Average Error: 2.1 → 0.3
Time: 5.3s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.32054013202638357 \cdot 10^{46} \lor \neg \left(z \le 5.73759123783464084 \cdot 10^{-90}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + {\left(a \cdot \left(z \cdot b\right)\right)}^{1}\\ \end{array}\]
\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 -4.32054013202638357 \cdot 10^{46} \lor \neg \left(z \le 5.73759123783464084 \cdot 10^{-90}\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 \left(z \cdot b\right)\right)}^{1}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r678199 = x;
        double r678200 = y;
        double r678201 = z;
        double r678202 = r678200 * r678201;
        double r678203 = r678199 + r678202;
        double r678204 = t;
        double r678205 = a;
        double r678206 = r678204 * r678205;
        double r678207 = r678203 + r678206;
        double r678208 = r678205 * r678201;
        double r678209 = b;
        double r678210 = r678208 * r678209;
        double r678211 = r678207 + r678210;
        return r678211;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r678212 = z;
        double r678213 = -4.3205401320263836e+46;
        bool r678214 = r678212 <= r678213;
        double r678215 = 5.737591237834641e-90;
        bool r678216 = r678212 <= r678215;
        double r678217 = !r678216;
        bool r678218 = r678214 || r678217;
        double r678219 = a;
        double r678220 = b;
        double r678221 = y;
        double r678222 = fma(r678219, r678220, r678221);
        double r678223 = t;
        double r678224 = x;
        double r678225 = fma(r678219, r678223, r678224);
        double r678226 = fma(r678222, r678212, r678225);
        double r678227 = r678221 * r678212;
        double r678228 = r678224 + r678227;
        double r678229 = r678223 * r678219;
        double r678230 = r678228 + r678229;
        double r678231 = r678212 * r678220;
        double r678232 = r678219 * r678231;
        double r678233 = 1.0;
        double r678234 = pow(r678232, r678233);
        double r678235 = r678230 + r678234;
        double r678236 = r678218 ? r678226 : r678235;
        return r678236;
}

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

Original2.1
Target0.3
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.3205401320263836e+46 or 5.737591237834641e-90 < 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. Simplified0.4

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

    if -4.3205401320263836e+46 < z < 5.737591237834641e-90

    1. Initial program 0.6

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

    3. Applied pow10.6

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

    4. Applied pow10.6

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

    5. Applied pow10.6

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

    6. Applied pow-prod-down0.6

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

    7. Applied pow-prod-down0.6

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

    8. Simplified0.2

      \[\leadsto \left(\left(x + y \cdot z\right) + t \cdot a\right) + {\color{blue}{\left(a \cdot \left(z \cdot b\right)\right)}}^{1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.32054013202638357 \cdot 10^{46} \lor \neg \left(z \le 5.73759123783464084 \cdot 10^{-90}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + {\left(a \cdot \left(z \cdot b\right)\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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)))