Average Error: 21.4 → 0.1
Time: 10.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.36496554640706019677869997344330658994 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 3.588070708117172201716955628167087287626 \cdot 10^{111}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.36496554640706019677869997344330658994 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 3.588070708117172201716955628167087287626 \cdot 10^{111}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\

\end{array}
double f(double x, double y) {
        double r959410 = x;
        double r959411 = r959410 * r959410;
        double r959412 = y;
        double r959413 = r959411 + r959412;
        double r959414 = sqrt(r959413);
        return r959414;
}
↓
double f(double x, double y) {
        double r959415 = x;
        double r959416 = -1.3649655464070602e+154;
        bool r959417 = r959415 <= r959416;
        double r959418 = y;
        double r959419 = r959418 / r959415;
        double r959420 = -0.5;
        double r959421 = r959419 * r959420;
        double r959422 = r959421 - r959415;
        double r959423 = 3.588070708117172e+111;
        bool r959424 = r959415 <= r959423;
        double r959425 = r959415 * r959415;
        double r959426 = r959425 + r959418;
        double r959427 = sqrt(r959426);
        double r959428 = 0.5;
        double r959429 = r959428 * r959419;
        double r959430 = r959415 + r959429;
        double r959431 = r959424 ? r959427 : r959430;
        double r959432 = r959417 ? r959422 : r959431;
        return r959432;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.4
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3649655464070602e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

    3. Simplified0

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{-1}{2} - x}\]

    if -1.3649655464070602e+154 < x < 3.588070708117172e+111

    1. Initial program 0.0

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

    if 3.588070708117172e+111 < x

    1. Initial program 51.0

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

    2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{x + \frac{1}{2} \cdot \frac{y}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.36496554640706019677869997344330658994 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 3.588070708117172201716955628167087287626 \cdot 10^{111}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.5823995511225407e57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

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