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 r629335 = x;
        double r629336 = y;
        double r629337 = z;
        double r629338 = r629336 * r629337;
        double r629339 = r629335 + r629338;
        double r629340 = t;
        double r629341 = a;
        double r629342 = r629340 * r629341;
        double r629343 = r629339 + r629342;
        double r629344 = r629341 * r629337;
        double r629345 = b;
        double r629346 = r629344 * r629345;
        double r629347 = r629343 + r629346;
        return r629347;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r629348 = b;
        double r629349 = -7.827307777042338e+22;
        bool r629350 = r629348 <= r629349;
        double r629351 = 5.979029897606914e-67;
        bool r629352 = r629348 <= r629351;
        double r629353 = !r629352;
        bool r629354 = r629350 || r629353;
        double r629355 = y;
        double r629356 = z;
        double r629357 = a;
        double r629358 = t;
        double r629359 = x;
        double r629360 = fma(r629357, r629358, r629359);
        double r629361 = fma(r629355, r629356, r629360);
        double r629362 = r629357 * r629356;
        double r629363 = r629362 * r629348;
        double r629364 = r629361 + r629363;
        double r629365 = fma(r629357, r629348, r629355);
        double r629366 = fma(r629365, r629356, r629360);
        double r629367 = r629354 ? r629364 : r629366;
        return r629367;
}

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)))