Average Error: 2.3 → 1.4
Time: 3.2s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 2.02733625987128582 \cdot 10^{-34}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + {\left(a \cdot \left(z \cdot b\right)\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;z \le 2.02733625987128582 \cdot 10^{-34}:\\
\;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + {\left(a \cdot \left(z \cdot b\right)\right)}^{1}\\

\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 r559347 = x;
        double r559348 = y;
        double r559349 = z;
        double r559350 = r559348 * r559349;
        double r559351 = r559347 + r559350;
        double r559352 = t;
        double r559353 = a;
        double r559354 = r559352 * r559353;
        double r559355 = r559351 + r559354;
        double r559356 = r559353 * r559349;
        double r559357 = b;
        double r559358 = r559356 * r559357;
        double r559359 = r559355 + r559358;
        return r559359;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r559360 = z;
        double r559361 = 2.027336259871286e-34;
        bool r559362 = r559360 <= r559361;
        double r559363 = x;
        double r559364 = y;
        double r559365 = r559364 * r559360;
        double r559366 = r559363 + r559365;
        double r559367 = t;
        double r559368 = a;
        double r559369 = r559367 * r559368;
        double r559370 = r559366 + r559369;
        double r559371 = b;
        double r559372 = r559360 * r559371;
        double r559373 = r559368 * r559372;
        double r559374 = 1.0;
        double r559375 = pow(r559373, r559374);
        double r559376 = r559370 + r559375;
        double r559377 = fma(r559368, r559371, r559364);
        double r559378 = fma(r559368, r559367, r559363);
        double r559379 = fma(r559377, r559360, r559378);
        double r559380 = r559362 ? r559376 : r559379;
        return r559380;
}

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

Original2.3
Target0.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 2.027336259871286e-34

    1. Initial program 1.7

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

    3. Applied pow11.7

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot \color{blue}{{b}^{1}}\]

    4. Applied pow11.7

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot \color{blue}{{z}^{1}}\right) \cdot {b}^{1}\]

    5. Applied pow11.7

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(\color{blue}{{a}^{1}} \cdot {z}^{1}\right) \cdot {b}^{1}\]

    6. Applied pow-prod-down1.7

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{{\left(a \cdot z\right)}^{1}} \cdot {b}^{1}\]

    7. Applied pow-prod-down1.7

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + \color{blue}{{\left(\left(a \cdot z\right) \cdot b\right)}^{1}}\]

    8. Simplified1.8

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + {\color{blue}{\left(a \cdot \left(z \cdot b\right)\right)}}^{1}\]

    if 2.027336259871286e-34 < z

    1. Initial program 4.2

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

    2. Simplified0.2

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 2.02733625987128582 \cdot 10^{-34}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + {\left(a \cdot \left(z \cdot b\right)\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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)))