Average Error: 21.7 → 0.1
Time: 10.1s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.299677200223385524664624994654815196296 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \mathbf{elif}\;x \le 7.41981782524246538584328907917534943779 \cdot 10^{111}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.299677200223385524664624994654815196296 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\

\mathbf{elif}\;x \le 7.41981782524246538584328907917534943779 \cdot 10^{111}:\\
\;\;\;\;\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 r266699 = x;
        double r266700 = r266699 * r266699;
        double r266701 = y;
        double r266702 = r266700 + r266701;
        double r266703 = sqrt(r266702);
        return r266703;
}
↓
double f(double x, double y) {
        double r266704 = x;
        double r266705 = -1.2996772002233855e+154;
        bool r266706 = r266704 <= r266705;
        double r266707 = 0.5;
        double r266708 = y;
        double r266709 = r266708 / r266704;
        double r266710 = fma(r266707, r266709, r266704);
        double r266711 = -r266710;
        double r266712 = 7.419817825242465e+111;
        bool r266713 = r266704 <= r266712;
        double r266714 = r266704 * r266704;
        double r266715 = r266714 + r266708;
        double r266716 = sqrt(r266715);
        double r266717 = r266713 ? r266716 : r266710;
        double r266718 = r266706 ? r266711 : r266717;
        return r266718;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.7
Target0.4
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 < -1.2996772002233855e+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

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)}\]

    if -1.2996772002233855e+154 < x < 7.419817825242465e+111

    1. Initial program 0.0

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

    if 7.419817825242465e+111 < x

    1. Initial program 50.8

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.299677200223385524664624994654815196296 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \mathbf{elif}\;x \le 7.41981782524246538584328907917534943779 \cdot 10^{111}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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)))