\right) \cdot b} \cdot \left(\sqrt[3]{a \cdot z} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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)))