Average Error: 31.1 → 17.8
Time: 2.7s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.372592495718595623920592719897374473902 \cdot 10^{113}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -7.06955799813848965953437685478940011158 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le -2.198761773812964703192136107756966469302 \cdot 10^{-195}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 6.594840577856811645577684374093823157435 \cdot 10^{73}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.372592495718595623920592719897374473902 \cdot 10^{113}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le -7.06955799813848965953437685478940011158 \cdot 10^{-161}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \le -2.198761773812964703192136107756966469302 \cdot 10^{-195}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 6.594840577856811645577684374093823157435 \cdot 10^{73}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

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

\end{array}
double f(double x, double y) {
        double r509943 = x;
        double r509944 = r509943 * r509943;
        double r509945 = y;
        double r509946 = r509945 * r509945;
        double r509947 = r509944 + r509946;
        double r509948 = sqrt(r509947);
        return r509948;
}
↓
double f(double x, double y) {
        double r509949 = x;
        double r509950 = -1.3725924957185956e+113;
        bool r509951 = r509949 <= r509950;
        double r509952 = -r509949;
        double r509953 = -7.06955799813849e-161;
        bool r509954 = r509949 <= r509953;
        double r509955 = r509949 * r509949;
        double r509956 = y;
        double r509957 = r509956 * r509956;
        double r509958 = r509955 + r509957;
        double r509959 = sqrt(r509958);
        double r509960 = -2.1987617738129647e-195;
        bool r509961 = r509949 <= r509960;
        double r509962 = 6.594840577856812e+73;
        bool r509963 = r509949 <= r509962;
        double r509964 = r509963 ? r509959 : r509949;
        double r509965 = r509961 ? r509956 : r509964;
        double r509966 = r509954 ? r509959 : r509965;
        double r509967 = r509951 ? r509952 : r509966;
        return r509967;
}

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
Target17.4
Herbie17.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659982632437974301616192301785 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.116557621183362039388201959321597704512 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -1.3725924957185956e+113

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 10.7

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

    3. Simplified10.7

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

    if -1.3725924957185956e+113 < x < -7.06955799813849e-161 or -2.1987617738129647e-195 < x < 6.594840577856812e+73

    1. Initial program 20.3

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

    if -7.06955799813849e-161 < x < -2.1987617738129647e-195

    1. Initial program 29.5

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

    2. Taylor expanded around 0 35.4

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

    if 6.594840577856812e+73 < x

    1. Initial program 48.0

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

    2. Taylor expanded around inf 12.9

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.372592495718595623920592719897374473902 \cdot 10^{113}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -7.06955799813848965953437685478940011158 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le -2.198761773812964703192136107756966469302 \cdot 10^{-195}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 6.594840577856811645577684374093823157435 \cdot 10^{73}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.123695082659983e145) (- x) (if (< x 1.11655762118336204e93) (sqrt (+ (* x x) (* y y))) x))

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