Average Error: 21.3 → 0.1
Time: 15.7s
Precision: 64
\[\sqrt{x \cdot x + y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.187498495601377232542915422132552769484 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\ \mathbf{elif}\;x \le 1.666665935899026866978309593021789775966 \cdot 10^{110}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]
\sqrt{x \cdot x + y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.187498495601377232542915422132552769484 \cdot 10^{152}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\

\mathbf{elif}\;x \le 1.666665935899026866978309593021789775966 \cdot 10^{110}:\\
\;\;\;\;\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 r25339480 = x;
        double r25339481 = r25339480 * r25339480;
        double r25339482 = y;
        double r25339483 = r25339481 + r25339482;
        double r25339484 = sqrt(r25339483);
        return r25339484;
}
↓
double f(double x, double y) {
        double r25339485 = x;
        double r25339486 = -3.187498495601377e+152;
        bool r25339487 = r25339485 <= r25339486;
        double r25339488 = -0.5;
        double r25339489 = y;
        double r25339490 = r25339489 / r25339485;
        double r25339491 = r25339488 * r25339490;
        double r25339492 = r25339491 - r25339485;
        double r25339493 = 1.666665935899027e+110;
        bool r25339494 = r25339485 <= r25339493;
        double r25339495 = fma(r25339485, r25339485, r25339489);
        double r25339496 = sqrt(r25339495);
        double r25339497 = 0.5;
        double r25339498 = r25339497 / r25339485;
        double r25339499 = fma(r25339498, r25339489, r25339485);
        double r25339500 = r25339494 ? r25339496 : r25339499;
        double r25339501 = r25339487 ? r25339492 : r25339500;
        return r25339501;
}

Error

Bits error versus x

Bits error versus y

Target

Original21.3
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.509769801047259255153812752081023359759 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122540716781541767466805967807 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.187498495601377e+152

    1. Initial program 63.2

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

    2. Simplified63.2

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

    3. Taylor expanded around -inf 0.0

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

    4. Simplified0.0

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

    if -3.187498495601377e+152 < x < 1.666665935899027e+110

    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 1.666665935899027e+110 < x

    1. Initial program 49.0

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

    2. Simplified49.0

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

    3. Taylor expanded around inf 0.5

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

    4. Simplified0.5

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.187498495601377232542915422132552769484 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\ \mathbf{elif}\;x \le 1.666665935899026866978309593021789775966 \cdot 10^{110}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :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)))