Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.4 → 0.2

Time: 14.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.344192746470904425870702492238984808211 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 3.146509911612528293591568472384734910948 \cdot 10^{98}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r23925335 = x;
        double r23925336 = r23925335 * r23925335;
        double r23925337 = y;
        double r23925338 = r23925336 + r23925337;
        double r23925339 = sqrt(r23925338);
        return r23925339;
}

↓

double f(double x, double y) {
        double r23925340 = x;
        double r23925341 = -1.3441927464709044e+154;
        bool r23925342 = r23925340 <= r23925341;
        double r23925343 = -0.5;
        double r23925344 = y;
        double r23925345 = r23925340 / r23925344;
        double r23925346 = r23925343 / r23925345;
        double r23925347 = r23925346 - r23925340;
        double r23925348 = 3.146509911612528e+98;
        bool r23925349 = r23925340 <= r23925348;
        double r23925350 = fma(r23925340, r23925340, r23925344);
        double r23925351 = sqrt(r23925350);
        double r23925352 = 0.5;
        double r23925353 = r23925344 / r23925340;
        double r23925354 = fma(r23925352, r23925353, r23925340);
        double r23925355 = r23925349 ? r23925351 : r23925354;
        double r23925356 = r23925342 ? r23925347 : r23925355;
        return r23925356;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.4
Target	0.5
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x < -1.3441927464709044e+154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded around -inf 0

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

   4. Simplified 0

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

   if -1.3441927464709044e+154 < x < 3.146509911612528e+98

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   if 3.146509911612528e+98 < x

   1. Initial program 47.4

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

   2. Simplified 47.4

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

   3. Taylor expanded around inf 0.7

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

   4. Simplified 0.7

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.344192746470904425870702492238984808211 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 3.146509911612528293591568472384734910948 \cdot 10^{98}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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)))