tanhf (example 3.4)

Average Error: 30.0 → 0.5

Time: 23.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02260320965798558526249628641835442977026:\\ \;\;\;\;\frac{\frac{1 \cdot \left(1 \cdot 1\right) - \left(\cos \left(2 \cdot x\right) \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot \cos x}{\sqrt[3]{\left(\left(1 + \cos x\right) \cdot \cos x\right) \cdot \left(\left(\left(1 + \cos x\right) \cdot \cos x\right) \cdot \left(\left(1 + \cos x\right) \cdot \cos x\right)\right)} + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;x \le 0.02069943920891310559784237455005495576188:\\ \;\;\;\;{x}^{5} \cdot \frac{1}{240} + x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1 \cdot \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) - \left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)\right)}{\left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right) + \left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)}}{\left(\cos x \cdot \cos x + \cos x \cdot 1\right) + 1 \cdot 1}}{\sin x}\\ \end{array}\]

C

double f(double x) {
        double r4140191 = 1.0;
        double r4140192 = x;
        double r4140193 = cos(r4140192);
        double r4140194 = r4140191 - r4140193;
        double r4140195 = sin(r4140192);
        double r4140196 = r4140194 / r4140195;
        return r4140196;
}

↓

double f(double x) {
        double r4140197 = x;
        double r4140198 = -0.022603209657985585;
        bool r4140199 = r4140197 <= r4140198;
        double r4140200 = 1.0;
        double r4140201 = r4140200 * r4140200;
        double r4140202 = r4140200 * r4140201;
        double r4140203 = 2.0;
        double r4140204 = r4140203 * r4140197;
        double r4140205 = cos(r4140204);
        double r4140206 = 0.5;
        double r4140207 = r4140205 * r4140206;
        double r4140208 = r4140207 + r4140206;
        double r4140209 = cos(r4140197);
        double r4140210 = r4140208 * r4140209;
        double r4140211 = r4140202 - r4140210;
        double r4140212 = r4140200 + r4140209;
        double r4140213 = r4140212 * r4140209;
        double r4140214 = r4140213 * r4140213;
        double r4140215 = r4140213 * r4140214;
        double r4140216 = cbrt(r4140215);
        double r4140217 = r4140216 + r4140201;
        double r4140218 = r4140211 / r4140217;
        double r4140219 = sin(r4140197);
        double r4140220 = r4140218 / r4140219;
        double r4140221 = 0.020699439208913106;
        bool r4140222 = r4140197 <= r4140221;
        double r4140223 = 5.0;
        double r4140224 = pow(r4140197, r4140223);
        double r4140225 = 0.004166666666666667;
        double r4140226 = r4140224 * r4140225;
        double r4140227 = 0.041666666666666664;
        double r4140228 = r4140197 * r4140197;
        double r4140229 = r4140227 * r4140228;
        double r4140230 = r4140206 + r4140229;
        double r4140231 = r4140197 * r4140230;
        double r4140232 = r4140226 + r4140231;
        double r4140233 = r4140201 * r4140201;
        double r4140234 = r4140233 * r4140233;
        double r4140235 = r4140200 * r4140234;
        double r4140236 = r4140197 + r4140197;
        double r4140237 = cos(r4140236);
        double r4140238 = r4140237 * r4140206;
        double r4140239 = r4140206 + r4140238;
        double r4140240 = r4140239 * r4140209;
        double r4140241 = r4140240 * r4140240;
        double r4140242 = r4140240 * r4140241;
        double r4140243 = r4140235 - r4140242;
        double r4140244 = r4140202 + r4140240;
        double r4140245 = r4140202 * r4140244;
        double r4140246 = r4140241 + r4140245;
        double r4140247 = r4140243 / r4140246;
        double r4140248 = r4140209 * r4140209;
        double r4140249 = r4140209 * r4140200;
        double r4140250 = r4140248 + r4140249;
        double r4140251 = r4140250 + r4140201;
        double r4140252 = r4140247 / r4140251;
        double r4140253 = r4140252 / r4140219;
        double r4140254 = r4140222 ? r4140232 : r4140253;
        double r4140255 = r4140199 ? r4140220 : r4140254;
        return r4140255;
}

Error

Bits error versus x

Target

Original	30.0
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if x < -0.022603209657985585

   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. Simplified 1.0

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

   6. Applied sqr-cos 1.0

      \[\leadsto \frac{\frac{1 \cdot \left(1 \cdot 1\right) - \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)} \cdot \cos x}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube 1.0

      \[\leadsto \frac{\frac{1 \cdot \left(1 \cdot 1\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x}{1 \cdot 1 + \color{blue}{\sqrt[3]{\left(\left(\cos x \cdot \cos x + 1 \cdot \cos x\right) \cdot \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right) \cdot \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}}{\sin x}\]

   9. Simplified 1.0

      \[\leadsto \frac{\frac{1 \cdot \left(1 \cdot 1\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x}{1 \cdot 1 + \sqrt[3]{\color{blue}{\left(\left(\left(\cos x + 1\right) \cdot \cos x\right) \cdot \left(\left(\cos x + 1\right) \cdot \cos x\right)\right) \cdot \left(\left(\cos x + 1\right) \cdot \cos x\right)}}}}{\sin x}\]

   if -0.022603209657985585 < x < 0.020699439208913106

   1. Initial program 59.9

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified 0.0

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

   if 0.020699439208913106 < 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. Simplified 1.0

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

   6. Applied sqr-cos 1.0

      \[\leadsto \frac{\frac{1 \cdot \left(1 \cdot 1\right) - \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)} \cdot \cos x}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\]
   7. Using strategy rm

   8. Applied flip3-- 1.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} - {\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x\right)}^{3}}{\left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right) + \left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x\right) \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x\right)\right)}}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\]

   9. Simplified 1.1

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) \cdot 1 - \left(\cos x \cdot \left(\frac{1}{2} \cdot \cos \left(x + x\right) + \frac{1}{2}\right)\right) \cdot \left(\left(\cos x \cdot \left(\frac{1}{2} \cdot \cos \left(x + x\right) + \frac{1}{2}\right)\right) \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \cos \left(x + x\right) + \frac{1}{2}\right)\right)\right)}}{\left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right)\right) + \left(\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x\right) \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right) \cdot \cos x\right)\right)}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}{\sin x}\]

   10. Simplified 1.1

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02260320965798558526249628641835442977026:\\ \;\;\;\;\frac{\frac{1 \cdot \left(1 \cdot 1\right) - \left(\cos \left(2 \cdot x\right) \cdot \frac{1}{2} + \frac{1}{2}\right) \cdot \cos x}{\sqrt[3]{\left(\left(1 + \cos x\right) \cdot \cos x\right) \cdot \left(\left(\left(1 + \cos x\right) \cdot \cos x\right) \cdot \left(\left(1 + \cos x\right) \cdot \cos x\right)\right)} + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;x \le 0.02069943920891310559784237455005495576188:\\ \;\;\;\;{x}^{5} \cdot \frac{1}{240} + x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1 \cdot \left(\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right)\right)\right) - \left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)\right)}{\left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right) \cdot \left(\left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right) + \left(1 \cdot \left(1 \cdot 1\right)\right) \cdot \left(1 \cdot \left(1 \cdot 1\right) + \left(\frac{1}{2} + \cos \left(x + x\right) \cdot \frac{1}{2}\right) \cdot \cos x\right)}}{\left(\cos x \cdot \cos x + \cos x \cdot 1\right) + 1 \cdot 1}}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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