Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.0 → 0.5

Time: 16.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -280311605560231821312:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \le 3.959780483839766565551118859398086167973 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r34062647 = x;
        double r34062648 = cosh(r34062647);
        double r34062649 = y;
        double r34062650 = r34062649 / r34062647;
        double r34062651 = r34062648 * r34062650;
        double r34062652 = z;
        double r34062653 = r34062651 / r34062652;
        return r34062653;
}

↓

double f(double x, double y, double z) {
        double r34062654 = y;
        double r34062655 = -2.8031160556023182e+20;
        bool r34062656 = r34062654 <= r34062655;
        double r34062657 = x;
        double r34062658 = cosh(r34062657);
        double r34062659 = z;
        double r34062660 = r34062654 / r34062659;
        double r34062661 = r34062660 / r34062657;
        double r34062662 = r34062658 * r34062661;
        double r34062663 = 3.9597804838397666e-53;
        bool r34062664 = r34062654 <= r34062663;
        double r34062665 = r34062658 * r34062654;
        double r34062666 = 1.0;
        double r34062667 = r34062666 / r34062657;
        double r34062668 = r34062665 * r34062667;
        double r34062669 = r34062668 / r34062659;
        double r34062670 = r34062664 ? r34062669 : r34062662;
        double r34062671 = r34062656 ? r34062662 : r34062670;
        return r34062671;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	8.0
Target	0.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if y < -2.8031160556023182e+20 or 3.9597804838397666e-53 < y

   1. Initial program 19.6

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

   3. Applied associate-*r/ 19.6

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

   4. Applied associate-/l/ 0.7

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

   6. Applied associate-/l* 0.6

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

   8. Applied div-inv 0.6

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

   9. Simplified 0.6

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

   if -2.8031160556023182e+20 < y < 3.9597804838397666e-53

   1. Initial program 0.4

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

   3. Applied div-inv 0.5

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

   4. Applied associate-*r* 0.5

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -280311605560231821312:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \le 3.959780483839766565551118859398086167973 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :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))