Average Error: 21.3 → 0.4
Time: 2.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot x\\

\end{array}
double f(double x, double y) {
        double r496935 = x;
        double r496936 = r496935 * r496935;
        double r496937 = y;
        double r496938 = r496936 + r496937;
        double r496939 = sqrt(r496938);
        return r496939;
}
↓
double f(double x, double y) {
        double r496940 = x;
        double r496941 = -1.3778733035656446e+154;
        bool r496942 = r496940 <= r496941;
        double r496943 = 0.5;
        double r496944 = y;
        double r496945 = r496944 / r496940;
        double r496946 = r496943 * r496945;
        double r496947 = r496940 + r496946;
        double r496948 = -r496947;
        double r496949 = 1.2922566123944575e+80;
        bool r496950 = r496940 <= r496949;
        double r496951 = fma(r496940, r496940, r496944);
        double r496952 = sqrt(r496951);
        double r496953 = 1.0;
        double r496954 = r496953 * r496940;
        double r496955 = r496950 ? r496952 : r496954;
        double r496956 = r496942 ? r496948 : r496955;
        return r496956;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.3
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3778733035656446e+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 -1.3778733035656446e+154 < x < 1.2922566123944575e+80

    1. Initial program 0.0

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

    3. Applied fma-def0.0

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

    if 1.2922566123944575e+80 < x

    1. Initial program 44.1

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

    3. Applied *-un-lft-identity44.1

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

    4. Applied sqrt-prod44.1

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

    5. Simplified44.1

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

    6. Simplified31.5

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

    7. Taylor expanded around inf 1.6

      \[\leadsto 1 \cdot \color{blue}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.37787330356564457 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.29225661239445747 \cdot 10^{80}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))