tanhf (example 3.4)

Average Error: 29.9 → 0.5

Time: 26.9s

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 r61809 = 1.0;
        double r61810 = x;
        double r61811 = cos(r61810);
        double r61812 = r61809 - r61811;
        double r61813 = sin(r61810);
        double r61814 = r61812 / r61813;
        return r61814;
}

↓

double f(double x) {
        double r61815 = x;
        double r61816 = -0.022316068015986585;
        bool r61817 = r61815 <= r61816;
        double r61818 = 1.0;
        double r61819 = 3.0;
        double r61820 = pow(r61818, r61819);
        double r61821 = cos(r61815);
        double r61822 = pow(r61821, r61819);
        double r61823 = r61820 - r61822;
        double r61824 = log(r61823);
        double r61825 = exp(r61824);
        double r61826 = r61821 + r61818;
        double r61827 = r61821 * r61826;
        double r61828 = r61818 * r61818;
        double r61829 = r61827 + r61828;
        double r61830 = sin(r61815);
        double r61831 = r61829 * r61830;
        double r61832 = r61825 / r61831;
        double r61833 = 0.022196995022368538;
        bool r61834 = r61815 <= r61833;
        double r61835 = 0.041666666666666664;
        double r61836 = pow(r61815, r61819);
        double r61837 = r61835 * r61836;
        double r61838 = 0.004166666666666667;
        double r61839 = 5.0;
        double r61840 = pow(r61815, r61839);
        double r61841 = r61838 * r61840;
        double r61842 = 0.5;
        double r61843 = r61842 * r61815;
        double r61844 = r61841 + r61843;
        double r61845 = r61837 + r61844;
        double r61846 = exp(1.0);
        double r61847 = r61818 - r61821;
        double r61848 = log(r61847);
        double r61849 = pow(r61846, r61848);
        double r61850 = 1.0;
        double r61851 = r61850 / r61830;
        double r61852 = r61849 * r61851;
        double r61853 = r61834 ? r61845 : r61852;
        double r61854 = r61817 ? r61832 : r61853;
        return r61854;
}

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)))