Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.6 → 0.4

Time: 14.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.90737577331802458 \cdot 10^{-57} \lor \neg \left(z \le 2.54423027784147743 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r479901 = x;
        double r479902 = cosh(r479901);
        double r479903 = y;
        double r479904 = r479903 / r479901;
        double r479905 = r479902 * r479904;
        double r479906 = z;
        double r479907 = r479905 / r479906;
        return r479907;
}

↓

double f(double x, double y, double z) {
        double r479908 = z;
        double r479909 = -5.907375773318025e-57;
        bool r479910 = r479908 <= r479909;
        double r479911 = 2.5442302778414774e-06;
        bool r479912 = r479908 <= r479911;
        double r479913 = !r479912;
        bool r479914 = r479910 || r479913;
        double r479915 = x;
        double r479916 = cosh(r479915);
        double r479917 = y;
        double r479918 = r479916 * r479917;
        double r479919 = r479915 * r479908;
        double r479920 = r479918 / r479919;
        double r479921 = r479908 / r479917;
        double r479922 = r479915 * r479921;
        double r479923 = r479916 / r479922;
        double r479924 = r479914 ? r479920 : r479923;
        return r479924;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.6
Target	0.4
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -5.907375773318025e-57 or 2.5442302778414774e-06 < z

   1. Initial program 10.7

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

   3. Applied associate-*r/ 10.7

      \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z}\]

   4. Applied associate-/l/ 0.4

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]

   5. Simplified 0.4

      \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{x \cdot z}}\]

   if -5.907375773318025e-57 < z < 2.5442302778414774e-06

   1. Initial program 0.3

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

   3. Applied associate-/l* 0.4

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\]

   4. Simplified 0.4

      \[\leadsto \frac{\cosh x}{\color{blue}{z \cdot \frac{x}{y}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 0.4

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

   7. Applied associate-*l* 0.4

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

   8. Simplified 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.90737577331802458 \cdot 10^{-57} \lor \neg \left(z \le 2.54423027784147743 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))