Average Error: 21.5 → 0.6
Time: 9.9s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.337826701582892488089574244217473576524 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.417169030606568251689508094083542147075 \cdot 10^{48}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.337826701582892488089574244217473576524 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 1.417169030606568251689508094083542147075 \cdot 10^{48}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r24551535 = x;
        double r24551536 = r24551535 * r24551535;
        double r24551537 = y;
        double r24551538 = r24551536 + r24551537;
        double r24551539 = sqrt(r24551538);
        return r24551539;
}
↓
double f(double x, double y) {
        double r24551540 = x;
        double r24551541 = -1.3378267015828925e+154;
        bool r24551542 = r24551540 <= r24551541;
        double r24551543 = -0.5;
        double r24551544 = y;
        double r24551545 = r24551540 / r24551544;
        double r24551546 = r24551543 / r24551545;
        double r24551547 = r24551546 - r24551540;
        double r24551548 = 1.4171690306065683e+48;
        bool r24551549 = r24551540 <= r24551548;
        double r24551550 = r24551540 * r24551540;
        double r24551551 = r24551550 + r24551544;
        double r24551552 = sqrt(r24551551);
        double r24551553 = r24551544 / r24551540;
        double r24551554 = 0.5;
        double r24551555 = r24551553 * r24551554;
        double r24551556 = r24551555 + r24551540;
        double r24551557 = r24551549 ? r24551552 : r24551556;
        double r24551558 = r24551542 ? r24551547 : r24551557;
        return r24551558;
}

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
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3378267015828925e+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 -1.3378267015828925e+154 < x < 1.4171690306065683e+48

    1. Initial program 0.0

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

    if 1.4171690306065683e+48 < x

    1. Initial program 39.0

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

    2. Taylor expanded around inf 2.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.337826701582892488089574244217473576524 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.417169030606568251689508094083542147075 \cdot 10^{48}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{2} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))