tanhf (example 3.4)

Average Error: 30.5 → 0.9

Time: 25.5s

Precision: 64

Math

\[\frac{1 - \cos x}{\sin x}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.03628702050037945631144609137663792353123:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(1 + \cos x\right) \cdot \cos x}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.622109525504273900412314200758512328093 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}\right)}}{\sin x}\\ \end{array}\]

C

double f(double x) {
        double r55420 = 1.0;
        double r55421 = x;
        double r55422 = cos(r55421);
        double r55423 = r55420 - r55422;
        double r55424 = sin(r55421);
        double r55425 = r55423 / r55424;
        return r55425;
}

↓

double f(double x) {
        double r55426 = 1.0;
        double r55427 = x;
        double r55428 = cos(r55427);
        double r55429 = r55426 - r55428;
        double r55430 = sin(r55427);
        double r55431 = r55429 / r55430;
        double r55432 = -0.036287020500379456;
        bool r55433 = r55431 <= r55432;
        double r55434 = 3.0;
        double r55435 = pow(r55426, r55434);
        double r55436 = pow(r55428, r55434);
        double r55437 = r55435 - r55436;
        double r55438 = r55426 * r55426;
        double r55439 = r55426 + r55428;
        double r55440 = r55439 * r55428;
        double r55441 = r55438 + r55440;
        double r55442 = r55437 / r55441;
        double r55443 = r55442 / r55430;
        double r55444 = 1.622109525504274e-06;
        bool r55445 = r55431 <= r55444;
        double r55446 = 0.041666666666666664;
        double r55447 = pow(r55427, r55434);
        double r55448 = r55446 * r55447;
        double r55449 = 0.004166666666666667;
        double r55450 = 5.0;
        double r55451 = pow(r55427, r55450);
        double r55452 = r55449 * r55451;
        double r55453 = 0.5;
        double r55454 = r55453 * r55427;
        double r55455 = r55452 + r55454;
        double r55456 = r55448 + r55455;
        double r55457 = r55428 * r55428;
        double r55458 = r55438 - r55457;
        double r55459 = r55458 / r55439;
        double r55460 = log(r55459);
        double r55461 = exp(r55460);
        double r55462 = r55461 / r55430;
        double r55463 = r55445 ? r55456 : r55462;
        double r55464 = r55433 ? r55443 : r55463;
        return r55464;
}

Error

Bits error versus x

Target

Original	30.5
Target	0.0
Herbie	0.9

\[\tan \left(\frac{x}{2}\right)\]

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.036287020500379456

   1. Initial program 0.7

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied add-exp-log 0.7

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{\sin x}\]
   4. Using strategy rm

   5. Applied flip3-- 0.8

      \[\leadsto \frac{e^{\log \color{blue}{\left(\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\right)}}}{\sin x}\]

   6. Applied log-div 0.8

      \[\leadsto \frac{e^{\color{blue}{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}}{\sin x}\]

   7. Applied exp-diff 0.8

      \[\leadsto \frac{\color{blue}{\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{e^{\log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}}}{\sin x}\]

   8. Simplified 0.8

      \[\leadsto \frac{\frac{\color{blue}{{1}^{3} - {\left(\cos x\right)}^{3}}}{e^{\log \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}}{\sin x}\]

   9. Simplified 0.8

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(1 + \cos x\right) \cdot \cos x}}}{\sin x}\]

   if -0.036287020500379456 < (/ (- 1.0 (cos x)) (sin x)) < 1.622109525504274e-06

   1. Initial program 59.5

      \[\frac{1 - \cos x}{\sin x}\]

   2. Taylor expanded around 0 0.7

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

   if 1.622109525504274e-06 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.1

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied add-exp-log 1.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{\sin x}\]
   4. Using strategy rm

   5. Applied flip-- 1.6

      \[\leadsto \frac{e^{\log \color{blue}{\left(\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}\right)}}}{\sin x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.03628702050037945631144609137663792353123:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(1 + \cos x\right) \cdot \cos x}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 1.622109525504273900412314200758512328093 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}\right)}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))