Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.8 → 0.6

Time: 14.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -29659565816422.921875 \lor \neg \left(z \le 6.561790096885940735873177644395925207708 \cdot 10^{-72}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r382228 = x;
        double r382229 = cosh(r382228);
        double r382230 = y;
        double r382231 = r382230 / r382228;
        double r382232 = r382229 * r382231;
        double r382233 = z;
        double r382234 = r382232 / r382233;
        return r382234;
}

↓

double f(double x, double y, double z) {
        double r382235 = z;
        double r382236 = -29659565816422.92;
        bool r382237 = r382235 <= r382236;
        double r382238 = 6.561790096885941e-72;
        bool r382239 = r382235 <= r382238;
        double r382240 = !r382239;
        bool r382241 = r382237 || r382240;
        double r382242 = x;
        double r382243 = cosh(r382242);
        double r382244 = y;
        double r382245 = r382242 * r382235;
        double r382246 = r382244 / r382245;
        double r382247 = r382243 * r382246;
        double r382248 = r382244 / r382242;
        double r382249 = r382243 * r382248;
        double r382250 = 1.0;
        double r382251 = r382250 / r382235;
        double r382252 = r382249 * r382251;
        double r382253 = r382241 ? r382247 : r382252;
        return r382253;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.8
Target	0.5
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -29659565816422.92 or 6.561790096885941e-72 < z

   1. Initial program 11.2

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

   3. Applied associate-/l* 11.5

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

   4. Simplified 9.3

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

   6. Applied div-inv 9.3

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

   7. Simplified 9.2

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

   8. Taylor expanded around 0 0.6

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

   if -29659565816422.92 < z < 6.561790096885941e-72

   1. Initial program 0.3

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

   3. Applied div-inv 0.5

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -29659565816422.921875 \lor \neg \left(z \le 6.561790096885940735873177644395925207708 \cdot 10^{-72}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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))