Average Error: 37.9 → 25.7
Time: 1.3s
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 r779301 = x;
        double r779302 = r779301 * r779301;
        double r779303 = y;
        double r779304 = r779303 * r779303;
        double r779305 = r779302 + r779304;
        double r779306 = z;
        double r779307 = r779306 * r779306;
        double r779308 = r779305 + r779307;
        double r779309 = sqrt(r779308);
        return r779309;
}
↓
double f(double x, double y, double z) {
        double r779310 = x;
        double r779311 = -2.009142108550562e+41;
        bool r779312 = r779310 <= r779311;
        double r779313 = -1.0;
        double r779314 = r779313 * r779310;
        double r779315 = 6.392742446151673e+128;
        bool r779316 = r779310 <= r779315;
        double r779317 = r779310 * r779310;
        double r779318 = y;
        double r779319 = r779318 * r779318;
        double r779320 = r779317 + r779319;
        double r779321 = z;
        double r779322 = r779321 * r779321;
        double r779323 = r779320 + r779322;
        double r779324 = sqrt(r779323);
        double r779325 = r779316 ? r779324 : r779310;
        double r779326 = r779312 ? r779314 : r779325;
        return r779326;
}

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))))