Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.4

Time: 12.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -813868602.68375670909881591796875 \lor \neg \left(z \le 2.464261935710089975344020916546644554505 \cdot 10^{44}\right):\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{1}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{z \cdot \frac{x}{y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r362080 = x;
        double r362081 = cosh(r362080);
        double r362082 = y;
        double r362083 = r362082 / r362080;
        double r362084 = r362081 * r362083;
        double r362085 = z;
        double r362086 = r362084 / r362085;
        return r362086;
}

↓

double f(double x, double y, double z) {
        double r362087 = z;
        double r362088 = -813868602.6837567;
        bool r362089 = r362087 <= r362088;
        double r362090 = 2.46426193571009e+44;
        bool r362091 = r362087 <= r362090;
        double r362092 = !r362091;
        bool r362093 = r362089 || r362092;
        double r362094 = x;
        double r362095 = cosh(r362094);
        double r362096 = y;
        double r362097 = r362095 * r362096;
        double r362098 = 1.0;
        double r362099 = r362087 * r362094;
        double r362100 = r362098 / r362099;
        double r362101 = r362097 * r362100;
        double r362102 = 0.5;
        double r362103 = -r362094;
        double r362104 = exp(r362103);
        double r362105 = exp(r362094);
        double r362106 = r362104 + r362105;
        double r362107 = r362102 * r362106;
        double r362108 = r362094 / r362096;
        double r362109 = r362087 * r362108;
        double r362110 = r362107 / r362109;
        double r362111 = r362093 ? r362101 : r362110;
        return r362111;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
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 < -813868602.6837567 or 2.46426193571009e+44 < z

   1. Initial program 13.0

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

   3. Applied associate-*r/ 13.0

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

   4. Applied associate-/l/ 0.3

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

   6. Applied div-inv 0.4

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

   if -813868602.6837567 < z < 2.46426193571009e+44

   1. Initial program 0.5

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

   2. Taylor expanded around inf 17.1

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

   3. Simplified 17.1

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

   5. Applied *-un-lft-identity 17.1

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

   6. Applied times-frac 0.5

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

   7. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -813868602.68375670909881591796875 \lor \neg \left(z \le 2.464261935710089975344020916546644554505 \cdot 10^{44}\right):\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{1}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{z \cdot \frac{x}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))