Average Error: 31.8 → 18.8
Time: 1.3s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8789989068572990460:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -4.2015690056388073 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.33367121357677706 \cdot 10^{-291}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 8.0509896767148372 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.3202809311828264 \cdot 10^{-48}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 3.9171532397504441 \cdot 10^{68}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8789989068572990460:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le -4.2015690056388073 \cdot 10^{-231}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \le 1.33367121357677706 \cdot 10^{-291}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 8.0509896767148372 \cdot 10^{-74}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;x \le 1.3202809311828264 \cdot 10^{-48}:\\
\;\;\;\;y\\

\mathbf{elif}\;x \le 3.9171532397504441 \cdot 10^{68}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

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

\end{array}
double f(double x, double y) {
        double r784557 = x;
        double r784558 = r784557 * r784557;
        double r784559 = y;
        double r784560 = r784559 * r784559;
        double r784561 = r784558 + r784560;
        double r784562 = sqrt(r784561);
        return r784562;
}
↓
double f(double x, double y) {
        double r784563 = x;
        double r784564 = -8.78998906857299e+18;
        bool r784565 = r784563 <= r784564;
        double r784566 = -1.0;
        double r784567 = r784566 * r784563;
        double r784568 = -4.2015690056388073e-231;
        bool r784569 = r784563 <= r784568;
        double r784570 = r784563 * r784563;
        double r784571 = y;
        double r784572 = r784571 * r784571;
        double r784573 = r784570 + r784572;
        double r784574 = sqrt(r784573);
        double r784575 = 1.333671213576777e-291;
        bool r784576 = r784563 <= r784575;
        double r784577 = 8.050989676714837e-74;
        bool r784578 = r784563 <= r784577;
        double r784579 = 1.3202809311828264e-48;
        bool r784580 = r784563 <= r784579;
        double r784581 = 3.917153239750444e+68;
        bool r784582 = r784563 <= r784581;
        double r784583 = r784582 ? r784574 : r784563;
        double r784584 = r784580 ? r784571 : r784583;
        double r784585 = r784578 ? r784574 : r784584;
        double r784586 = r784576 ? r784571 : r784585;
        double r784587 = r784569 ? r784574 : r784586;
        double r784588 = r784565 ? r784567 : r784587;
        return r784588;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.8
Target17.3
Herbie18.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -8.78998906857299e+18

    1. Initial program 42.6

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

    2. Taylor expanded around -inf 13.9

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

    if -8.78998906857299e+18 < x < -4.2015690056388073e-231 or 1.333671213576777e-291 < x < 8.050989676714837e-74 or 1.3202809311828264e-48 < x < 3.917153239750444e+68

    1. Initial program 20.2

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

    if -4.2015690056388073e-231 < x < 1.333671213576777e-291 or 8.050989676714837e-74 < x < 1.3202809311828264e-48

    1. Initial program 28.0

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

    2. Taylor expanded around 0 36.9

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

    if 3.917153239750444e+68 < x

    1. Initial program 48.1

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

    2. Taylor expanded around inf 12.1

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8789989068572990460:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -4.2015690056388073 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.33367121357677706 \cdot 10^{-291}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 8.0509896767148372 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le 1.3202809311828264 \cdot 10^{-48}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 3.9171532397504441 \cdot 10^{68}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))