Average Error: 21.7 → 0.2
Time: 1.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33708786850011456 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 6.06492474519930152 \cdot 10^{100}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\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}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

\end{array}
double f(double x, double y) {
        double r634961 = x;
        double r634962 = r634961 * r634961;
        double r634963 = y;
        double r634964 = r634962 + r634963;
        double r634965 = sqrt(r634964);
        return r634965;
}
↓
double f(double x, double y) {
        double r634966 = x;
        double r634967 = -1.3370878685001146e+154;
        bool r634968 = r634966 <= r634967;
        double r634969 = 0.5;
        double r634970 = y;
        double r634971 = r634970 / r634966;
        double r634972 = r634969 * r634971;
        double r634973 = r634966 + r634972;
        double r634974 = -r634973;
        double r634975 = 6.0649247451993015e+100;
        bool r634976 = r634966 <= r634975;
        double r634977 = r634966 * r634966;
        double r634978 = r634977 + r634970;
        double r634979 = sqrt(r634978);
        double r634980 = fma(r634969, r634971, r634966);
        double r634981 = r634976 ? r634979 : r634980;
        double r634982 = r634968 ? r634974 : r634981;
        return r634982;
}

Error

Bits error versus x

Bits error versus y

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. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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)))