Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.1 → 0.2

Time: 1.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.3694640831062883 \cdot 10^{154}:\\ \;\;\;\;1 \cdot \left(-1 \cdot x\right)\\ \mathbf{elif}\;x \le 8.4390817817310158 \cdot 10^{104}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r586105 = x;
        double r586106 = r586105 * r586105;
        double r586107 = y;
        double r586108 = r586106 + r586107;
        double r586109 = sqrt(r586108);
        return r586109;
}

↓

double f(double x, double y) {
        double r586110 = x;
        double r586111 = -1.3694640831062883e+154;
        bool r586112 = r586110 <= r586111;
        double r586113 = 1.0;
        double r586114 = -1.0;
        double r586115 = r586114 * r586110;
        double r586116 = r586113 * r586115;
        double r586117 = 8.439081781731016e+104;
        bool r586118 = r586110 <= r586117;
        double r586119 = r586110 * r586110;
        double r586120 = y;
        double r586121 = r586119 + r586120;
        double r586122 = sqrt(r586121);
        double r586123 = 0.5;
        double r586124 = r586120 / r586110;
        double r586125 = fma(r586123, r586124, r586110);
        double r586126 = r586118 ? r586122 : r586125;
        double r586127 = r586112 ? r586116 : r586126;
        return r586127;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.1
Target	0.5
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.5823995511225407 \cdot 10^{57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.3694640831062883e+154

   1. Initial program 64.0

      \[\sqrt{x \cdot x + y}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 64.0

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

   4. Applied sqrt-prod 64.0

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

   5. Simplified 64.0

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

   6. Simplified 31.0

      \[\leadsto 1 \cdot \color{blue}{\mathsf{hypot}\left(x, {y}^{\frac{1}{2}}\right)}\]

   7. Taylor expanded around -inf 0

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

   if -1.3694640831062883e+154 < x < 8.439081781731016e+104

   1. Initial program 0.0

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

   if 8.439081781731016e+104 < x

   1. Initial program 49.3

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

   2. Taylor expanded around inf 0.9

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

   3. Simplified 0.9

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3694640831062883 \cdot 10^{154}:\\ \;\;\;\;1 \cdot \left(-1 \cdot x\right)\\ \mathbf{elif}\;x \le 8.4390817817310158 \cdot 10^{104}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{y}{x}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :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)))