tanhf (example 3.4)

Average Error: 29.7 → 0.7

Time: 17.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0161525551537130101:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 8.0836609335251401 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x} + 1 \cdot 1}}{\sin x}\\ \end{array}\]

C

double f(double x) {
        double r45354 = 1.0;
        double r45355 = x;
        double r45356 = cos(r45355);
        double r45357 = r45354 - r45356;
        double r45358 = sin(r45355);
        double r45359 = r45357 / r45358;
        return r45359;
}

↓

double f(double x) {
        double r45360 = 1.0;
        double r45361 = x;
        double r45362 = cos(r45361);
        double r45363 = r45360 - r45362;
        double r45364 = sin(r45361);
        double r45365 = r45363 / r45364;
        double r45366 = -0.01615255515371301;
        bool r45367 = r45365 <= r45366;
        double r45368 = 3.0;
        double r45369 = pow(r45360, r45368);
        double r45370 = pow(r45362, r45368);
        double r45371 = r45369 - r45370;
        double r45372 = exp(r45371);
        double r45373 = log(r45372);
        double r45374 = r45360 + r45362;
        double r45375 = r45362 * r45374;
        double r45376 = r45360 * r45360;
        double r45377 = r45375 + r45376;
        double r45378 = r45373 / r45377;
        double r45379 = r45378 / r45364;
        double r45380 = 8.08366093352514e-05;
        bool r45381 = r45365 <= r45380;
        double r45382 = 0.041666666666666664;
        double r45383 = pow(r45361, r45368);
        double r45384 = r45382 * r45383;
        double r45385 = 0.004166666666666667;
        double r45386 = 5.0;
        double r45387 = pow(r45361, r45386);
        double r45388 = r45385 * r45387;
        double r45389 = 0.5;
        double r45390 = r45389 * r45361;
        double r45391 = r45388 + r45390;
        double r45392 = r45384 + r45391;
        double r45393 = 2.0;
        double r45394 = pow(r45362, r45393);
        double r45395 = r45376 - r45394;
        double r45396 = r45395 / r45363;
        double r45397 = r45362 * r45396;
        double r45398 = r45397 + r45376;
        double r45399 = r45371 / r45398;
        double r45400 = r45399 / r45364;
        double r45401 = r45381 ? r45392 : r45400;
        double r45402 = r45367 ? r45379 : r45401;
        return r45402;
}

Error

Bits error versus x

Target

Original	29.7
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)) < -0.01615255515371301

   1. Initial program 0.8

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

   3. Applied flip3-- 0.9

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

   4. Simplified 0.9

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

   6. Applied add-log-exp 1.0

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

   7. Applied add-log-exp 1.0

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

   8. Applied diff-log 1.0

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

   9. Simplified 0.9

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

   if -0.01615255515371301 < (/ (- 1.0 (cos x)) (sin x)) < 8.08366093352514e-05

   1. Initial program 59.7

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

   2. Taylor expanded around 0 0.4

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

   if 8.08366093352514e-05 < (/ (- 1.0 (cos x)) (sin x))

   1. Initial program 1.0

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

   3. Applied flip3-- 1.1

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

   4. Simplified 1.1

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

   6. Applied flip-+ 1.1

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

   7. Simplified 1.1

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0161525551537130101:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 8.0836609335251401 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \frac{1 \cdot 1 - {\left(\cos x\right)}^{2}}{1 - \cos x} + 1 \cdot 1}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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