tanhf (example 3.4)

Average Error: 31.0 → 0.5

Time: 22.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2927191 = 1.0;
        double r2927192 = x;
        double r2927193 = cos(r2927192);
        double r2927194 = r2927191 - r2927193;
        double r2927195 = sin(r2927192);
        double r2927196 = r2927194 / r2927195;
        return r2927196;
}

↓

double f(double x) {
        double r2927197 = x;
        double r2927198 = -0.022089907824363444;
        bool r2927199 = r2927197 <= r2927198;
        double r2927200 = 1.0;
        double r2927201 = sin(r2927197);
        double r2927202 = 1.0;
        double r2927203 = cos(r2927197);
        double r2927204 = r2927202 - r2927203;
        double r2927205 = r2927201 / r2927204;
        double r2927206 = r2927200 / r2927205;
        double r2927207 = 0.020402141047047902;
        bool r2927208 = r2927197 <= r2927207;
        double r2927209 = r2927197 * r2927197;
        double r2927210 = 0.041666666666666664;
        double r2927211 = r2927209 * r2927210;
        double r2927212 = 0.5;
        double r2927213 = r2927211 + r2927212;
        double r2927214 = r2927197 * r2927213;
        double r2927215 = 5.0;
        double r2927216 = pow(r2927197, r2927215);
        double r2927217 = 0.004166666666666667;
        double r2927218 = r2927216 * r2927217;
        double r2927219 = r2927214 + r2927218;
        double r2927220 = r2927208 ? r2927219 : r2927206;
        double r2927221 = r2927199 ? r2927206 : r2927220;
        return r2927221;
}

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. Using strategy rm

   3. Applied add-exp-log 60.0

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

   4. Taylor expanded around 0 0.0

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

   5. Simplified 0.0

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

4. Final simplification 0.5

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

Reproduce

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

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

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