Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.4 → 0.1

Time: 8.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.36496554640706019677869997344330658994 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 3.588070708117172201716955628167087287626 \cdot 10^{111}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r425161 = x;
        double r425162 = r425161 * r425161;
        double r425163 = y;
        double r425164 = r425162 + r425163;
        double r425165 = sqrt(r425164);
        return r425165;
}

↓

double f(double x, double y) {
        double r425166 = x;
        double r425167 = -1.3649655464070602e+154;
        bool r425168 = r425166 <= r425167;
        double r425169 = y;
        double r425170 = r425169 / r425166;
        double r425171 = 0.5;
        double r425172 = fma(r425170, r425171, r425166);
        double r425173 = -r425172;
        double r425174 = 3.588070708117172e+111;
        bool r425175 = r425166 <= r425174;
        double r425176 = fma(r425166, r425166, r425169);
        double r425177 = sqrt(r425176);
        double r425178 = r425175 ? r425177 : r425172;
        double r425179 = r425168 ? r425173 : r425178;
        return r425179;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.4
Target	0.5
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 < -1.3649655464070602e+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

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)}\]

   if -1.3649655464070602e+154 < x < 3.588070708117172e+111

   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 3.588070708117172e+111 < x

   1. Initial program 51.0

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

   2. Simplified 51.0

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

   3. Taylor expanded around inf 0.6

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

   4. Simplified 0.6

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.36496554640706019677869997344330658994 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 3.588070708117172201716955628167087287626 \cdot 10^{111}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.5823995511225407e57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

  (sqrt (+ (* x x) y)))