Average Error: 21.1 → 0.1
Time: 24.8s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\ \mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\

\mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r345433 = x;
        double r345434 = r345433 * r345433;
        double r345435 = y;
        double r345436 = r345434 + r345435;
        double r345437 = sqrt(r345436);
        return r345437;
}
↓
double f(double x, double y) {
        double r345438 = x;
        double r345439 = -1.3349869326014939e+154;
        bool r345440 = r345438 <= r345439;
        double r345441 = -0.5;
        double r345442 = y;
        double r345443 = r345442 / r345438;
        double r345444 = r345441 * r345443;
        double r345445 = r345444 - r345438;
        double r345446 = 1.4388934535207275e+123;
        bool r345447 = r345438 <= r345446;
        double r345448 = r345438 * r345438;
        double r345449 = r345448 + r345442;
        double r345450 = sqrt(r345449);
        double r345451 = 0.5;
        double r345452 = r345451 * r345443;
        double r345453 = r345438 + r345452;
        double r345454 = r345447 ? r345450 : r345453;
        double r345455 = r345440 ? r345445 : r345454;
        return r345455;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.1
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 < -1.3349869326014939e+154

    1. Initial program 64.0

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

    3. Applied add-sqr-sqrt64.0

      \[\leadsto \sqrt{\color{blue}{\sqrt{x \cdot x + y} \cdot \sqrt{x \cdot x + y}}}\]

    4. Applied sqrt-prod64.0

      \[\leadsto \color{blue}{\sqrt{\sqrt{x \cdot x + y}} \cdot \sqrt{\sqrt{x \cdot x + y}}}\]

    5. Taylor expanded around -inf 0

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

    6. Simplified0

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

    if -1.3349869326014939e+154 < x < 1.4388934535207275e+123

    1. Initial program 0.0

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

    if 1.4388934535207275e+123 < x

    1. Initial program 53.4

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

    2. Taylor expanded around inf 0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\ \mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))