Average Error: 21.4 → 0.0
Time: 1.8s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.32224429827678141 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.1579375708996944 \cdot 10^{147}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.32224429827678141 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 3.1579375708996944 \cdot 10^{147}:\\
\;\;\;\;\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 r419845 = x;
        double r419846 = r419845 * r419845;
        double r419847 = y;
        double r419848 = r419846 + r419847;
        double r419849 = sqrt(r419848);
        return r419849;
}
↓
double f(double x, double y) {
        double r419850 = x;
        double r419851 = -1.3222442982767814e+154;
        bool r419852 = r419850 <= r419851;
        double r419853 = 0.5;
        double r419854 = y;
        double r419855 = r419854 / r419850;
        double r419856 = r419853 * r419855;
        double r419857 = r419850 + r419856;
        double r419858 = -r419857;
        double r419859 = 3.1579375708996944e+147;
        bool r419860 = r419850 <= r419859;
        double r419861 = r419850 * r419850;
        double r419862 = r419861 + r419854;
        double r419863 = sqrt(r419862);
        double r419864 = fma(r419853, r419855, r419850);
        double r419865 = r419860 ? r419863 : r419864;
        double r419866 = r419852 ? r419858 : r419865;
        return r419866;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.4
Target0.6
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3222442982767814e+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 -1.3222442982767814e+154 < x < 3.1579375708996944e+147

    1. Initial program 0.0

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

    if 3.1579375708996944e+147 < x

    1. Initial program 61.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.32224429827678141 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 3.1579375708996944 \cdot 10^{147}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))