Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.5 → 0.4

Time: 17.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.378113244966585428960316749794424299186 \cdot 10^{-37} \lor \neg \left(z \le 180493624317508243685376\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r334933 = x;
        double r334934 = cosh(r334933);
        double r334935 = y;
        double r334936 = r334935 / r334933;
        double r334937 = r334934 * r334936;
        double r334938 = z;
        double r334939 = r334937 / r334938;
        return r334939;
}

↓

double f(double x, double y, double z) {
        double r334940 = z;
        double r334941 = -3.3781132449665854e-37;
        bool r334942 = r334940 <= r334941;
        double r334943 = 1.8049362431750824e+23;
        bool r334944 = r334940 <= r334943;
        double r334945 = !r334944;
        bool r334946 = r334942 || r334945;
        double r334947 = x;
        double r334948 = cosh(r334947);
        double r334949 = y;
        double r334950 = r334948 * r334949;
        double r334951 = r334947 * r334940;
        double r334952 = r334950 / r334951;
        double r334953 = r334948 / r334947;
        double r334954 = r334949 / r334940;
        double r334955 = r334953 * r334954;
        double r334956 = r334946 ? r334952 : r334955;
        return r334956;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.5
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 z < -3.3781132449665854e-37 or 1.8049362431750824e+23 < z

   1. Initial program 11.3

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

   3. Applied associate-*r/ 11.3

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

   4. Applied associate-/l/ 0.4

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

   5. Simplified 0.4

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

   if -3.3781132449665854e-37 < z < 1.8049362431750824e+23

   1. Initial program 0.4

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

   3. Applied associate-*r/ 0.4

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

   4. Applied associate-/l/ 19.6

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

   5. Simplified 19.6

      \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{x \cdot z}}\]
   6. Using strategy rm

   7. Applied times-frac 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.378113244966585428960316749794424299186 \cdot 10^{-37} \lor \neg \left(z \le 180493624317508243685376\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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