Average Error: 20.7 → 0.1
Time: 1.9s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.839483098530447 \cdot 10^{148}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 9.20903655101720352 \cdot 10^{124}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.839483098530447 \cdot 10^{148}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 9.20903655101720352 \cdot 10^{124}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r525956 = x;
        double r525957 = r525956 * r525956;
        double r525958 = y;
        double r525959 = r525957 + r525958;
        double r525960 = sqrt(r525959);
        return r525960;
}
↓
double f(double x, double y) {
        double r525961 = x;
        double r525962 = -7.839483098530447e+148;
        bool r525963 = r525961 <= r525962;
        double r525964 = 0.5;
        double r525965 = y;
        double r525966 = r525965 / r525961;
        double r525967 = r525964 * r525966;
        double r525968 = r525961 + r525967;
        double r525969 = -r525968;
        double r525970 = 9.209036551017204e+124;
        bool r525971 = r525961 <= r525970;
        double r525972 = r525961 * r525961;
        double r525973 = r525972 + r525965;
        double r525974 = sqrt(r525973);
        double r525975 = r525971 ? r525974 : r525968;
        double r525976 = r525963 ? r525969 : r525975;
        return r525976;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.7
Target0.4
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -7.839483098530447e+148

    1. Initial program 61.9

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

    2. Taylor expanded around -inf 0

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

    if -7.839483098530447e+148 < x < 9.209036551017204e+124

    1. Initial program 0.0

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

    if 9.209036551017204e+124 < x

    1. Initial program 52.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.839483098530447 \cdot 10^{148}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 9.20903655101720352 \cdot 10^{124}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))