Average Error: 21.0 → 0.2
Time: 12.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r293253 = x;
        double r293254 = r293253 * r293253;
        double r293255 = y;
        double r293256 = r293254 + r293255;
        double r293257 = sqrt(r293256);
        return r293257;
}
↓
double f(double x, double y) {
        double r293258 = x;
        double r293259 = -1.3221955759293222e+154;
        bool r293260 = r293258 <= r293259;
        double r293261 = y;
        double r293262 = r293261 / r293258;
        double r293263 = -0.5;
        double r293264 = r293262 * r293263;
        double r293265 = r293264 - r293258;
        double r293266 = 1.892549585482312e+97;
        bool r293267 = r293258 <= r293266;
        double r293268 = r293258 * r293258;
        double r293269 = r293268 + r293261;
        double r293270 = sqrt(r293269);
        double r293271 = 0.5;
        double r293272 = r293271 * r293262;
        double r293273 = r293258 + r293272;
        double r293274 = r293267 ? r293270 : r293273;
        double r293275 = r293260 ? r293265 : r293274;
        return r293275;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.0
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3221955759293222e+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 -1.3221955759293222e+154 < x < 1.892549585482312e+97

    1. Initial program 0.0

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

    if 1.892549585482312e+97 < x

    1. Initial program 47.8

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

    2. Taylor expanded around inf 0.8

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.322195575929322175161499122447085220085 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 1.892549585482311918236295649622823641354 \cdot 10^{97}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

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