Average Error: 38.3 → 25.8
Time: 6.3s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.801328071339668 \cdot 10^{105}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \le 2.42014104061242399 \cdot 10^{123}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.801328071339668 \cdot 10^{105}:\\
\;\;\;\;-z\\

\mathbf{elif}\;z \le 2.42014104061242399 \cdot 10^{123}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r650104 = x;
        double r650105 = r650104 * r650104;
        double r650106 = y;
        double r650107 = r650106 * r650106;
        double r650108 = r650105 + r650107;
        double r650109 = z;
        double r650110 = r650109 * r650109;
        double r650111 = r650108 + r650110;
        double r650112 = sqrt(r650111);
        return r650112;
}
↓
double f(double x, double y, double z) {
        double r650113 = z;
        double r650114 = -1.8013280713396678e+105;
        bool r650115 = r650113 <= r650114;
        double r650116 = -r650113;
        double r650117 = 2.420141040612424e+123;
        bool r650118 = r650113 <= r650117;
        double r650119 = x;
        double r650120 = y;
        double r650121 = r650120 * r650120;
        double r650122 = fma(r650119, r650119, r650121);
        double r650123 = fma(r650113, r650113, r650122);
        double r650124 = sqrt(r650123);
        double r650125 = r650118 ? r650124 : r650113;
        double r650126 = r650115 ? r650116 : r650125;
        return r650126;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original38.3
Target25.7
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.8013280713396678e+105

    1. Initial program 54.5

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

    2. Simplified54.5

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

    3. Taylor expanded around -inf 18.3

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

    4. Simplified18.3

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

    if -1.8013280713396678e+105 < z < 2.420141040612424e+123

    1. Initial program 29.9

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

    2. Simplified29.9

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

    if 2.420141040612424e+123 < z

    1. Initial program 57.5

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

    2. Simplified57.5

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

    3. Taylor expanded around inf 15.9

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

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.801328071339668 \cdot 10^{105}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \le 2.42014104061242399 \cdot 10^{123}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

  :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))))