Average Error: 21.1 → 0.1
Time: 12.1s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\
\;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\

\mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\
\;\;\;\;\sqrt{x \cdot x + y}\\

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

\end{array}
double f(double x, double y) {
        double r353561 = x;
        double r353562 = r353561 * r353561;
        double r353563 = y;
        double r353564 = r353562 + r353563;
        double r353565 = sqrt(r353564);
        return r353565;
}
↓
double f(double x, double y) {
        double r353566 = x;
        double r353567 = -1.3349869326014939e+154;
        bool r353568 = r353566 <= r353567;
        double r353569 = y;
        double r353570 = r353569 / r353566;
        double r353571 = 0.5;
        double r353572 = fma(r353570, r353571, r353566);
        double r353573 = -r353572;
        double r353574 = 1.4388934535207275e+123;
        bool r353575 = r353566 <= r353574;
        double r353576 = r353566 * r353566;
        double r353577 = r353576 + r353569;
        double r353578 = sqrt(r353577);
        double r353579 = r353575 ? r353578 : r353572;
        double r353580 = r353568 ? r353573 : r353579;
        return r353580;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.1
Target0.5
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 < -1.3349869326014939e+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

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

    if -1.3349869326014939e+154 < x < 1.4388934535207275e+123

    1. Initial program 0.0

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

    if 1.4388934535207275e+123 < x

    1. Initial program 53.4

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.334986932601493855851749327767836382071 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 1.438893453520727542249422121009742042751 \cdot 10^{123}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))