Average Error: 19.7 → 0.4
Time: 11.4s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3262598264970426 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 9.943288082924311 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3262598264970426 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 9.943288082924311 \cdot 10^{+67}:\\
\;\;\;\;\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 r16036273 = x;
        double r16036274 = r16036273 * r16036273;
        double r16036275 = y;
        double r16036276 = r16036274 + r16036275;
        double r16036277 = sqrt(r16036276);
        return r16036277;
}
↓
double f(double x, double y) {
        double r16036278 = x;
        double r16036279 = -1.3262598264970426e+154;
        bool r16036280 = r16036278 <= r16036279;
        double r16036281 = -0.5;
        double r16036282 = y;
        double r16036283 = r16036278 / r16036282;
        double r16036284 = r16036281 / r16036283;
        double r16036285 = r16036284 - r16036278;
        double r16036286 = 9.943288082924311e+67;
        bool r16036287 = r16036278 <= r16036286;
        double r16036288 = fma(r16036278, r16036278, r16036282);
        double r16036289 = sqrt(r16036288);
        double r16036290 = 0.5;
        double r16036291 = r16036290 / r16036278;
        double r16036292 = fma(r16036291, r16036282, r16036278);
        double r16036293 = r16036287 ? r16036289 : r16036292;
        double r16036294 = r16036280 ? r16036285 : r16036293;
        return r16036294;
}

Error

Bits error versus x

Bits error versus y

Target

Original19.7
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(\frac{1}{2} \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3262598264970426e+154

    1. Initial program 59.6

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

    2. Simplified59.6

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

    3. Taylor expanded around -inf 0.0

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

    4. Simplified0.0

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

    if -1.3262598264970426e+154 < x < 9.943288082924311e+67

    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 9.943288082924311e+67 < x

    1. Initial program 38.6

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

    2. Simplified38.6

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

    3. Taylor expanded around inf 1.6

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

    4. Simplified1.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3262598264970426 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 9.943288082924311 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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)))