Average Error: 21.2 → 0.1
Time: 6.2s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.336927375966215258326160736033930548748 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 6.791589890324519170947728943797958496879 \cdot 10^{101}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.336927375966215258326160736033930548748 \cdot 10^{154}:\\
\;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\

\mathbf{elif}\;x \le 6.791589890324519170947728943797958496879 \cdot 10^{101}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r384522 = x;
        double r384523 = r384522 * r384522;
        double r384524 = y;
        double r384525 = r384523 + r384524;
        double r384526 = sqrt(r384525);
        return r384526;
}
↓
double f(double x, double y) {
        double r384527 = x;
        double r384528 = -1.3369273759662153e+154;
        bool r384529 = r384527 <= r384528;
        double r384530 = y;
        double r384531 = r384530 / r384527;
        double r384532 = -0.5;
        double r384533 = r384531 * r384532;
        double r384534 = r384533 - r384527;
        double r384535 = 6.791589890324519e+101;
        bool r384536 = r384527 <= r384535;
        double r384537 = r384527 * r384527;
        double r384538 = r384537 + r384530;
        double r384539 = sqrt(r384538);
        double r384540 = 0.5;
        double r384541 = r384540 * r384531;
        double r384542 = r384527 + r384541;
        double r384543 = r384536 ? r384539 : r384542;
        double r384544 = r384529 ? r384534 : r384543;
        return r384544;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.2
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 < -1.3369273759662153e+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

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

    if -1.3369273759662153e+154 < x < 6.791589890324519e+101

    1. Initial program 0.0

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

    if 6.791589890324519e+101 < x

    1. Initial program 47.6

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

    2. Taylor expanded around inf 0.6

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.336927375966215258326160736033930548748 \cdot 10^{154}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{-1}{2} - x\\ \mathbf{elif}\;x \le 6.791589890324519170947728943797958496879 \cdot 10^{101}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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)))