Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.5 → 0.4

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.978520408058128474686070432017731650003 \cdot 10^{51} \lor \neg \left(y \le 3539127650907089920\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r407711 = x;
        double r407712 = cosh(r407711);
        double r407713 = y;
        double r407714 = r407713 / r407711;
        double r407715 = r407712 * r407714;
        double r407716 = z;
        double r407717 = r407715 / r407716;
        return r407717;
}

↓

double f(double x, double y, double z) {
        double r407718 = y;
        double r407719 = -1.9785204080581285e+51;
        bool r407720 = r407718 <= r407719;
        double r407721 = 3.53912765090709e+18;
        bool r407722 = r407718 <= r407721;
        double r407723 = !r407722;
        bool r407724 = r407720 || r407723;
        double r407725 = x;
        double r407726 = cosh(r407725);
        double r407727 = z;
        double r407728 = r407725 * r407727;
        double r407729 = r407718 / r407728;
        double r407730 = r407726 * r407729;
        double r407731 = sqrt(r407726);
        double r407732 = r407718 / r407725;
        double r407733 = r407731 * r407732;
        double r407734 = r407731 * r407733;
        double r407735 = r407734 / r407727;
        double r407736 = r407724 ? r407730 : r407735;
        return r407736;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.5
Target	0.4
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if y < -1.9785204080581285e+51 or 3.53912765090709e+18 < y

   1. Initial program 24.4

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

   3. Applied *-un-lft-identity 24.4

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

   4. Applied times-frac 24.4

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

   5. Simplified 24.4

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

   6. Simplified 0.3

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

   if -1.9785204080581285e+51 < y < 3.53912765090709e+18

   1. Initial program 0.5

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

   3. Applied add-sqr-sqrt 0.5

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

   4. Applied associate-*l* 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.978520408058128474686070432017731650003 \cdot 10^{51} \lor \neg \left(y \le 3539127650907089920\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))