Average Error: 37.8 → 25.5
Time: 1.3s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.24117253240271209 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 6.6019422836665007 \cdot 10^{109}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.24117253240271209 \cdot 10^{119}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le 6.6019422836665007 \cdot 10^{109}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x, double y, double z) {
        double r698797 = x;
        double r698798 = r698797 * r698797;
        double r698799 = y;
        double r698800 = r698799 * r698799;
        double r698801 = r698798 + r698800;
        double r698802 = z;
        double r698803 = r698802 * r698802;
        double r698804 = r698801 + r698803;
        double r698805 = sqrt(r698804);
        return r698805;
}
↓
double f(double x, double y, double z) {
        double r698806 = x;
        double r698807 = -4.241172532402712e+119;
        bool r698808 = r698806 <= r698807;
        double r698809 = -1.0;
        double r698810 = r698809 * r698806;
        double r698811 = 6.601942283666501e+109;
        bool r698812 = r698806 <= r698811;
        double r698813 = r698806 * r698806;
        double r698814 = y;
        double r698815 = r698814 * r698814;
        double r698816 = r698813 + r698815;
        double r698817 = z;
        double r698818 = r698817 * r698817;
        double r698819 = r698816 + r698818;
        double r698820 = sqrt(r698819);
        double r698821 = r698812 ? r698820 : r698806;
        double r698822 = r698808 ? r698810 : r698821;
        return r698822;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.8
Target25.3
Herbie25.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.241172532402712e+119

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 17.8

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

    if -4.241172532402712e+119 < x < 6.601942283666501e+109

    1. Initial program 29.2

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

    if 6.601942283666501e+109 < x

    1. Initial program 55.8

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

    2. Taylor expanded around inf 17.5

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

  4. Final simplification25.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.24117253240271209 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 6.6019422836665007 \cdot 10^{109}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))