Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.4

Time: 19.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -184150434362061826180164289036288 \lor \neg \left(y \le 3325101390191379852297270305226752\right):\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{x}}{z \cdot 2}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r321188 = x;
        double r321189 = cosh(r321188);
        double r321190 = y;
        double r321191 = r321190 / r321188;
        double r321192 = r321189 * r321191;
        double r321193 = z;
        double r321194 = r321192 / r321193;
        return r321194;
}

↓

double f(double x, double y, double z) {
        double r321195 = y;
        double r321196 = -1.8415043436206183e+32;
        bool r321197 = r321195 <= r321196;
        double r321198 = 3.32510139019138e+33;
        bool r321199 = r321195 <= r321198;
        double r321200 = !r321199;
        bool r321201 = r321197 || r321200;
        double r321202 = x;
        double r321203 = cosh(r321202);
        double r321204 = r321203 * r321195;
        double r321205 = z;
        double r321206 = r321204 / r321205;
        double r321207 = r321206 / r321202;
        double r321208 = exp(r321202);
        double r321209 = -r321202;
        double r321210 = exp(r321209);
        double r321211 = r321208 + r321210;
        double r321212 = r321195 / r321202;
        double r321213 = r321211 * r321212;
        double r321214 = 2.0;
        double r321215 = r321205 * r321214;
        double r321216 = r321213 / r321215;
        double r321217 = r321201 ? r321207 : r321216;
        return r321217;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	0.4
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if y < -1.8415043436206183e+32 or 3.32510139019138e+33 < y

   1. Initial program 24.5

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

   3. Applied associate-*r/ 24.5

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

   4. Applied associate-/l/ 0.4

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

   6. Applied associate-/r* 0.4

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

   if -1.8415043436206183e+32 < y < 3.32510139019138e+33

   1. Initial program 0.4

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

   3. Applied cosh-def 0.4

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

   4. Applied associate-*l/ 0.4

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

   5. Applied associate-/l/ 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -184150434362061826180164289036288 \lor \neg \left(y \le 3325101390191379852297270305226752\right):\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{x}}{z \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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))