Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.4

Time: 12.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -27156977852.05674 \lor \neg \left(z \le 1.53951426725456284 \cdot 10^{-49}\right):\\ \;\;\;\;\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z \cdot \frac{x}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r594507 = x;
        double r594508 = cosh(r594507);
        double r594509 = y;
        double r594510 = r594509 / r594507;
        double r594511 = r594508 * r594510;
        double r594512 = z;
        double r594513 = r594511 / r594512;
        return r594513;
}

↓

double f(double x, double y, double z) {
        double r594514 = z;
        double r594515 = -27156977852.05674;
        bool r594516 = r594514 <= r594515;
        double r594517 = 1.5395142672545628e-49;
        bool r594518 = r594514 <= r594517;
        double r594519 = !r594518;
        bool r594520 = r594516 || r594519;
        double r594521 = x;
        double r594522 = cosh(r594521);
        double r594523 = sqrt(r594522);
        double r594524 = y;
        double r594525 = r594521 * r594514;
        double r594526 = r594524 / r594525;
        double r594527 = r594523 * r594526;
        double r594528 = r594523 * r594527;
        double r594529 = r594521 / r594524;
        double r594530 = r594514 * r594529;
        double r594531 = r594522 / r594530;
        double r594532 = r594520 ? r594528 : r594531;
        return r594532;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
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 < -27156977852.05674 or 1.5395142672545628e-49 < z

   1. Initial program 11.5

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

   3. Applied *-un-lft-identity 11.5

      \[\leadsto \frac{\cosh x \cdot \frac{y}{x}}{\color{blue}{1 \cdot z}}\]

   4. Applied times-frac 11.5

      \[\leadsto \color{blue}{\frac{\cosh x}{1} \cdot \frac{\frac{y}{x}}{z}}\]

   5. Simplified 11.5

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

   6. Simplified 0.4

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

   8. Applied add-sqr-sqrt 0.5

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

   9. Applied associate-*l* 0.5

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

   if -27156977852.05674 < z < 1.5395142672545628e-49

   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

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -27156977852.05674 \lor \neg \left(z \le 1.53951426725456284 \cdot 10^{-49}\right):\\ \;\;\;\;\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z \cdot \frac{x}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))