Average Error: 19.4 → 0.3
Time: 9.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3359776254393746 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.8410325872827527 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3359776254393746 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 2.8410325872827527 \cdot 10^{+74}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

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

\end{array}
double f(double x, double y) {
        double r19085263 = x;
        double r19085264 = r19085263 * r19085263;
        double r19085265 = y;
        double r19085266 = r19085264 + r19085265;
        double r19085267 = sqrt(r19085266);
        return r19085267;
}
↓
double f(double x, double y) {
        double r19085268 = x;
        double r19085269 = -1.3359776254393746e+154;
        bool r19085270 = r19085268 <= r19085269;
        double r19085271 = -0.5;
        double r19085272 = y;
        double r19085273 = r19085268 / r19085272;
        double r19085274 = r19085271 / r19085273;
        double r19085275 = r19085274 - r19085268;
        double r19085276 = 2.8410325872827527e+74;
        bool r19085277 = r19085268 <= r19085276;
        double r19085278 = fma(r19085268, r19085268, r19085272);
        double r19085279 = sqrt(r19085278);
        double r19085280 = 0.5;
        double r19085281 = r19085280 / r19085268;
        double r19085282 = fma(r19085281, r19085272, r19085268);
        double r19085283 = r19085277 ? r19085279 : r19085282;
        double r19085284 = r19085270 ? r19085275 : r19085283;
        return r19085284;
}

Error

Bits error versus x

Bits error versus y

Target

Original19.4
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(\frac{1}{2} \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3359776254393746e+154

    1. Initial program 59.5

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

    2. Simplified59.5

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

    3. Taylor expanded around -inf 0

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

    4. Simplified0

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

    if -1.3359776254393746e+154 < x < 2.8410325872827527e+74

    1. Initial program 0.0

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

    2. Simplified0.0

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

    if 2.8410325872827527e+74 < x

    1. Initial program 40.5

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

    2. Simplified40.5

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

    3. Taylor expanded around inf 1.2

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

    4. Simplified1.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3359776254393746 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.8410325872827527 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 1/2 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 1/2 (/ y x)) x)))

  (sqrt (+ (* x x) y)))