Average Error: 21.1 → 0.2
Time: 1.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3694640831062883 \cdot 10^{154}:\\ \;\;\;\;1 \cdot \left(-1 \cdot x\right)\\ \mathbf{elif}\;x \le 8.4390817817310158 \cdot 10^{104}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3694640831062883 \cdot 10^{154}:\\
\;\;\;\;1 \cdot \left(-1 \cdot x\right)\\

\mathbf{elif}\;x \le 8.4390817817310158 \cdot 10^{104}:\\
\;\;\;\;\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 r518231 = x;
        double r518232 = r518231 * r518231;
        double r518233 = y;
        double r518234 = r518232 + r518233;
        double r518235 = sqrt(r518234);
        return r518235;
}
↓
double f(double x, double y) {
        double r518236 = x;
        double r518237 = -1.3694640831062883e+154;
        bool r518238 = r518236 <= r518237;
        double r518239 = 1.0;
        double r518240 = -1.0;
        double r518241 = r518240 * r518236;
        double r518242 = r518239 * r518241;
        double r518243 = 8.439081781731016e+104;
        bool r518244 = r518236 <= r518243;
        double r518245 = r518236 * r518236;
        double r518246 = y;
        double r518247 = r518245 + r518246;
        double r518248 = sqrt(r518247);
        double r518249 = 0.5;
        double r518250 = r518246 / r518236;
        double r518251 = fma(r518249, r518250, r518236);
        double r518252 = r518244 ? r518248 : r518251;
        double r518253 = r518238 ? r518242 : r518252;
        return r518253;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.1
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 < -1.3694640831062883e+154

    1. Initial program 64.0

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

    3. Applied *-un-lft-identity64.0

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

    4. Applied sqrt-prod64.0

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

    5. Simplified64.0

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

    6. Simplified31.0

      \[\leadsto 1 \cdot \color{blue}{\mathsf{hypot}\left(x, {y}^{\frac{1}{2}}\right)}\]

    7. Taylor expanded around -inf 0

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

    if -1.3694640831062883e+154 < x < 8.439081781731016e+104

    1. Initial program 0.0

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

    if 8.439081781731016e+104 < x

    1. Initial program 49.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3694640831062883 \cdot 10^{154}:\\ \;\;\;\;1 \cdot \left(-1 \cdot x\right)\\ \mathbf{elif}\;x \le 8.4390817817310158 \cdot 10^{104}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))