Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 19.8 → 0.7

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.3323185489366894 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.3070827329489974 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r20199644 = x;
        double r20199645 = r20199644 * r20199644;
        double r20199646 = y;
        double r20199647 = r20199645 + r20199646;
        double r20199648 = sqrt(r20199647);
        return r20199648;
}

↓

double f(double x, double y) {
        double r20199649 = x;
        double r20199650 = -1.3323185489366894e+154;
        bool r20199651 = r20199649 <= r20199650;
        double r20199652 = -0.5;
        double r20199653 = y;
        double r20199654 = r20199649 / r20199653;
        double r20199655 = r20199652 / r20199654;
        double r20199656 = r20199655 - r20199649;
        double r20199657 = 1.3070827329489974e+38;
        bool r20199658 = r20199649 <= r20199657;
        double r20199659 = fma(r20199649, r20199649, r20199653);
        double r20199660 = sqrt(r20199659);
        double r20199661 = 0.5;
        double r20199662 = r20199661 / r20199649;
        double r20199663 = fma(r20199662, r20199653, r20199649);
        double r20199664 = r20199658 ? r20199660 : r20199663;
        double r20199665 = r20199651 ? r20199656 : r20199664;
        return r20199665;
}

Error

Bits error versus x

Bits error versus y

Target

Original	19.8
Target	0.5
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;x \lt -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(\frac{1}{2} \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x \lt 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{y}{x} + x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.3323185489366894e+154

   1. Initial program 59.6

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

   2. Simplified 59.6

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

   3. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-\left(x + \frac{1}{2} \cdot \frac{y}{x}\right)}\]

   4. Simplified 0

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

   if -1.3323185489366894e+154 < x < 1.3070827329489974e+38

   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 1.3070827329489974e+38 < x

   1. Initial program 36.4

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

   2. Simplified 36.4

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

   3. Taylor expanded around inf 2.6

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

   4. Simplified 2.6

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3323185489366894 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{-1}{2}}{\frac{x}{y}} - x\\ \mathbf{elif}\;x \le 1.3070827329489974 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{2}}{x}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 1/2 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 1/2 (/ y x)) x)))

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