Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 1.0

Time: 20.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.165545862005594721917182648161868440768 \cdot 10^{84} \lor \neg \left(y \le 1485.903697517288492235820740461349487305\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{y}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r331651 = x;
        double r331652 = cosh(r331651);
        double r331653 = y;
        double r331654 = r331653 / r331651;
        double r331655 = r331652 * r331654;
        double r331656 = z;
        double r331657 = r331655 / r331656;
        return r331657;
}

↓

double f(double x, double y, double z) {
        double r331658 = y;
        double r331659 = -3.1655458620055947e+84;
        bool r331660 = r331658 <= r331659;
        double r331661 = 1485.9036975172885;
        bool r331662 = r331658 <= r331661;
        double r331663 = !r331662;
        bool r331664 = r331660 || r331663;
        double r331665 = 0.5;
        double r331666 = x;
        double r331667 = r331666 * r331658;
        double r331668 = z;
        double r331669 = r331667 / r331668;
        double r331670 = r331666 * r331668;
        double r331671 = r331658 / r331670;
        double r331672 = fma(r331665, r331669, r331671);
        double r331673 = cosh(r331666);
        double r331674 = r331658 / r331666;
        double r331675 = r331673 * r331674;
        double r331676 = r331675 / r331668;
        double r331677 = r331664 ? r331672 : r331676;
        return r331677;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	0.4
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if y < -3.1655458620055947e+84 or 1485.9036975172885 < y

   1. Initial program 25.0

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

   2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right)}{x \cdot z}}\]

   3. Simplified 0.4

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{x \cdot z}}\]

   4. Taylor expanded around 0 1.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]

   5. Simplified 1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{y}{x \cdot z}\right)}\]

   if -3.1655458620055947e+84 < y < 1485.9036975172885

   1. Initial program 0.9

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.165545862005594721917182648161868440768 \cdot 10^{84} \lor \neg \left(y \le 1485.903697517288492235820740461349487305\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{y}{x \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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