Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.8 → 0.4

Time: 14.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -45025619.10583603382110595703125 \lor \neg \left(y \le 3.23545020010918295138184517262131343092 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \cosh x}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r397598 = x;
        double r397599 = cosh(r397598);
        double r397600 = y;
        double r397601 = r397600 / r397598;
        double r397602 = r397599 * r397601;
        double r397603 = z;
        double r397604 = r397602 / r397603;
        return r397604;
}

↓

double f(double x, double y, double z) {
        double r397605 = y;
        double r397606 = -45025619.105836034;
        bool r397607 = r397605 <= r397606;
        double r397608 = 3.235450200109183e-53;
        bool r397609 = r397605 <= r397608;
        double r397610 = !r397609;
        bool r397611 = r397607 || r397610;
        double r397612 = 0.5;
        double r397613 = x;
        double r397614 = exp(r397613);
        double r397615 = -r397613;
        double r397616 = exp(r397615);
        double r397617 = r397614 + r397616;
        double r397618 = r397612 * r397617;
        double r397619 = r397605 * r397618;
        double r397620 = z;
        double r397621 = r397620 * r397613;
        double r397622 = r397619 / r397621;
        double r397623 = r397605 / r397613;
        double r397624 = cosh(r397613);
        double r397625 = r397623 * r397624;
        double r397626 = r397625 / r397620;
        double r397627 = r397611 ? r397622 : r397626;
        return r397627;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.8
Target	0.5
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -45025619.105836034 or 3.235450200109183e-53 < y

   1. Initial program 19.4

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

   2. Taylor expanded around inf 0.6

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

   3. Simplified 0.6

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

   if -45025619.105836034 < y < 3.235450200109183e-53

   1. Initial program 0.3

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

   3. Applied *-commutative 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -45025619.10583603382110595703125 \lor \neg \left(y \le 3.23545020010918295138184517262131343092 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \cosh x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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))