Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.1 → 0.3

Time: 8.8s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -8.54460814469498844 \cdot 10^{108} \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.08528837221755491 \cdot 10^{303}\right):\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r394318 = x;
        double r394319 = cosh(r394318);
        double r394320 = y;
        double r394321 = r394320 / r394318;
        double r394322 = r394319 * r394321;
        double r394323 = z;
        double r394324 = r394322 / r394323;
        return r394324;
}

↓

double f(double x, double y, double z) {
        double r394325 = x;
        double r394326 = cosh(r394325);
        double r394327 = y;
        double r394328 = r394327 / r394325;
        double r394329 = r394326 * r394328;
        double r394330 = z;
        double r394331 = r394329 / r394330;
        double r394332 = -8.544608144694988e+108;
        bool r394333 = r394331 <= r394332;
        double r394334 = 1.0852883722175549e+303;
        bool r394335 = r394331 <= r394334;
        double r394336 = !r394335;
        bool r394337 = r394333 || r394336;
        double r394338 = exp(r394325);
        double r394339 = -r394325;
        double r394340 = exp(r394339);
        double r394341 = r394338 + r394340;
        double r394342 = r394341 * r394327;
        double r394343 = r394342 / r394330;
        double r394344 = 2.0;
        double r394345 = r394344 * r394325;
        double r394346 = r394343 / r394345;
        double r394347 = r394337 ? r394346 : r394331;
        return r394347;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	8.1
Target	0.5
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 (/ (* (cosh x) (/ y x)) z) < -8.544608144694988e+108 or 1.0852883722175549e+303 < (/ (* (cosh x) (/ y x)) z)

   1. Initial program 29.4

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

   3. Applied cosh-def 29.4

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

   4. Applied frac-times 29.4

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

   5. Applied associate-/l/ 11.0

      \[\leadsto \color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}}\]
   6. Using strategy rm

   7. Applied associate-/r* 0.4

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

   if -8.544608144694988e+108 < (/ (* (cosh x) (/ y x)) z) < 1.0852883722175549e+303

   1. Initial program 0.2

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -8.54460814469498844 \cdot 10^{108} \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.08528837221755491 \cdot 10^{303}\right):\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))