Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.1 → 0.3

Time: 8.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.1772508290348614 \cdot 10^{-30} \lor \neg \left(z \le 1686.5899740750706\right):\\ \;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r578987 = x;
        double r578988 = cosh(r578987);
        double r578989 = y;
        double r578990 = r578989 / r578987;
        double r578991 = r578988 * r578990;
        double r578992 = z;
        double r578993 = r578991 / r578992;
        return r578993;
}

↓

double f(double x, double y, double z) {
        double r578994 = z;
        double r578995 = -4.1772508290348614e-30;
        bool r578996 = r578994 <= r578995;
        double r578997 = 1686.5899740750706;
        bool r578998 = r578994 <= r578997;
        double r578999 = !r578998;
        bool r579000 = r578996 || r578999;
        double r579001 = y;
        double r579002 = x;
        double r579003 = exp(r579002);
        double r579004 = -r579002;
        double r579005 = exp(r579004);
        double r579006 = r579003 + r579005;
        double r579007 = r579001 * r579006;
        double r579008 = 2.0;
        double r579009 = r579008 * r579002;
        double r579010 = r578994 * r579009;
        double r579011 = r579007 / r579010;
        double r579012 = cosh(r579002);
        double r579013 = r579001 / r579002;
        double r579014 = r579012 * r579013;
        double r579015 = r579014 / r578994;
        double r579016 = r579000 ? r579011 : r579015;
        return r579016;
}

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 z < -4.1772508290348614e-30 or 1686.5899740750706 < z

   1. Initial program 11.9

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

   3. Applied cosh-def 11.9

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

   4. Applied frac-times 11.9

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

   5. Applied associate-/l/ 0.3

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

   7. Applied *-commutative 0.3

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

   if -4.1772508290348614e-30 < z < 1686.5899740750706

   1. Initial program 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.1772508290348614 \cdot 10^{-30} \lor \neg \left(z \le 1686.5899740750706\right):\\ \;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))