Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.0 → 0.4

Time: 22.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -40510205393232588192078396191825461248:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;y \le 7.039456669498942289348721752920549611426 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r11881180 = x;
        double r11881181 = cosh(r11881180);
        double r11881182 = y;
        double r11881183 = r11881182 / r11881180;
        double r11881184 = r11881181 * r11881183;
        double r11881185 = z;
        double r11881186 = r11881184 / r11881185;
        return r11881186;
}

↓

double f(double x, double y, double z) {
        double r11881187 = y;
        double r11881188 = -4.051020539323259e+37;
        bool r11881189 = r11881187 <= r11881188;
        double r11881190 = x;
        double r11881191 = cosh(r11881190);
        double r11881192 = r11881191 * r11881187;
        double r11881193 = z;
        double r11881194 = r11881190 * r11881193;
        double r11881195 = r11881192 / r11881194;
        double r11881196 = 7.039456669498942e-17;
        bool r11881197 = r11881187 <= r11881196;
        double r11881198 = r11881192 / r11881190;
        double r11881199 = r11881198 / r11881193;
        double r11881200 = r11881197 ? r11881199 : r11881195;
        double r11881201 = r11881189 ? r11881195 : r11881200;
        return r11881201;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	8.0
Target	0.4
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -4.051020539323259e+37 or 7.039456669498942e-17 < y

   1. Initial program 22.7

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

   3. Applied associate-*r/ 22.7

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

   4. Applied associate-/l/ 0.3

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

   if -4.051020539323259e+37 < y < 7.039456669498942e-17

   1. Initial program 0.4

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

   3. Applied div-inv 0.5

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

   4. Applied associate-*r* 0.5

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

   6. Applied un-div-inv 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -40510205393232588192078396191825461248:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;y \le 7.039456669498942289348721752920549611426 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))