tanhf (example 3.4)

Average Error: 31.0 → 0.5

Time: 21.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0220899078243634444662646387769200373441:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02040214104704790240574219239988451590762:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{1}{240}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24} + \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \end{array}\]

C

double f(double x) {
        double r2649231 = 1.0;
        double r2649232 = x;
        double r2649233 = cos(r2649232);
        double r2649234 = r2649231 - r2649233;
        double r2649235 = sin(r2649232);
        double r2649236 = r2649234 / r2649235;
        return r2649236;
}

↓

double f(double x) {
        double r2649237 = x;
        double r2649238 = -0.022089907824363444;
        bool r2649239 = r2649237 <= r2649238;
        double r2649240 = 1.0;
        double r2649241 = sin(r2649237);
        double r2649242 = 1.0;
        double r2649243 = cos(r2649237);
        double r2649244 = r2649242 - r2649243;
        double r2649245 = r2649241 / r2649244;
        double r2649246 = r2649240 / r2649245;
        double r2649247 = 0.020402141047047902;
        bool r2649248 = r2649237 <= r2649247;
        double r2649249 = 5.0;
        double r2649250 = pow(r2649237, r2649249);
        double r2649251 = 0.004166666666666667;
        double r2649252 = r2649237 * r2649237;
        double r2649253 = 0.041666666666666664;
        double r2649254 = r2649252 * r2649253;
        double r2649255 = 0.5;
        double r2649256 = r2649254 + r2649255;
        double r2649257 = r2649237 * r2649256;
        double r2649258 = fma(r2649250, r2649251, r2649257);
        double r2649259 = r2649248 ? r2649258 : r2649246;
        double r2649260 = r2649239 ? r2649246 : r2649259;
        return r2649260;
}

Error

Bits error versus x

Target

Original	31.0
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if x < -0.022089907824363444 or 0.020402141047047902 < x

   1. Initial program 0.9

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

   3. Applied clear-num 1.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]

   if -0.022089907824363444 < x < 0.020402141047047902

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, \frac{1}{240}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24} + \frac{1}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0220899078243634444662646387769200373441:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02040214104704790240574219239988451590762:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{1}{240}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24} + \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))