Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.1 → 0.2

Time: 8.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r27175940 = x;
        double r27175941 = r27175940 * r27175940;
        double r27175942 = y;
        double r27175943 = r27175941 + r27175942;
        double r27175944 = sqrt(r27175943);
        return r27175944;
}

↓

double f(double x, double y) {
        double r27175945 = x;
        double r27175946 = -8.430368293804642e+153;
        bool r27175947 = r27175945 <= r27175946;
        double r27175948 = y;
        double r27175949 = -0.5;
        double r27175950 = r27175948 * r27175949;
        double r27175951 = r27175950 / r27175945;
        double r27175952 = r27175951 - r27175945;
        double r27175953 = 2.2470376765740678e+83;
        bool r27175954 = r27175945 <= r27175953;
        double r27175955 = fma(r27175945, r27175945, r27175948);
        double r27175956 = sqrt(r27175955);
        double r27175957 = 0.5;
        double r27175958 = r27175957 / r27175945;
        double r27175959 = fma(r27175958, r27175948, r27175945);
        double r27175960 = r27175954 ? r27175956 : r27175959;
        double r27175961 = r27175947 ? r27175952 : r27175960;
        return r27175961;
}

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

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

   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

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

4. Final simplification 0.2

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

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)))