Average Error: 2.2 → 0.3
Time: 4.1s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -78273077770423380213760 \lor \neg \left(b \le 5.979029897606913827193515324797296998831 \cdot 10^{-67}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, t, x\right)\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]
\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 -78273077770423380213760 \lor \neg \left(b \le 5.979029897606913827193515324797296998831 \cdot 10^{-67}\right):\\
\;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, t, x\right)\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 r717764 = x;
        double r717765 = y;
        double r717766 = z;
        double r717767 = r717765 * r717766;
        double r717768 = r717764 + r717767;
        double r717769 = t;
        double r717770 = a;
        double r717771 = r717769 * r717770;
        double r717772 = r717768 + r717771;
        double r717773 = r717770 * r717766;
        double r717774 = b;
        double r717775 = r717773 * r717774;
        double r717776 = r717772 + r717775;
        return r717776;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r717777 = b;
        double r717778 = -7.827307777042338e+22;
        bool r717779 = r717777 <= r717778;
        double r717780 = 5.979029897606914e-67;
        bool r717781 = r717777 <= r717780;
        double r717782 = !r717781;
        bool r717783 = r717779 || r717782;
        double r717784 = y;
        double r717785 = z;
        double r717786 = a;
        double r717787 = t;
        double r717788 = x;
        double r717789 = fma(r717786, r717787, r717788);
        double r717790 = fma(r717784, r717785, r717789);
        double r717791 = r717786 * r717785;
        double r717792 = r717791 * r717777;
        double r717793 = r717790 + r717792;
        double r717794 = fma(r717786, r717777, r717784);
        double r717795 = fma(r717794, r717785, r717789);
        double r717796 = r717783 ? r717793 : r717795;
        return r717796;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888128:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.758974318836428710669076838657752600596 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -7.827307777042338e+22 or 5.979029897606914e-67 < b

    1. Initial program 0.6

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

    if -7.827307777042338e+22 < b < 5.979029897606914e-67

    1. Initial program 3.7

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

    2. Simplified0.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -78273077770423380213760 \lor \neg \left(b \le 5.979029897606913827193515324797296998831 \cdot 10^{-67}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, t, x\right)\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))