Average Error: 21.4 → 0.1
Time: 2.0s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33991033865496575 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.9126873135626368 \cdot 10^{105}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33991033865496575 \cdot 10^{154}:\\
\;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\

\mathbf{elif}\;x \le 5.9126873135626368 \cdot 10^{105}:\\
\;\;\;\;\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 r446474 = x;
        double r446475 = r446474 * r446474;
        double r446476 = y;
        double r446477 = r446475 + r446476;
        double r446478 = sqrt(r446477);
        return r446478;
}
↓
double f(double x, double y) {
        double r446479 = x;
        double r446480 = -1.3399103386549657e+154;
        bool r446481 = r446479 <= r446480;
        double r446482 = 0.5;
        double r446483 = y;
        double r446484 = r446483 / r446479;
        double r446485 = r446482 * r446484;
        double r446486 = r446479 + r446485;
        double r446487 = -r446486;
        double r446488 = 5.912687313562637e+105;
        bool r446489 = r446479 <= r446488;
        double r446490 = r446479 * r446479;
        double r446491 = r446490 + r446483;
        double r446492 = sqrt(r446491);
        double r446493 = fma(r446482, r446484, r446479);
        double r446494 = r446489 ? r446492 : r446493;
        double r446495 = r446481 ? r446487 : r446494;
        return r446495;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.4
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 < -1.3399103386549657e+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 -1.3399103386549657e+154 < x < 5.912687313562637e+105

    1. Initial program 0.0

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

    if 5.912687313562637e+105 < x

    1. Initial program 50.0

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33991033865496575 \cdot 10^{154}:\\ \;\;\;\;-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \le 5.9126873135626368 \cdot 10^{105}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))