Average Error: 37.3 → 25.2
Time: 5.2s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.643648655796742516992848026301703974375 \cdot 10^{126}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \le 5.367114864082888858240124126679124813161 \cdot 10^{82}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, y, x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.643648655796742516992848026301703974375 \cdot 10^{126}:\\
\;\;\;\;-z\\

\mathbf{elif}\;z \le 5.367114864082888858240124126679124813161 \cdot 10^{82}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, y, x \cdot x\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;z\\

\end{array}
double f(double x, double y, double z) {
        double r22023585 = x;
        double r22023586 = r22023585 * r22023585;
        double r22023587 = y;
        double r22023588 = r22023587 * r22023587;
        double r22023589 = r22023586 + r22023588;
        double r22023590 = z;
        double r22023591 = r22023590 * r22023590;
        double r22023592 = r22023589 + r22023591;
        double r22023593 = sqrt(r22023592);
        return r22023593;
}
↓
double f(double x, double y, double z) {
        double r22023594 = z;
        double r22023595 = -1.6436486557967425e+126;
        bool r22023596 = r22023594 <= r22023595;
        double r22023597 = -r22023594;
        double r22023598 = 5.367114864082889e+82;
        bool r22023599 = r22023594 <= r22023598;
        double r22023600 = y;
        double r22023601 = x;
        double r22023602 = r22023601 * r22023601;
        double r22023603 = fma(r22023600, r22023600, r22023602);
        double r22023604 = fma(r22023594, r22023594, r22023603);
        double r22023605 = sqrt(r22023604);
        double r22023606 = r22023599 ? r22023605 : r22023594;
        double r22023607 = r22023596 ? r22023597 : r22023606;
        return r22023607;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original37.3
Target25.0
Herbie25.2
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.6436486557967425e+126

    1. Initial program 58.6

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    2. Simplified58.6

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, y, x \cdot x\right)\right)}}\]

    3. Taylor expanded around -inf 15.7

      \[\leadsto \color{blue}{-1 \cdot z}\]

    4. Simplified15.7

      \[\leadsto \color{blue}{-z}\]

    if -1.6436486557967425e+126 < z < 5.367114864082889e+82

    1. Initial program 28.7

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    2. Simplified28.7

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, y, x \cdot x\right)\right)}}\]

    if 5.367114864082889e+82 < z

    1. Initial program 51.9

      \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]

    2. Simplified51.9

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, y, x \cdot x\right)\right)}}\]

    3. Taylor expanded around inf 19.9

      \[\leadsto \color{blue}{z}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.643648655796742516992848026301703974375 \cdot 10^{126}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \le 5.367114864082888858240124126679124813161 \cdot 10^{82}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, y, x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"

  :herbie-target
  (if (< z -6.396479394109776e+136) (- z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

  (sqrt (+ (+ (* x x) (* y y)) (* z z))))