tanhf (example 3.4)

Average Error: 30.4 → 0.6

Time: 6.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.004893422048095925:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 2.98402005876572314 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{\sin x}{1 - \cos x}\right)}^{1}}\\ \end{array}\]

C

double f(double x) {
        double r37415 = 1.0;
        double r37416 = x;
        double r37417 = cos(r37416);
        double r37418 = r37415 - r37417;
        double r37419 = sin(r37416);
        double r37420 = r37418 / r37419;
        return r37420;
}

↓

double f(double x) {
        double r37421 = 1.0;
        double r37422 = x;
        double r37423 = cos(r37422);
        double r37424 = r37421 - r37423;
        double r37425 = sin(r37422);
        double r37426 = r37424 / r37425;
        double r37427 = -0.0048934220480959245;
        bool r37428 = r37426 <= r37427;
        double r37429 = exp(r37426);
        double r37430 = log(r37429);
        double r37431 = 2.984020058765723e-05;
        bool r37432 = r37426 <= r37431;
        double r37433 = 0.041666666666666664;
        double r37434 = 3.0;
        double r37435 = pow(r37422, r37434);
        double r37436 = r37433 * r37435;
        double r37437 = 0.004166666666666667;
        double r37438 = 5.0;
        double r37439 = pow(r37422, r37438);
        double r37440 = r37437 * r37439;
        double r37441 = 0.5;
        double r37442 = r37441 * r37422;
        double r37443 = r37440 + r37442;
        double r37444 = r37436 + r37443;
        double r37445 = 1.0;
        double r37446 = r37425 / r37424;
        double r37447 = pow(r37446, r37445);
        double r37448 = r37445 / r37447;
        double r37449 = r37432 ? r37444 : r37448;
        double r37450 = r37428 ? r37430 : r37449;
        return r37450;
}

Error

Bits error versus x

Target

Original	30.4
Target	0.0
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0048934220480959245

   1. Initial program 0.9

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

   3. Applied add-log-exp 1.0

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

   if -0.0048934220480959245 < (/ (- 1.0 (cos x)) (sin x)) < 2.984020058765723e-05

   1. Initial program 59.9

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

   2. Taylor expanded around 0 0.1

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

   if 2.984020058765723e-05 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.1

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

   3. Applied clear-num 1.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
   4. Using strategy rm

   5. Applied pow1 1.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.004893422048095925:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 2.98402005876572314 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{\sin x}{1 - \cos x}\right)}^{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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