Average Error: 21.4 → 0.5
Time: 1.5s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3369975420138414 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.0395017264775639 \cdot 10^{51}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3369975420138414 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 1.0395017264775639 \cdot 10^{51}:\\
\;\;\;\;\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 r442640 = x;
        double r442641 = r442640 * r442640;
        double r442642 = y;
        double r442643 = r442641 + r442642;
        double r442644 = sqrt(r442643);
        return r442644;
}
↓
double f(double x, double y) {
        double r442645 = x;
        double r442646 = -1.3369975420138414e+154;
        bool r442647 = r442645 <= r442646;
        double r442648 = 0.5;
        double r442649 = y;
        double r442650 = r442649 / r442645;
        double r442651 = r442648 * r442650;
        double r442652 = r442645 + r442651;
        double r442653 = -r442652;
        double r442654 = 1.0395017264775639e+51;
        bool r442655 = r442645 <= r442654;
        double r442656 = r442645 * r442645;
        double r442657 = r442656 + r442649;
        double r442658 = sqrt(r442657);
        double r442659 = fma(r442648, r442650, r442645);
        double r442660 = r442655 ? r442658 : r442659;
        double r442661 = r442647 ? r442653 : r442660;
        return r442661;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.4
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3369975420138414e+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 -1.3369975420138414e+154 < x < 1.0395017264775639e+51

    1. Initial program 0.0

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

    if 1.0395017264775639e+51 < x

    1. Initial program 39.5

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

    2. Taylor expanded around inf 1.8

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

    3. Simplified1.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3369975420138414 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 1.0395017264775639 \cdot 10^{51}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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)))