Average Error: 22.0 → 0.1
Time: 1.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33520561977810183 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 8.45687055249789216 \cdot 10^{117}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33520561977810183 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 8.45687055249789216 \cdot 10^{117}:\\
\;\;\;\;\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 r511751 = x;
        double r511752 = r511751 * r511751;
        double r511753 = y;
        double r511754 = r511752 + r511753;
        double r511755 = sqrt(r511754);
        return r511755;
}
↓
double f(double x, double y) {
        double r511756 = x;
        double r511757 = -1.3352056197781018e+154;
        bool r511758 = r511756 <= r511757;
        double r511759 = 0.5;
        double r511760 = y;
        double r511761 = r511760 / r511756;
        double r511762 = r511759 * r511761;
        double r511763 = r511756 + r511762;
        double r511764 = -r511763;
        double r511765 = 8.456870552497892e+117;
        bool r511766 = r511756 <= r511765;
        double r511767 = r511756 * r511756;
        double r511768 = r511767 + r511760;
        double r511769 = sqrt(r511768);
        double r511770 = fma(r511759, r511761, r511756);
        double r511771 = r511766 ? r511769 : r511770;
        double r511772 = r511758 ? r511764 : r511771;
        return r511772;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.0
Target0.6
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3352056197781018e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 0.0

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

    if -1.3352056197781018e+154 < x < 8.456870552497892e+117

    1. Initial program 0.0

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

    if 8.456870552497892e+117 < x

    1. Initial program 52.8

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33520561977810183 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 8.45687055249789216 \cdot 10^{117}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))