Average Error: 31.1 → 18.9
Time: 1.2s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.1561596166685901 \cdot 10^{125}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 2.33673518569970664 \cdot 10^{-296}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 1.0193327448038136 \cdot 10^{95}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.1561596166685901 \cdot 10^{125}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le -3.80996693730795831 \cdot 10^{-103}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \le 2.33673518569970664 \cdot 10^{-296}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 1.0193327448038136 \cdot 10^{95}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

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

\end{array}
double f(double x, double y) {
        double r675434 = x;
        double r675435 = r675434 * r675434;
        double r675436 = y;
        double r675437 = r675436 * r675436;
        double r675438 = r675435 + r675437;
        double r675439 = sqrt(r675438);
        return r675439;
}
↓
double f(double x, double y) {
        double r675440 = x;
        double r675441 = -8.15615961666859e+125;
        bool r675442 = r675440 <= r675441;
        double r675443 = -1.0;
        double r675444 = r675443 * r675440;
        double r675445 = -3.8099669373079583e-103;
        bool r675446 = r675440 <= r675445;
        double r675447 = r675440 * r675440;
        double r675448 = y;
        double r675449 = r675448 * r675448;
        double r675450 = r675447 + r675449;
        double r675451 = sqrt(r675450);
        double r675452 = 2.3367351856997066e-296;
        bool r675453 = r675440 <= r675452;
        double r675454 = 1.0193327448038136e+95;
        bool r675455 = r675440 <= r675454;
        double r675456 = r675455 ? r675451 : r675440;
        double r675457 = r675453 ? r675448 : r675456;
        double r675458 = r675446 ? r675451 : r675457;
        double r675459 = r675442 ? r675444 : r675458;
        return r675459;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.1
Target17.5
Herbie18.9
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -8.15615961666859e+125

    1. Initial program 55.6

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around -inf 9.1

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

    if -8.15615961666859e+125 < x < -3.8099669373079583e-103 or 2.3367351856997066e-296 < x < 1.0193327448038136e+95

    1. Initial program 18.9

      \[\sqrt{x \cdot x + y \cdot y}\]

    if -3.8099669373079583e-103 < x < 2.3367351856997066e-296

    1. Initial program 27.8

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around 0 36.1

      \[\leadsto \color{blue}{y}\]

    if 1.0193327448038136e+95 < x

    1. Initial program 50.2

      \[\sqrt{x \cdot x + y \cdot y}\]

    2. Taylor expanded around inf 10.4

      \[\leadsto \color{blue}{x}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.1561596166685901 \cdot 10^{125}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 2.33673518569970664 \cdot 10^{-296}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 1.0193327448038136 \cdot 10^{95}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.123695082659983e+145) (- x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

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