Average Error: 2.2 → 0.1
Time: 3.8s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -0.336004926004555393 \lor \neg \left(z \le 139.52054939017245\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + a \cdot \left(z \cdot b\right)\\ \end{array}\]
\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 -0.336004926004555393 \lor \neg \left(z \le 139.52054939017245\right):\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r548222 = x;
        double r548223 = y;
        double r548224 = z;
        double r548225 = r548223 * r548224;
        double r548226 = r548222 + r548225;
        double r548227 = t;
        double r548228 = a;
        double r548229 = r548227 * r548228;
        double r548230 = r548226 + r548229;
        double r548231 = r548228 * r548224;
        double r548232 = b;
        double r548233 = r548231 * r548232;
        double r548234 = r548230 + r548233;
        return r548234;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r548235 = z;
        double r548236 = -0.3360049260045554;
        bool r548237 = r548235 <= r548236;
        double r548238 = 139.52054939017245;
        bool r548239 = r548235 <= r548238;
        double r548240 = !r548239;
        bool r548241 = r548237 || r548240;
        double r548242 = a;
        double r548243 = b;
        double r548244 = y;
        double r548245 = fma(r548242, r548243, r548244);
        double r548246 = t;
        double r548247 = x;
        double r548248 = fma(r548242, r548246, r548247);
        double r548249 = fma(r548245, r548235, r548248);
        double r548250 = r548244 * r548235;
        double r548251 = r548247 + r548250;
        double r548252 = r548246 * r548242;
        double r548253 = r548251 + r548252;
        double r548254 = r548235 * r548243;
        double r548255 = r548242 * r548254;
        double r548256 = r548253 + r548255;
        double r548257 = r548241 ? r548249 : r548256;
        return r548257;
}

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
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -0.3360049260045554 or 139.52054939017245 < z

    1. Initial program 5.2

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

    2. Simplified0.1

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

    if -0.3360049260045554 < z < 139.52054939017245

    1. Initial program 0.3

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

    3. Applied associate-*l*0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.336004926004555393 \lor \neg \left(z \le 139.52054939017245\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + a \cdot \left(z \cdot b\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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)))