tanhf (example 3.4)

Average Error: 30.4 → 0.6

Time: 6.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.004893422048095925:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{1 - \cos x}{\sin x}}\right)\right)\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 2.98402005876572314 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r62663 = 1.0;
        double r62664 = x;
        double r62665 = cos(r62664);
        double r62666 = r62663 - r62665;
        double r62667 = sin(r62664);
        double r62668 = r62666 / r62667;
        return r62668;
}

↓

double f(double x) {
        double r62669 = 1.0;
        double r62670 = x;
        double r62671 = cos(r62670);
        double r62672 = r62669 - r62671;
        double r62673 = sin(r62670);
        double r62674 = r62672 / r62673;
        double r62675 = -0.0048934220480959245;
        bool r62676 = r62674 <= r62675;
        double r62677 = exp(r62674);
        double r62678 = log1p(r62677);
        double r62679 = expm1(r62678);
        double r62680 = log(r62679);
        double r62681 = 2.984020058765723e-05;
        bool r62682 = r62674 <= r62681;
        double r62683 = 0.041666666666666664;
        double r62684 = 3.0;
        double r62685 = pow(r62670, r62684);
        double r62686 = 0.004166666666666667;
        double r62687 = 5.0;
        double r62688 = pow(r62670, r62687);
        double r62689 = 0.5;
        double r62690 = r62689 * r62670;
        double r62691 = fma(r62686, r62688, r62690);
        double r62692 = fma(r62683, r62685, r62691);
        double r62693 = expm1(r62674);
        double r62694 = log1p(r62693);
        double r62695 = r62682 ? r62692 : r62694;
        double r62696 = r62676 ? r62680 : r62695;
        return r62696;
}

Error

Bits error versus x

Target

Original	30.4
Target	0.0
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0048934220480959245

   1. Initial program 0.9

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

   3. Applied add-log-exp 1.0

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

   5. Applied expm1-log1p-u 1.1

      \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{1 - \cos x}{\sin x}}\right)\right)\right)}\]

   if -0.0048934220480959245 < (/ (- 1.0 (cos x)) (sin x)) < 2.984020058765723e-05

   1. Initial program 59.9

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

   2. Taylor expanded around 0 0.1

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

   3. Simplified 0.1

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

   if 2.984020058765723e-05 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.1

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

   3. Applied log1p-expm1-u 1.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.004893422048095925:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{1 - \cos x}{\sin x}}\right)\right)\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 2.98402005876572314 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \cos x}{\sin x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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