Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.4 → 0.4

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.352540539223698945459720582374994240123 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 4.721377972444830028288106556093954402552 \cdot 10^{71}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r325682 = x;
        double r325683 = r325682 * r325682;
        double r325684 = y;
        double r325685 = r325683 + r325684;
        double r325686 = sqrt(r325685);
        return r325686;
}

↓

double f(double x, double y) {
        double r325687 = x;
        double r325688 = -1.352540539223699e+154;
        bool r325689 = r325687 <= r325688;
        double r325690 = y;
        double r325691 = r325690 / r325687;
        double r325692 = 0.5;
        double r325693 = fma(r325691, r325692, r325687);
        double r325694 = -r325693;
        double r325695 = 4.72137797244483e+71;
        bool r325696 = r325687 <= r325695;
        double r325697 = fma(r325687, r325687, r325690);
        double r325698 = sqrt(r325697);
        double r325699 = r325696 ? r325698 : r325693;
        double r325700 = r325689 ? r325694 : r325699;
        return r325700;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.4
Target	0.5
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if x < -1.352540539223699e+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 -1.352540539223699e+154 < x < 4.72137797244483e+71

   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 4.72137797244483e+71 < x

   1. Initial program 42.1

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

   2. Simplified 42.1

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

   3. Taylor expanded around inf 1.6

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

   4. Simplified 1.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.352540539223698945459720582374994240123 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 4.721377972444830028288106556093954402552 \cdot 10^{71}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))