tanhf (example 3.4)

Average Error: 30.3 → 1.0

Time: 19.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0391915361306189591328319465901586227119:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 5.238158504786694352056697354669267951977 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left({x}^{5}, \frac{1}{240}, x \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{1}{\sin x}}\right)}^{\left(1 - \cos x\right)}\right)\\ \end{array}\]

C

double f(double x) {
        double r37351 = 1.0;
        double r37352 = x;
        double r37353 = cos(r37352);
        double r37354 = r37351 - r37353;
        double r37355 = sin(r37352);
        double r37356 = r37354 / r37355;
        return r37356;
}

↓

double f(double x) {
        double r37357 = 1.0;
        double r37358 = x;
        double r37359 = cos(r37358);
        double r37360 = r37357 - r37359;
        double r37361 = sin(r37358);
        double r37362 = r37360 / r37361;
        double r37363 = -0.03919153613061896;
        bool r37364 = r37362 <= r37363;
        double r37365 = 1.0;
        double r37366 = r37357 + r37359;
        double r37367 = r37365 / r37366;
        double r37368 = r37366 * r37360;
        double r37369 = r37365 / r37361;
        double r37370 = r37368 * r37369;
        double r37371 = r37367 * r37370;
        double r37372 = 0.0005238158504786694;
        bool r37373 = r37362 <= r37372;
        double r37374 = 0.041666666666666664;
        double r37375 = 3.0;
        double r37376 = pow(r37358, r37375);
        double r37377 = 5.0;
        double r37378 = pow(r37358, r37377);
        double r37379 = 0.004166666666666667;
        double r37380 = 0.5;
        double r37381 = r37358 * r37380;
        double r37382 = fma(r37378, r37379, r37381);
        double r37383 = fma(r37374, r37376, r37382);
        double r37384 = exp(r37369);
        double r37385 = pow(r37384, r37360);
        double r37386 = log(r37385);
        double r37387 = r37373 ? r37383 : r37386;
        double r37388 = r37364 ? r37371 : r37387;
        return r37388;
}

Error

Bits error versus x

Target

Original	30.3
Target	0.0
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if (/ (- 1.0 (cos x)) (sin x)) < -0.03919153613061896

   1. Initial program 0.7

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

   3. Applied clear-num 0.8

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
   4. Using strategy rm

   5. Applied flip-- 1.2

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}}\]

   6. Applied associate-/r/ 1.3

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

   7. Applied add-cube-cbrt 1.3

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

   8. Applied times-frac 1.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x}} \cdot \frac{\sqrt[3]{1}}{1 + \cos x}}\]

   9. Simplified 0.9

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

   10. Simplified 0.9

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

   if -0.03919153613061896 < (/ (- 1.0 (cos x)) (sin x)) < 0.0005238158504786694

   1. Initial program 59.2

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

   2. Taylor expanded around 0 0.9

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

   3. Simplified 0.9

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

   if 0.0005238158504786694 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 0.9

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

   3. Applied clear-num 1.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
   4. Using strategy rm

   5. Applied add-log-exp 1.0

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

   6. Simplified 1.1

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0391915361306189591328319465901586227119:\\ \;\;\;\;\frac{1}{1 + \cos x} \cdot \left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right)\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 5.238158504786694352056697354669267951977 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left({x}^{5}, \frac{1}{240}, x \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{1}{\sin x}}\right)}^{\left(1 - \cos x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))