Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.3

Time: 3.7s

Precision: 64

Math

\[\frac{x \cdot \frac{\sin y}{y}}{z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -53734252884500.3828125 \lor \neg \left(z \le 141625740.855926513671875\right):\\ \;\;\;\;\left(x \cdot \frac{\sin y}{y}\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\sin y}{y} \cdot \frac{1}{z}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r393889 = x;
        double r393890 = y;
        double r393891 = sin(r393890);
        double r393892 = r393891 / r393890;
        double r393893 = r393889 * r393892;
        double r393894 = z;
        double r393895 = r393893 / r393894;
        return r393895;
}

↓

double f(double x, double y, double z) {
        double r393896 = z;
        double r393897 = -53734252884500.38;
        bool r393898 = r393896 <= r393897;
        double r393899 = 141625740.8559265;
        bool r393900 = r393896 <= r393899;
        double r393901 = !r393900;
        bool r393902 = r393898 || r393901;
        double r393903 = x;
        double r393904 = y;
        double r393905 = sin(r393904);
        double r393906 = r393905 / r393904;
        double r393907 = r393903 * r393906;
        double r393908 = 1.0;
        double r393909 = r393908 / r393896;
        double r393910 = r393907 * r393909;
        double r393911 = r393906 * r393909;
        double r393912 = r393903 * r393911;
        double r393913 = r393902 ? r393910 : r393912;
        return r393913;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.7
Target	0.3
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -53734252884500.38 or 141625740.8559265 < z

   1. Initial program 0.1

      \[\frac{x \cdot \frac{\sin y}{y}}{z}\]
   2. Using strategy rm

   3. Applied div-inv 0.2

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sin y}{y}\right) \cdot \frac{1}{z}}\]

   if -53734252884500.38 < z < 141625740.8559265

   1. Initial program 5.5

      \[\frac{x \cdot \frac{\sin y}{y}}{z}\]
   2. Using strategy rm

   3. Applied div-inv 5.7

      \[\leadsto \color{blue}{\left(x \cdot \frac{\sin y}{y}\right) \cdot \frac{1}{z}}\]
   4. Using strategy rm

   5. Applied associate-*l* 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -53734252884500.3828125 \lor \neg \left(z \le 141625740.855926513671875\right):\\ \;\;\;\;\left(x \cdot \frac{\sin y}{y}\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\sin y}{y} \cdot \frac{1}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))