Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.5

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.35750180424885101 \cdot 10^{-17} \lor \neg \left(y \le 1.2052909119810524 \cdot 10^{57}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r416335 = x;
        double r416336 = cosh(r416335);
        double r416337 = y;
        double r416338 = r416337 / r416335;
        double r416339 = r416336 * r416338;
        double r416340 = z;
        double r416341 = r416339 / r416340;
        return r416341;
}

↓

double f(double x, double y, double z) {
        double r416342 = y;
        double r416343 = -1.357501804248851e-17;
        bool r416344 = r416342 <= r416343;
        double r416345 = 1.2052909119810524e+57;
        bool r416346 = r416342 <= r416345;
        double r416347 = !r416346;
        bool r416348 = r416344 || r416347;
        double r416349 = x;
        double r416350 = cosh(r416349);
        double r416351 = z;
        double r416352 = r416349 * r416351;
        double r416353 = r416342 / r416352;
        double r416354 = r416350 * r416353;
        double r416355 = r416342 / r416349;
        double r416356 = r416355 / r416351;
        double r416357 = r416350 * r416356;
        double r416358 = r416348 ? r416354 : r416357;
        return r416358;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	0.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if y < -1.357501804248851e-17 or 1.2052909119810524e+57 < y

   1. Initial program 22.2

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

   3. Applied *-un-lft-identity 22.2

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

   4. Applied times-frac 22.1

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

   5. Simplified 22.1

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

   6. Simplified 0.3

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

   if -1.357501804248851e-17 < y < 1.2052909119810524e+57

   1. Initial program 0.6

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

   3. Applied *-un-lft-identity 0.6

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

   4. Applied times-frac 0.6

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

   5. Simplified 0.6

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

   6. Simplified 10.3

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

   8. Applied associate-/r* 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.35750180424885101 \cdot 10^{-17} \lor \neg \left(y \le 1.2052909119810524 \cdot 10^{57}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))