tanhf (example 3.4)

Average Error: 30.4 → 0.6

Time: 7.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001308568773005096318520767972870544326724 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 4.957178312169810962668929968898134941213 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \end{array}\]

C

double f(double x) {
        double r63706 = 1.0;
        double r63707 = x;
        double r63708 = cos(r63707);
        double r63709 = r63706 - r63708;
        double r63710 = sin(r63707);
        double r63711 = r63709 / r63710;
        return r63711;
}

↓

double f(double x) {
        double r63712 = 1.0;
        double r63713 = x;
        double r63714 = cos(r63713);
        double r63715 = r63712 - r63714;
        double r63716 = sin(r63713);
        double r63717 = r63715 / r63716;
        double r63718 = -0.0013085687730050963;
        bool r63719 = r63717 <= r63718;
        double r63720 = 4.957178312169811e-08;
        bool r63721 = r63717 <= r63720;
        double r63722 = !r63721;
        bool r63723 = r63719 || r63722;
        double r63724 = exp(r63717);
        double r63725 = log(r63724);
        double r63726 = 0.041666666666666664;
        double r63727 = 3.0;
        double r63728 = pow(r63713, r63727);
        double r63729 = 0.004166666666666667;
        double r63730 = 5.0;
        double r63731 = pow(r63713, r63730);
        double r63732 = 0.5;
        double r63733 = r63732 * r63713;
        double r63734 = fma(r63729, r63731, r63733);
        double r63735 = fma(r63726, r63728, r63734);
        double r63736 = r63723 ? r63725 : r63735;
        return r63736;
}

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 2 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0013085687730050963 or 4.957178312169811e-08 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.1

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

   3. Applied add-log-exp 1.2

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

   if -0.0013085687730050963 < (/ (- 1.0 (cos x)) (sin x)) < 4.957178312169811e-08

   1. Initial program 60.2

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified 0.0

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.001308568773005096318520767972870544326724 \lor \neg \left(\frac{1 - \cos x}{\sin x} \le 4.957178312169810962668929968898134941213 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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)))