Average Error: 31.7 → 18.1
Time: 1.7s
Precision: 64
\[\sqrt{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot y\\ \mathbf{elif}\;y \le -1.7874287404230692 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 5.8993776144081826 \cdot 10^{-308}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \le 2.4345784437110915 \cdot 10^{-199}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;y \le 1.7970794289179904 \cdot 10^{-166}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \le 1.0688990210562475 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;y \le 2.23402097896517821 \cdot 10^{109}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]
\sqrt{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.4362841072323272 \cdot 10^{150}:\\
\;\;\;\;-1 \cdot y\\

\mathbf{elif}\;y \le -1.7874287404230692 \cdot 10^{-225}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 5.8993776144081826 \cdot 10^{-308}:\\
\;\;\;\;x\\

\mathbf{elif}\;y \le 2.4345784437110915 \cdot 10^{-199}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;y \le 1.7970794289179904 \cdot 10^{-166}:\\
\;\;\;\;x\\

\mathbf{elif}\;y \le 1.0688990210562475 \cdot 10^{-162}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;y \le 2.23402097896517821 \cdot 10^{109}:\\
\;\;\;\;\sqrt{x \cdot x + y \cdot y}\\

\mathbf{else}:\\
\;\;\;\;y\\

\end{array}
double f(double x, double y) {
        double r954461 = x;
        double r954462 = r954461 * r954461;
        double r954463 = y;
        double r954464 = r954463 * r954463;
        double r954465 = r954462 + r954464;
        double r954466 = sqrt(r954465);
        return r954466;
}
↓
double f(double x, double y) {
        double r954467 = y;
        double r954468 = -3.436284107232327e+150;
        bool r954469 = r954467 <= r954468;
        double r954470 = -1.0;
        double r954471 = r954470 * r954467;
        double r954472 = -1.7874287404230692e-225;
        bool r954473 = r954467 <= r954472;
        double r954474 = x;
        double r954475 = r954474 * r954474;
        double r954476 = r954467 * r954467;
        double r954477 = r954475 + r954476;
        double r954478 = sqrt(r954477);
        double r954479 = 5.899377614408183e-308;
        bool r954480 = r954467 <= r954479;
        double r954481 = 2.4345784437110915e-199;
        bool r954482 = r954467 <= r954481;
        double r954483 = r954470 * r954474;
        double r954484 = 1.7970794289179904e-166;
        bool r954485 = r954467 <= r954484;
        double r954486 = 1.0688990210562475e-162;
        bool r954487 = r954467 <= r954486;
        double r954488 = 2.2340209789651782e+109;
        bool r954489 = r954467 <= r954488;
        double r954490 = r954489 ? r954478 : r954467;
        double r954491 = r954487 ? r954483 : r954490;
        double r954492 = r954485 ? r954474 : r954491;
        double r954493 = r954482 ? r954483 : r954492;
        double r954494 = r954480 ? r954474 : r954493;
        double r954495 = r954473 ? r954478 : r954494;
        double r954496 = r954469 ? r954471 : r954495;
        return r954496;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original31.7
Target17.7
Herbie18.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if y < -3.436284107232327e+150

    1. Initial program 62.6

      \[\sqrt{x \cdot x + y \cdot y}\]
    2. Using strategy rm

    3. Applied flip-+64.0

      \[\leadsto \sqrt{\color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(y \cdot y\right) \cdot \left(y \cdot y\right)}{x \cdot x - y \cdot y}}}\]

    4. Simplified64.0

      \[\leadsto \sqrt{\frac{\color{blue}{\left(-{y}^{3}\right) \cdot y + {x}^{4}}}{x \cdot x - y \cdot y}}\]

    5. Taylor expanded around -inf 8.8

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

    if -3.436284107232327e+150 < y < -1.7874287404230692e-225 or 1.0688990210562475e-162 < y < 2.2340209789651782e+109

    1. Initial program 17.2

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

    if -1.7874287404230692e-225 < y < 5.899377614408183e-308 or 2.4345784437110915e-199 < y < 1.7970794289179904e-166

    1. Initial program 31.5

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

    2. Taylor expanded around inf 33.7

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

    if 5.899377614408183e-308 < y < 2.4345784437110915e-199 or 1.7970794289179904e-166 < y < 1.0688990210562475e-162

    1. Initial program 31.3

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

    2. Taylor expanded around -inf 35.2

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

    if 2.2340209789651782e+109 < y

    1. Initial program 53.4

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

    2. Taylor expanded around 0 9.8

      \[\leadsto \color{blue}{y}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot y\\ \mathbf{elif}\;y \le -1.7874287404230692 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 5.8993776144081826 \cdot 10^{-308}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \le 2.4345784437110915 \cdot 10^{-199}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;y \le 1.7970794289179904 \cdot 10^{-166}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \le 1.0688990210562475 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;y \le 2.23402097896517821 \cdot 10^{109}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))