tanhf (example 3.4)

Average Error: 29.8 → 0.7

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0118953516307934663:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 2.2776287241343477 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\ \end{array}\]

C

double f(double x) {
        double r54660 = 1.0;
        double r54661 = x;
        double r54662 = cos(r54661);
        double r54663 = r54660 - r54662;
        double r54664 = sin(r54661);
        double r54665 = r54663 / r54664;
        return r54665;
}

↓

double f(double x) {
        double r54666 = 1.0;
        double r54667 = x;
        double r54668 = cos(r54667);
        double r54669 = r54666 - r54668;
        double r54670 = sin(r54667);
        double r54671 = r54669 / r54670;
        double r54672 = -0.011895351630793466;
        bool r54673 = r54671 <= r54672;
        double r54674 = exp(r54671);
        double r54675 = log(r54674);
        double r54676 = 2.2776287241343477e-05;
        bool r54677 = r54671 <= r54676;
        double r54678 = 0.041666666666666664;
        double r54679 = 3.0;
        double r54680 = pow(r54667, r54679);
        double r54681 = r54678 * r54680;
        double r54682 = 0.004166666666666667;
        double r54683 = 5.0;
        double r54684 = pow(r54667, r54683);
        double r54685 = r54682 * r54684;
        double r54686 = 0.5;
        double r54687 = r54686 * r54667;
        double r54688 = r54685 + r54687;
        double r54689 = r54681 + r54688;
        double r54690 = pow(r54666, r54679);
        double r54691 = pow(r54668, r54679);
        double r54692 = r54690 - r54691;
        double r54693 = exp(r54692);
        double r54694 = log(r54693);
        double r54695 = 2.0;
        double r54696 = pow(r54668, r54695);
        double r54697 = r54666 * r54666;
        double r54698 = r54696 - r54697;
        double r54699 = r54668 - r54666;
        double r54700 = r54698 / r54699;
        double r54701 = r54668 * r54700;
        double r54702 = r54701 + r54697;
        double r54703 = r54694 / r54702;
        double r54704 = r54703 / r54670;
        double r54705 = r54677 ? r54689 : r54704;
        double r54706 = r54673 ? r54675 : r54705;
        return r54706;
}

Error

Bits error versus x

Target

Original	29.8
Target	0.0
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.011895351630793466

   1. Initial program 0.8

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

   3. Applied add-log-exp 1.0

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

   if -0.011895351630793466 < (/ (- 1.0 (cos x)) (sin x)) < 2.2776287241343477e-05

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.2

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

   if 2.2776287241343477e-05 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.1

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

   3. Applied flip3-- 1.2

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

   4. Simplified 1.2

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

   6. Applied add-log-exp 1.2

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

   7. Applied add-log-exp 1.2

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

   8. Applied diff-log 1.3

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

   9. Simplified 1.2

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

   11. Applied flip-+ 1.2

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

   12. Simplified 1.2

       \[\leadsto \frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{\color{blue}{{\left(\cos x\right)}^{2} - 1 \cdot 1}}{\cos x - 1} + 1 \cdot 1}}{\sin x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0118953516307934663:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 2.2776287241343477 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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