tanhf (example 3.4)

Average Error: 30.3 → 0.8

Time: 7.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0350875129794391352:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.00153222520976908894:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r63384 = 1.0;
        double r63385 = x;
        double r63386 = cos(r63385);
        double r63387 = r63384 - r63386;
        double r63388 = sin(r63385);
        double r63389 = r63387 / r63388;
        return r63389;
}

↓

double f(double x) {
        double r63390 = 1.0;
        double r63391 = x;
        double r63392 = cos(r63391);
        double r63393 = r63390 - r63392;
        double r63394 = sin(r63391);
        double r63395 = r63393 / r63394;
        double r63396 = -0.035087512979439135;
        bool r63397 = r63395 <= r63396;
        double r63398 = 3.0;
        double r63399 = pow(r63390, r63398);
        double r63400 = pow(r63392, r63398);
        double r63401 = r63399 - r63400;
        double r63402 = r63390 + r63392;
        double r63403 = r63390 * r63390;
        double r63404 = fma(r63392, r63402, r63403);
        double r63405 = r63404 * r63394;
        double r63406 = r63401 / r63405;
        double r63407 = 0.001532225209769089;
        bool r63408 = r63395 <= r63407;
        double r63409 = 0.041666666666666664;
        double r63410 = pow(r63391, r63398);
        double r63411 = 0.004166666666666667;
        double r63412 = 5.0;
        double r63413 = pow(r63391, r63412);
        double r63414 = 0.5;
        double r63415 = r63414 * r63391;
        double r63416 = fma(r63411, r63413, r63415);
        double r63417 = fma(r63409, r63410, r63416);
        double r63418 = expm1(r63395);
        double r63419 = log1p(r63418);
        double r63420 = r63408 ? r63417 : r63419;
        double r63421 = r63397 ? r63406 : r63420;
        return r63421;
}

Error

Bits error versus x

Target

Original	30.3
Target	0
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.035087512979439135

   1. Initial program 0.7

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

   3. Applied clear-num 0.8

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

   5. Applied add-log-exp 1.0

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

   7. Applied associate-/r/ 1.0

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

   8. Applied exp-prod 1.0

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

   9. Applied log-pow 0.9

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

   10. Simplified 0.8

       \[\leadsto \left(1 - \cos x\right) \cdot \color{blue}{\frac{1}{\sin x}}\]
   11. Using strategy rm

   12. Applied flip3-- 0.9

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

   13. Applied frac-times 0.9

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

   14. Simplified 0.9

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

   15. Simplified 0.8

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

   if -0.035087512979439135 < (/ (- 1.0 (cos x)) (sin x)) < 0.001532225209769089

   1. Initial program 59.4

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

   2. Taylor expanded around 0 0.7

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

   3. Simplified 0.7

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

   if 0.001532225209769089 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.0

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

   3. Applied log1p-expm1-u 1.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0350875129794391352:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.00153222520976908894:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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