Average Error: 21.0 → 0.0
Time: 1.3s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.34679050822059381152104109136094934248 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.483080572797596756164012838819236522397 \cdot 10^{140}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.34679050822059381152104109136094934248 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 7.483080572797596756164012838819236522397 \cdot 10^{140}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

\end{array}
double f(double x, double y) {
        double r509137 = x;
        double r509138 = r509137 * r509137;
        double r509139 = y;
        double r509140 = r509138 + r509139;
        double r509141 = sqrt(r509140);
        return r509141;
}
↓
double f(double x, double y) {
        double r509142 = x;
        double r509143 = -1.3467905082205938e+154;
        bool r509144 = r509142 <= r509143;
        double r509145 = 0.5;
        double r509146 = y;
        double r509147 = r509146 / r509142;
        double r509148 = r509145 * r509147;
        double r509149 = r509142 + r509148;
        double r509150 = -r509149;
        double r509151 = 7.483080572797597e+140;
        bool r509152 = r509142 <= r509151;
        double r509153 = fma(r509142, r509142, r509146);
        double r509154 = sqrt(r509153);
        double r509155 = fma(r509145, r509147, r509142);
        double r509156 = r509152 ? r509154 : r509155;
        double r509157 = r509144 ? r509150 : r509156;
        return r509157;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.0
Target0.5
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3467905082205938e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

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

    if -1.3467905082205938e+154 < x < 7.483080572797597e+140

    1. Initial program 0.0

      \[\sqrt{x \cdot x + y}\]
    2. Using strategy rm

    3. Applied fma-def0.0

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, x, y\right)}}\]

    if 7.483080572797597e+140 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.34679050822059381152104109136094934248 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 7.483080572797596756164012838819236522397 \cdot 10^{140}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))