Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.6 → 0.5

Time: 10.7s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r512574 = x;
        double r512575 = cosh(r512574);
        double r512576 = y;
        double r512577 = r512576 / r512574;
        double r512578 = r512575 * r512577;
        double r512579 = z;
        double r512580 = r512578 / r512579;
        return r512580;
}

↓

double f(double x, double y, double z) {
        double r512581 = z;
        double r512582 = -5.907375773318025e-57;
        bool r512583 = r512581 <= r512582;
        double r512584 = 1.0700912275797494e+36;
        bool r512585 = r512581 <= r512584;
        double r512586 = !r512585;
        bool r512587 = r512583 || r512586;
        double r512588 = x;
        double r512589 = cosh(r512588);
        double r512590 = y;
        double r512591 = r512589 * r512590;
        double r512592 = 1.0;
        double r512593 = r512592 / r512588;
        double r512594 = r512593 / r512581;
        double r512595 = r512591 * r512594;
        double r512596 = r512590 / r512588;
        double r512597 = r512589 * r512596;
        double r512598 = r512592 / r512581;
        double r512599 = r512597 * r512598;
        double r512600 = r512587 ? r512595 : r512599;
        return r512600;
}

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 +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))