Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 1.3

Time: 15.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 24.59938632289427218324817658867686986923:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r326331 = x;
        double r326332 = y;
        double r326333 = sin(r326332);
        double r326334 = r326333 / r326332;
        double r326335 = r326331 * r326334;
        double r326336 = z;
        double r326337 = r326335 / r326336;
        return r326337;
}

↓

double f(double x, double y, double z) {
        double r326338 = x;
        double r326339 = 24.599386322894272;
        bool r326340 = r326338 <= r326339;
        double r326341 = z;
        double r326342 = y;
        double r326343 = sin(r326342);
        double r326344 = r326343 / r326342;
        double r326345 = r326341 / r326344;
        double r326346 = r326338 / r326345;
        double r326347 = r326338 * r326343;
        double r326348 = r326347 / r326342;
        double r326349 = r326348 / r326341;
        double r326350 = r326340 ? r326346 : r326349;
        return r326350;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.7
Target	0.3
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < 24.599386322894272

   1. Initial program 3.4

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

   3. Applied clear-num 3.9

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot \frac{\sin y}{y}}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 3.9

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

   6. Applied times-frac 2.3

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

   7. Applied associate-/r* 1.7

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

   8. Simplified 1.6

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

   if 24.599386322894272 < x

   1. Initial program 0.2

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

   3. Applied pow1 0.2

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

   4. Applied pow1 0.2

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

   5. Applied pow-prod-down 0.2

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

   6. Simplified 0.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 24.59938632289427218324817658867686986923:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))