Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.1 → 0.3

Time: 12.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -8.54460814469498844 \cdot 10^{108} \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.08528837221755491 \cdot 10^{303}\right):\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r587652 = x;
        double r587653 = cosh(r587652);
        double r587654 = y;
        double r587655 = r587654 / r587652;
        double r587656 = r587653 * r587655;
        double r587657 = z;
        double r587658 = r587656 / r587657;
        return r587658;
}

↓

double f(double x, double y, double z) {
        double r587659 = x;
        double r587660 = cosh(r587659);
        double r587661 = y;
        double r587662 = r587661 / r587659;
        double r587663 = r587660 * r587662;
        double r587664 = z;
        double r587665 = r587663 / r587664;
        double r587666 = -8.544608144694988e+108;
        bool r587667 = r587665 <= r587666;
        double r587668 = 1.0852883722175549e+303;
        bool r587669 = r587665 <= r587668;
        double r587670 = !r587669;
        bool r587671 = r587667 || r587670;
        double r587672 = exp(r587659);
        double r587673 = -r587659;
        double r587674 = exp(r587673);
        double r587675 = r587672 + r587674;
        double r587676 = r587675 * r587661;
        double r587677 = r587676 / r587664;
        double r587678 = 2.0;
        double r587679 = r587678 * r587659;
        double r587680 = r587677 / r587679;
        double r587681 = r587671 ? r587680 : r587665;
        return r587681;
}

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 (/ (* (cosh x) (/ y x)) z) < -8.544608144694988e+108 or 1.0852883722175549e+303 < (/ (* (cosh x) (/ y x)) z)

   1. Initial program 29.4

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

   3. Applied cosh-def 29.4

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

   4. Applied frac-times 29.4

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

   5. Applied associate-/l/ 11.0

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

   7. Applied associate-/r* 0.4

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

   if -8.544608144694988e+108 < (/ (* (cosh x) (/ y x)) z) < 1.0852883722175549e+303

   1. Initial program 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -8.54460814469498844 \cdot 10^{108} \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.08528837221755491 \cdot 10^{303}\right):\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

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))