Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.4

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -52672591011673712 \lor \neg \left(z \le 1.05182668461472934 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r502764 = x;
        double r502765 = cosh(r502764);
        double r502766 = y;
        double r502767 = r502766 / r502764;
        double r502768 = r502765 * r502767;
        double r502769 = z;
        double r502770 = r502768 / r502769;
        return r502770;
}

↓

double f(double x, double y, double z) {
        double r502771 = z;
        double r502772 = -5.267259101167371e+16;
        bool r502773 = r502771 <= r502772;
        double r502774 = 1.0518266846147293e-55;
        bool r502775 = r502771 <= r502774;
        double r502776 = !r502775;
        bool r502777 = r502773 || r502776;
        double r502778 = y;
        double r502779 = x;
        double r502780 = exp(r502779);
        double r502781 = 0.5;
        double r502782 = r502781 / r502780;
        double r502783 = fma(r502780, r502781, r502782);
        double r502784 = r502778 * r502783;
        double r502785 = r502771 * r502779;
        double r502786 = r502784 / r502785;
        double r502787 = cosh(r502779);
        double r502788 = r502778 / r502779;
        double r502789 = r502787 * r502788;
        double r502790 = 1.0;
        double r502791 = r502790 / r502771;
        double r502792 = r502789 * r502791;
        double r502793 = r502777 ? r502786 : r502792;
        return r502793;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	0.4
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -5.267259101167371e+16 or 1.0518266846147293e-55 < z

   1. Initial program 11.1

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

   2. Taylor expanded around inf 0.5

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

   3. Simplified 10.0

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}\]
   4. Using strategy rm

   5. Applied frac-times 0.5

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

   if -5.267259101167371e+16 < z < 1.0518266846147293e-55

   1. Initial program 0.3

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

   3. Applied div-inv 0.4

      \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -52672591011673712 \lor \neg \left(z \le 1.05182668461472934 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{y \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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))