Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.4 → 0.1

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.33627491053925200483497266921088186961 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\ \mathbf{elif}\;x \le 2.216128794604703682435523417006564161481 \cdot 10^{123}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r51921236 = x;
        double r51921237 = r51921236 * r51921236;
        double r51921238 = y;
        double r51921239 = r51921237 + r51921238;
        double r51921240 = sqrt(r51921239);
        return r51921240;
}

↓

double f(double x, double y) {
        double r51921241 = x;
        double r51921242 = -1.336274910539252e+154;
        bool r51921243 = r51921241 <= r51921242;
        double r51921244 = -0.5;
        double r51921245 = y;
        double r51921246 = r51921245 / r51921241;
        double r51921247 = r51921244 * r51921246;
        double r51921248 = r51921247 - r51921241;
        double r51921249 = 2.2161287946047037e+123;
        bool r51921250 = r51921241 <= r51921249;
        double r51921251 = fma(r51921241, r51921241, r51921245);
        double r51921252 = sqrt(r51921251);
        double r51921253 = 0.5;
        double r51921254 = fma(r51921246, r51921253, r51921241);
        double r51921255 = r51921250 ? r51921252 : r51921254;
        double r51921256 = r51921243 ? r51921248 : r51921255;
        return r51921256;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.4
Target	0.6
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if x < -1.336274910539252e+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 0.0

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

   4. Simplified 0.0

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

   if -1.336274910539252e+154 < x < 2.2161287946047037e+123

   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 2.2161287946047037e+123 < x

   1. Initial program 53.4

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

   2. Simplified 53.4

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

   3. Taylor expanded around inf 0.4

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

   4. Simplified 0.4

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33627491053925200483497266921088186961 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{y}{x} - x\\ \mathbf{elif}\;x \le 2.216128794604703682435523417006564161481 \cdot 10^{123}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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)))