Average Error: 19.8 → 0.5
Time: 9.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3298271535226257 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 3.890409410163301 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3298271535226257 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\

\mathbf{elif}\;x \le 3.890409410163301 \cdot 10^{+59}:\\
\;\;\;\;\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 r23152938 = x;
        double r23152939 = r23152938 * r23152938;
        double r23152940 = y;
        double r23152941 = r23152939 + r23152940;
        double r23152942 = sqrt(r23152941);
        return r23152942;
}
↓
double f(double x, double y) {
        double r23152943 = x;
        double r23152944 = -1.3298271535226257e+154;
        bool r23152945 = r23152943 <= r23152944;
        double r23152946 = -0.5;
        double r23152947 = y;
        double r23152948 = r23152943 / r23152947;
        double r23152949 = r23152946 / r23152948;
        double r23152950 = r23152949 - r23152943;
        double r23152951 = 3.890409410163301e+59;
        bool r23152952 = r23152943 <= r23152951;
        double r23152953 = fma(r23152943, r23152943, r23152947);
        double r23152954 = sqrt(r23152953);
        double r23152955 = 0.5;
        double r23152956 = r23152955 / r23152943;
        double r23152957 = fma(r23152956, r23152947, r23152943);
        double r23152958 = r23152952 ? r23152954 : r23152957;
        double r23152959 = r23152945 ? r23152950 : r23152958;
        return r23152959;
}

Error

Bits error versus x

Bits error versus y

Target

Original19.8
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(\frac{1}{2} \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3298271535226257e+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 -1.3298271535226257e+154 < x < 3.890409410163301e+59

    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 3.890409410163301e+59 < x

    1. Initial program 38.7

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

    2. Simplified38.7

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

    3. Taylor expanded around inf 1.9

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

    4. Simplified1.9

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3298271535226257 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 3.890409410163301 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))