tanhf (example 3.4)

Average Error: 29.9 → 0.5

Time: 26.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02231606801598658498808624983666959451512:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02219699502236853819181483515876607270911:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\log \left(1 - \cos x\right)\right)} \cdot \frac{1}{\sin x}\\ \end{array}\]

C

double f(double x) {
        double r71120 = 1.0;
        double r71121 = x;
        double r71122 = cos(r71121);
        double r71123 = r71120 - r71122;
        double r71124 = sin(r71121);
        double r71125 = r71123 / r71124;
        return r71125;
}

↓

double f(double x) {
        double r71126 = x;
        double r71127 = -0.022316068015986585;
        bool r71128 = r71126 <= r71127;
        double r71129 = 1.0;
        double r71130 = 3.0;
        double r71131 = pow(r71129, r71130);
        double r71132 = cos(r71126);
        double r71133 = pow(r71132, r71130);
        double r71134 = r71131 - r71133;
        double r71135 = log(r71134);
        double r71136 = exp(r71135);
        double r71137 = r71132 + r71129;
        double r71138 = r71132 * r71137;
        double r71139 = r71129 * r71129;
        double r71140 = r71138 + r71139;
        double r71141 = sin(r71126);
        double r71142 = r71140 * r71141;
        double r71143 = r71136 / r71142;
        double r71144 = 0.022196995022368538;
        bool r71145 = r71126 <= r71144;
        double r71146 = 0.041666666666666664;
        double r71147 = pow(r71126, r71130);
        double r71148 = r71146 * r71147;
        double r71149 = 0.004166666666666667;
        double r71150 = 5.0;
        double r71151 = pow(r71126, r71150);
        double r71152 = r71149 * r71151;
        double r71153 = 0.5;
        double r71154 = r71153 * r71126;
        double r71155 = r71152 + r71154;
        double r71156 = r71148 + r71155;
        double r71157 = exp(1.0);
        double r71158 = r71129 - r71132;
        double r71159 = log(r71158);
        double r71160 = pow(r71157, r71159);
        double r71161 = 1.0;
        double r71162 = r71161 / r71141;
        double r71163 = r71160 * r71162;
        double r71164 = r71145 ? r71156 : r71163;
        double r71165 = r71128 ? r71143 : r71164;
        return r71165;
}

Error

Bits error versus x

Target

Original	29.9
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if x < -0.022316068015986585

   1. Initial program 0.9

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

   3. Applied add-exp-log 0.9

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

   5. Applied flip3-- 1.0

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

   6. Applied log-div 1.1

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

   7. Applied exp-diff 1.0

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

   8. Applied associate-/l/ 1.0

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

   9. Simplified 1.0

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

   if -0.022316068015986585 < x < 0.022196995022368538

   1. Initial program 59.9

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

   2. Taylor expanded around 0 0.0

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

   if 0.022196995022368538 < x

   1. Initial program 0.9

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

   3. Applied add-exp-log 0.9

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

   5. Applied pow1 0.9

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

   6. Applied log-pow 0.9

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

   7. Applied exp-prod 1.0

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

   8. Simplified 1.0

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

   10. Applied div-inv 1.0

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02231606801598658498808624983666959451512:\\ \;\;\;\;\frac{e^{\log \left({1}^{3} - {\left(\cos x\right)}^{3}\right)}}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02219699502236853819181483515876607270911:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\log \left(1 - \cos x\right)\right)} \cdot \frac{1}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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