Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.1 → 0.0

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.321181503567267682308379168130116570047 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 1.726425002480424766694997354741713243289 \cdot 10^{131}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r316003 = x;
        double r316004 = r316003 * r316003;
        double r316005 = y;
        double r316006 = r316004 + r316005;
        double r316007 = sqrt(r316006);
        return r316007;
}

↓

double f(double x, double y) {
        double r316008 = x;
        double r316009 = -1.3211815035672677e+154;
        bool r316010 = r316008 <= r316009;
        double r316011 = y;
        double r316012 = r316011 / r316008;
        double r316013 = 0.5;
        double r316014 = fma(r316012, r316013, r316008);
        double r316015 = -r316014;
        double r316016 = 1.7264250024804248e+131;
        bool r316017 = r316008 <= r316016;
        double r316018 = fma(r316008, r316008, r316011);
        double r316019 = sqrt(r316018);
        double r316020 = r316017 ? r316019 : r316014;
        double r316021 = r316010 ? r316015 : r316020;
        return r316021;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.1
Target	0.5
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 < -1.3211815035672677e+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 -1.3211815035672677e+154 < x < 1.7264250024804248e+131

   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 1.7264250024804248e+131 < x

   1. Initial program 56.4

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

   2. Simplified 56.4

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.321181503567267682308379168130116570047 \cdot 10^{154}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \mathbf{elif}\;x \le 1.726425002480424766694997354741713243289 \cdot 10^{131}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, \frac{1}{2}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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)))