Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.6 → 0.5

Time: 10.1s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.90737577331802458 \cdot 10^{-57} \lor \neg \left(z \le 1.0700912275797494 \cdot 10^{36}\right):\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{\frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r581176 = x;
        double r581177 = cosh(r581176);
        double r581178 = y;
        double r581179 = r581178 / r581176;
        double r581180 = r581177 * r581179;
        double r581181 = z;
        double r581182 = r581180 / r581181;
        return r581182;
}

↓

double f(double x, double y, double z) {
        double r581183 = z;
        double r581184 = -5.907375773318025e-57;
        bool r581185 = r581183 <= r581184;
        double r581186 = 1.0700912275797494e+36;
        bool r581187 = r581183 <= r581186;
        double r581188 = !r581187;
        bool r581189 = r581185 || r581188;
        double r581190 = x;
        double r581191 = cosh(r581190);
        double r581192 = y;
        double r581193 = r581191 * r581192;
        double r581194 = 1.0;
        double r581195 = r581194 / r581190;
        double r581196 = r581195 / r581183;
        double r581197 = r581193 * r581196;
        double r581198 = r581192 / r581190;
        double r581199 = r581191 * r581198;
        double r581200 = r581194 / r581183;
        double r581201 = r581199 * r581200;
        double r581202 = r581189 ? r581197 : r581201;
        return r581202;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.6
Target	0.4
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if z < -5.907375773318025e-57 or 1.0700912275797494e+36 < z

   1. Initial program 11.2

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

   3. Applied div-inv 11.2

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

   4. Applied associate-*r* 11.2

      \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}}{z}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 11.2

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

   7. Applied times-frac 0.6

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

   8. Simplified 0.6

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

   if -5.907375773318025e-57 < z < 1.0700912275797494e+36

   1. Initial program 0.4

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
   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 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.90737577331802458 \cdot 10^{-57} \lor \neg \left(z \le 1.0700912275797494 \cdot 10^{36}\right):\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{\frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \end{array}\]

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))