Average Error: 2.0 → 0.9
Time: 3.9s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.3683211863065988 \cdot 10^{160} \lor \neg \left(z \le 3.7182254012398825 \cdot 10^{-136}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.3683211863065988 \cdot 10^{160} \lor \neg \left(z \le 3.7182254012398825 \cdot 10^{-136}\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) + \left(a \cdot z\right) \cdot b\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r753543 = x;
        double r753544 = y;
        double r753545 = z;
        double r753546 = r753544 * r753545;
        double r753547 = r753543 + r753546;
        double r753548 = t;
        double r753549 = a;
        double r753550 = r753548 * r753549;
        double r753551 = r753547 + r753550;
        double r753552 = r753549 * r753545;
        double r753553 = b;
        double r753554 = r753552 * r753553;
        double r753555 = r753551 + r753554;
        return r753555;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r753556 = z;
        double r753557 = -2.3683211863065988e+160;
        bool r753558 = r753556 <= r753557;
        double r753559 = 3.7182254012398825e-136;
        bool r753560 = r753556 <= r753559;
        double r753561 = !r753560;
        bool r753562 = r753558 || r753561;
        double r753563 = a;
        double r753564 = b;
        double r753565 = y;
        double r753566 = fma(r753563, r753564, r753565);
        double r753567 = t;
        double r753568 = x;
        double r753569 = fma(r753563, r753567, r753568);
        double r753570 = fma(r753566, r753556, r753569);
        double r753571 = r753565 * r753556;
        double r753572 = r753568 + r753571;
        double r753573 = r753567 * r753563;
        double r753574 = r753572 + r753573;
        double r753575 = r753563 * r753556;
        double r753576 = r753575 * r753564;
        double r753577 = r753574 + r753576;
        double r753578 = r753562 ? r753570 : r753577;
        return r753578;
}

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

Original2.0
Target0.4
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.3683211863065988e+160 or 3.7182254012398825e-136 < z

    1. Initial program 4.1

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

    2. Simplified1.1

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

    if -2.3683211863065988e+160 < z < 3.7182254012398825e-136

    1. Initial program 0.8

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.3683211863065988 \cdot 10^{160} \lor \neg \left(z \le 3.7182254012398825 \cdot 10^{-136}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))