Average Error: 21.4 → 0.1
Time: 4.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.36386960173928627200627038857438869043 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.91813475462953226773563982297224491741 \cdot 10^{118}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.36386960173928627200627038857438869043 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.91813475462953226773563982297224491741 \cdot 10^{118}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r303554 = x;
        double r303555 = r303554 * r303554;
        double r303556 = y;
        double r303557 = r303555 + r303556;
        double r303558 = sqrt(r303557);
        return r303558;
}
↓
double f(double x, double y) {
        double r303559 = x;
        double r303560 = -1.3638696017392863e+154;
        bool r303561 = r303559 <= r303560;
        double r303562 = 0.5;
        double r303563 = y;
        double r303564 = r303563 / r303559;
        double r303565 = r303562 * r303564;
        double r303566 = r303559 + r303565;
        double r303567 = -r303566;
        double r303568 = 1.9181347546295323e+118;
        bool r303569 = r303559 <= r303568;
        double r303570 = r303559 * r303559;
        double r303571 = r303570 + r303563;
        double r303572 = sqrt(r303571);
        double r303573 = r303569 ? r303572 : r303566;
        double r303574 = r303561 ? r303567 : r303573;
        return r303574;
}

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 < -1.3638696017392863e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0

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

    if -1.3638696017392863e+154 < x < 1.9181347546295323e+118

    1. Initial program 0.0

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

    if 1.9181347546295323e+118 < x

    1. Initial program 52.4

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

    2. Taylor expanded around inf 0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.36386960173928627200627038857438869043 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.91813475462953226773563982297224491741 \cdot 10^{118}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(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)))