Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.3 → 0.3

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1852666491.27304912 \lor \neg \left(z \le 309596806417.428772\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot y\right)}{z}}{x}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r575950 = x;
        double r575951 = cosh(r575950);
        double r575952 = y;
        double r575953 = r575952 / r575950;
        double r575954 = r575951 * r575953;
        double r575955 = z;
        double r575956 = r575954 / r575955;
        return r575956;
}

↓

double f(double x, double y, double z) {
        double r575957 = z;
        double r575958 = -1852666491.273049;
        bool r575959 = r575957 <= r575958;
        double r575960 = 309596806417.4288;
        bool r575961 = r575957 <= r575960;
        double r575962 = !r575961;
        bool r575963 = r575959 || r575962;
        double r575964 = x;
        double r575965 = cosh(r575964);
        double r575966 = y;
        double r575967 = r575965 * r575966;
        double r575968 = r575957 * r575964;
        double r575969 = r575967 / r575968;
        double r575970 = sqrt(r575965);
        double r575971 = r575970 * r575966;
        double r575972 = r575970 * r575971;
        double r575973 = r575972 / r575957;
        double r575974 = r575973 / r575964;
        double r575975 = r575963 ? r575969 : r575974;
        return r575975;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.3
Target	0.4
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if z < -1852666491.273049 or 309596806417.4288 < z

   1. Initial program 11.5

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

   3. Applied associate-*r/ 11.5

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

   4. Applied associate-/l/ 0.3

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

   if -1852666491.273049 < z < 309596806417.4288

   1. Initial program 0.3

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

   3. Applied associate-*r/ 0.3

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

   4. Applied associate-/l/ 19.0

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

   6. Applied associate-/r* 0.3

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

   8. Applied add-sqr-sqrt 0.3

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

   9. Applied associate-*l* 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1852666491.27304912 \lor \neg \left(z \le 309596806417.428772\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot y\right)}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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))