Average Error: 21.7 → 0.2
Time: 1.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33708786850011456 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 6.06492474519930152 \cdot 10^{100}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33708786850011456 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 6.06492474519930152 \cdot 10^{100}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r498940 = x;
        double r498941 = r498940 * r498940;
        double r498942 = y;
        double r498943 = r498941 + r498942;
        double r498944 = sqrt(r498943);
        return r498944;
}
↓
double f(double x, double y) {
        double r498945 = x;
        double r498946 = -1.3370878685001146e+154;
        bool r498947 = r498945 <= r498946;
        double r498948 = 0.5;
        double r498949 = y;
        double r498950 = r498949 / r498945;
        double r498951 = r498948 * r498950;
        double r498952 = r498945 + r498951;
        double r498953 = -r498952;
        double r498954 = 6.0649247451993015e+100;
        bool r498955 = r498945 <= r498954;
        double r498956 = r498945 * r498945;
        double r498957 = r498956 + r498949;
        double r498958 = sqrt(r498957);
        double r498959 = r498955 ? r498958 : r498952;
        double r498960 = r498947 ? r498953 : r498959;
        return r498960;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.7
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3370878685001146e+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 -1.3370878685001146e+154 < x < 6.0649247451993015e+100

    1. Initial program 0.0

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

    if 6.0649247451993015e+100 < x

    1. Initial program 48.3

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

    2. Taylor expanded around inf 0.9

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33708786850011456 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 6.06492474519930152 \cdot 10^{100}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{2} \cdot \frac{y}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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)))