Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.5 → 0.0

Time: 7.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.352732562381551610191380298409879181488 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 8.914808959129563024978640207181522528947 \cdot 10^{145}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r18962288 = x;
        double r18962289 = r18962288 * r18962288;
        double r18962290 = y;
        double r18962291 = r18962289 + r18962290;
        double r18962292 = sqrt(r18962291);
        return r18962292;
}

↓

double f(double x, double y) {
        double r18962293 = x;
        double r18962294 = -1.3527325623815516e+154;
        bool r18962295 = r18962293 <= r18962294;
        double r18962296 = -0.5;
        double r18962297 = y;
        double r18962298 = r18962293 / r18962297;
        double r18962299 = r18962296 / r18962298;
        double r18962300 = r18962299 - r18962293;
        double r18962301 = 8.914808959129563e+145;
        bool r18962302 = r18962293 <= r18962301;
        double r18962303 = fma(r18962293, r18962293, r18962297);
        double r18962304 = sqrt(r18962303);
        double r18962305 = r18962297 / r18962293;
        double r18962306 = 0.5;
        double r18962307 = fma(r18962305, r18962306, r18962293);
        double r18962308 = r18962302 ? r18962304 : r18962307;
        double r18962309 = r18962295 ? r18962300 : r18962308;
        return r18962309;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.5
Target	0.4
Herbie	0.0

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

Derivation

1. Split input into 3 regimes

2. if x < -1.3527325623815516e+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 -1.3527325623815516e+154 < x < 8.914808959129563e+145

   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 8.914808959129563e+145 < x

   1. Initial program 61.0

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

   2. Simplified 61.0

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

   3. Taylor expanded around inf 0.1

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

   4. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.352732562381551610191380298409879181488 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 8.914808959129563024978640207181522528947 \cdot 10^{145}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))