Average Error: 22.0 → 0.4
Time: 2.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.173401499259793093477351368213376433201 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.028244907719024672501462088231661919293 \cdot 10^{74}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.173401499259793093477351368213376433201 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 3.028244907719024672501462088231661919293 \cdot 10^{74}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r292481 = x;
        double r292482 = r292481 * r292481;
        double r292483 = y;
        double r292484 = r292482 + r292483;
        double r292485 = sqrt(r292484);
        return r292485;
}
↓
double f(double x, double y) {
        double r292486 = x;
        double r292487 = -1.1734014992597931e+154;
        bool r292488 = r292486 <= r292487;
        double r292489 = 0.5;
        double r292490 = y;
        double r292491 = r292490 / r292486;
        double r292492 = r292489 * r292491;
        double r292493 = r292486 + r292492;
        double r292494 = -r292493;
        double r292495 = 3.0282449077190247e+74;
        bool r292496 = r292486 <= r292495;
        double r292497 = r292486 * r292486;
        double r292498 = r292497 + r292490;
        double r292499 = sqrt(r292498);
        double r292500 = r292496 ? r292499 : r292493;
        double r292501 = r292488 ? r292494 : r292500;
        return r292501;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.0
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.1734014992597931e+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 -1.1734014992597931e+154 < x < 3.0282449077190247e+74

    1. Initial program 0.0

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

    if 3.0282449077190247e+74 < x

    1. Initial program 43.4

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

    2. Taylor expanded around inf 1.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.173401499259793093477351368213376433201 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.028244907719024672501462088231661919293 \cdot 10^{74}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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)))