Average Error: 21.5 → 0.0
Time: 11.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.352732562381551610191380298409879181488 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 8.914808959129563024978640207181522528947 \cdot 10^{145}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.352732562381551610191380298409879181488 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 8.914808959129563024978640207181522528947 \cdot 10^{145}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r28494266 = x;
        double r28494267 = r28494266 * r28494266;
        double r28494268 = y;
        double r28494269 = r28494267 + r28494268;
        double r28494270 = sqrt(r28494269);
        return r28494270;
}
↓
double f(double x, double y) {
        double r28494271 = x;
        double r28494272 = -1.3527325623815516e+154;
        bool r28494273 = r28494271 <= r28494272;
        double r28494274 = -0.5;
        double r28494275 = y;
        double r28494276 = r28494271 / r28494275;
        double r28494277 = r28494274 / r28494276;
        double r28494278 = r28494277 - r28494271;
        double r28494279 = 8.914808959129563e+145;
        bool r28494280 = r28494271 <= r28494279;
        double r28494281 = r28494271 * r28494271;
        double r28494282 = r28494281 + r28494275;
        double r28494283 = sqrt(r28494282);
        double r28494284 = r28494275 / r28494271;
        double r28494285 = 0.5;
        double r28494286 = r28494284 * r28494285;
        double r28494287 = r28494286 + r28494271;
        double r28494288 = r28494280 ? r28494283 : r28494287;
        double r28494289 = r28494273 ? r28494278 : r28494288;
        return r28494289;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.5
Target0.4
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3527325623815516e+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

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

    if -1.3527325623815516e+154 < x < 8.914808959129563e+145

    1. Initial program 0.0

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

    if 8.914808959129563e+145 < x

    1. Initial program 61.0

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

    2. Taylor expanded around inf 0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.352732562381551610191380298409879181488 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 8.914808959129563024978640207181522528947 \cdot 10^{145}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :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)))