Average Error: 37.9 → 25.7
Time: 1.5s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.00914210855056183 \cdot 10^{41}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 6.3927424461516727 \cdot 10^{128}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.00914210855056183 \cdot 10^{41}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le 6.3927424461516727 \cdot 10^{128}:\\
\;\;\;\;\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 r722725 = x;
        double r722726 = r722725 * r722725;
        double r722727 = y;
        double r722728 = r722727 * r722727;
        double r722729 = r722726 + r722728;
        double r722730 = z;
        double r722731 = r722730 * r722730;
        double r722732 = r722729 + r722731;
        double r722733 = sqrt(r722732);
        return r722733;
}
↓
double f(double x, double y, double z) {
        double r722734 = x;
        double r722735 = -2.009142108550562e+41;
        bool r722736 = r722734 <= r722735;
        double r722737 = -1.0;
        double r722738 = r722737 * r722734;
        double r722739 = 6.392742446151673e+128;
        bool r722740 = r722734 <= r722739;
        double r722741 = r722734 * r722734;
        double r722742 = y;
        double r722743 = r722742 * r722742;
        double r722744 = r722741 + r722743;
        double r722745 = z;
        double r722746 = r722745 * r722745;
        double r722747 = r722744 + r722746;
        double r722748 = sqrt(r722747);
        double r722749 = r722740 ? r722748 : r722734;
        double r722750 = r722736 ? r722738 : r722749;
        return r722750;
}

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

Original37.9
Target25.8
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.009142108550562e+41

    1. Initial program 50.2

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

    2. Taylor expanded around -inf 22.8

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

    if -2.009142108550562e+41 < x < 6.392742446151673e+128

    1. Initial program 28.9

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

    if 6.392742446151673e+128 < x

    1. Initial program 58.8

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

    2. Taylor expanded around inf 15.8

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.00914210855056183 \cdot 10^{41}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 6.3927424461516727 \cdot 10^{128}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))