Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.1 → 0.2

Time: 4.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.430368293804642349759278332177315240505 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.247037676574067808708350046782507690014 \cdot 10^{83}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r17105302 = x;
        double r17105303 = r17105302 * r17105302;
        double r17105304 = y;
        double r17105305 = r17105303 + r17105304;
        double r17105306 = sqrt(r17105305);
        return r17105306;
}

↓

double f(double x, double y) {
        double r17105307 = x;
        double r17105308 = -8.430368293804642e+153;
        bool r17105309 = r17105307 <= r17105308;
        double r17105310 = -0.5;
        double r17105311 = y;
        double r17105312 = r17105307 / r17105311;
        double r17105313 = r17105310 / r17105312;
        double r17105314 = r17105313 - r17105307;
        double r17105315 = 2.2470376765740678e+83;
        bool r17105316 = r17105307 <= r17105315;
        double r17105317 = fma(r17105307, r17105307, r17105311);
        double r17105318 = sqrt(r17105317);
        double r17105319 = r17105311 / r17105307;
        double r17105320 = 0.5;
        double r17105321 = fma(r17105319, r17105320, r17105307);
        double r17105322 = r17105316 ? r17105318 : r17105321;
        double r17105323 = r17105309 ? r17105314 : r17105322;
        return r17105323;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.1
Target	0.4
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 < -8.430368293804642e+153

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   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 -8.430368293804642e+153 < x < 2.2470376765740678e+83

   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 2.2470376765740678e+83 < x

   1. Initial program 43.9

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

   2. Simplified 43.9

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

   3. Taylor expanded around inf 1.0

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

   4. Simplified 1.0

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.430368293804642349759278332177315240505 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 2.247037676574067808708350046782507690014 \cdot 10^{83}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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)))