tanhf (example 3.4)

Average Error: 29.5 → 0.6

Time: 7.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.01102096909057656733132990467538547818549:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 7.852408454565184712722891724467899621231 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \end{array}\]

C

double f(double x) {
        double r44216 = 1.0;
        double r44217 = x;
        double r44218 = cos(r44217);
        double r44219 = r44216 - r44218;
        double r44220 = sin(r44217);
        double r44221 = r44219 / r44220;
        return r44221;
}

↓

double f(double x) {
        double r44222 = 1.0;
        double r44223 = x;
        double r44224 = cos(r44223);
        double r44225 = r44222 - r44224;
        double r44226 = sin(r44223);
        double r44227 = r44225 / r44226;
        double r44228 = -0.011020969090576567;
        bool r44229 = r44227 <= r44228;
        double r44230 = exp(r44227);
        double r44231 = sqrt(r44230);
        double r44232 = log(r44231);
        double r44233 = r44232 + r44232;
        double r44234 = 7.852408454565185e-05;
        bool r44235 = r44227 <= r44234;
        double r44236 = 0.041666666666666664;
        double r44237 = 3.0;
        double r44238 = pow(r44223, r44237);
        double r44239 = r44236 * r44238;
        double r44240 = 0.004166666666666667;
        double r44241 = 5.0;
        double r44242 = pow(r44223, r44241);
        double r44243 = r44240 * r44242;
        double r44244 = 0.5;
        double r44245 = r44244 * r44223;
        double r44246 = r44243 + r44245;
        double r44247 = r44239 + r44246;
        double r44248 = pow(r44222, r44237);
        double r44249 = pow(r44224, r44237);
        double r44250 = r44248 - r44249;
        double r44251 = 2.0;
        double r44252 = pow(r44224, r44251);
        double r44253 = r44222 * r44222;
        double r44254 = r44252 - r44253;
        double r44255 = r44224 - r44222;
        double r44256 = r44254 / r44255;
        double r44257 = r44224 * r44256;
        double r44258 = r44257 + r44253;
        double r44259 = r44258 * r44226;
        double r44260 = r44250 / r44259;
        double r44261 = r44235 ? r44247 : r44260;
        double r44262 = r44229 ? r44233 : r44261;
        return r44262;
}

Error

Bits error versus x

Target

Original	29.5
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)) < -0.011020969090576567

   1. Initial program 0.8

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

   3. Applied add-log-exp 0.9

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

   5. Applied add-sqr-sqrt 1.1

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

   6. Applied log-prod 1.1

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

   if -0.011020969090576567 < (/ (- 1.0 (cos x)) (sin x)) < 7.852408454565185e-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 7.852408454565185e-05 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 0.9

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

   3. Applied flip3-- 1.0

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

   4. Applied associate-/l/ 1.0

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

   5. Simplified 1.0

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

   7. Applied flip-+ 1.0

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

   8. Simplified 1.0

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.01102096909057656733132990467538547818549:\\ \;\;\;\;\log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right) + \log \left(\sqrt{e^{\frac{1 - \cos x}{\sin x}}}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 7.852408454565184712722891724467899621231 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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