Average Error: 31.6 → 17.1
Time: 16.5s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.039547558546271631628934502111036051442 \cdot 10^{104}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -3.240034389533803828723034610715031267776 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{y \cdot y + x \cdot x}\\ \mathbf{elif}\;x \le 2.715396324398381567593668375558902625339 \cdot 10^{-222}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 2.683247713446899747987853789973400736076 \cdot 10^{123}:\\ \;\;\;\;\sqrt{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.039547558546271631628934502111036051442 \cdot 10^{104}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le -3.240034389533803828723034610715031267776 \cdot 10^{-262}:\\
\;\;\;\;\sqrt{y \cdot y + x \cdot x}\\

\mathbf{elif}\;x \le 2.715396324398381567593668375558902625339 \cdot 10^{-222}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 2.683247713446899747987853789973400736076 \cdot 10^{123}:\\
\;\;\;\;\sqrt{y \cdot y + x \cdot x}\\

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

\end{array}
double f(double x, double y) {
        double r41730952 = x;
        double r41730953 = r41730952 * r41730952;
        double r41730954 = y;
        double r41730955 = r41730954 * r41730954;
        double r41730956 = r41730953 + r41730955;
        double r41730957 = sqrt(r41730956);
        return r41730957;
}
↓
double f(double x, double y) {
        double r41730958 = x;
        double r41730959 = -8.039547558546272e+104;
        bool r41730960 = r41730958 <= r41730959;
        double r41730961 = -r41730958;
        double r41730962 = -3.240034389533804e-262;
        bool r41730963 = r41730958 <= r41730962;
        double r41730964 = y;
        double r41730965 = r41730964 * r41730964;
        double r41730966 = r41730958 * r41730958;
        double r41730967 = r41730965 + r41730966;
        double r41730968 = sqrt(r41730967);
        double r41730969 = 2.7153963243983816e-222;
        bool r41730970 = r41730958 <= r41730969;
        double r41730971 = 2.6832477134469e+123;
        bool r41730972 = r41730958 <= r41730971;
        double r41730973 = r41730972 ? r41730968 : r41730958;
        double r41730974 = r41730970 ? r41730964 : r41730973;
        double r41730975 = r41730963 ? r41730968 : r41730974;
        double r41730976 = r41730960 ? r41730961 : r41730975;
        return r41730976;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.6
Target17.2
Herbie17.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659982632437974301616192301785 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.116557621183362039388201959321597704512 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -8.039547558546272e+104

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 9.5

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

    3. Simplified9.5

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

    if -8.039547558546272e+104 < x < -3.240034389533804e-262 or 2.7153963243983816e-222 < x < 2.6832477134469e+123

    1. Initial program 19.0

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

    if -3.240034389533804e-262 < x < 2.7153963243983816e-222

    1. Initial program 32.1

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

    2. Taylor expanded around 0 31.4

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

    if 2.6832477134469e+123 < x

    1. Initial program 55.3

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

    2. Taylor expanded around inf 8.3

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.039547558546271631628934502111036051442 \cdot 10^{104}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -3.240034389533803828723034610715031267776 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{y \cdot y + x \cdot x}\\ \mathbf{elif}\;x \le 2.715396324398381567593668375558902625339 \cdot 10^{-222}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 2.683247713446899747987853789973400736076 \cdot 10^{123}:\\ \;\;\;\;\sqrt{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"

  :herbie-target
  (if (< x -1.1236950826599826e+145) (- x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

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