Average Error: 29.4 → 1.0
Time: 8.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0444958089144920146:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, 1, \cos x \cdot \left(\cos x + 1\right)\right) \cdot \left(1 - \cos x\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0033326395732868408:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0444958089144920146:\\
\;\;\;\;\frac{\mathsf{fma}\left(1, 1, \cos x \cdot \left(\cos x + 1\right)\right) \cdot \left(1 - \cos x\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0033326395732868408:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{1}^{3} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\

\end{array}
double f(double x) {
        double r78328 = 1.0;
        double r78329 = x;
        double r78330 = cos(r78329);
        double r78331 = r78328 - r78330;
        double r78332 = sin(r78329);
        double r78333 = r78331 / r78332;
        return r78333;
}
↓
double f(double x) {
        double r78334 = 1.0;
        double r78335 = x;
        double r78336 = cos(r78335);
        double r78337 = r78334 - r78336;
        double r78338 = sin(r78335);
        double r78339 = r78337 / r78338;
        double r78340 = -0.044495808914492015;
        bool r78341 = r78339 <= r78340;
        double r78342 = r78336 + r78334;
        double r78343 = r78336 * r78342;
        double r78344 = fma(r78334, r78334, r78343);
        double r78345 = r78344 * r78337;
        double r78346 = r78334 * r78336;
        double r78347 = fma(r78336, r78336, r78346);
        double r78348 = fma(r78334, r78334, r78347);
        double r78349 = r78348 * r78338;
        double r78350 = r78345 / r78349;
        double r78351 = 0.003332639573286841;
        bool r78352 = r78339 <= r78351;
        double r78353 = 0.041666666666666664;
        double r78354 = 3.0;
        double r78355 = pow(r78335, r78354);
        double r78356 = 0.004166666666666667;
        double r78357 = 5.0;
        double r78358 = pow(r78335, r78357);
        double r78359 = 0.5;
        double r78360 = r78359 * r78335;
        double r78361 = fma(r78356, r78358, r78360);
        double r78362 = fma(r78353, r78355, r78361);
        double r78363 = pow(r78334, r78354);
        double r78364 = pow(r78336, r78354);
        double r78365 = expm1(r78364);
        double r78366 = log1p(r78365);
        double r78367 = r78363 - r78366;
        double r78368 = r78367 / r78349;
        double r78369 = r78352 ? r78362 : r78368;
        double r78370 = r78341 ? r78350 : r78369;
        return r78370;
}

Error

Bits error versus x

Target

Original29.4
Target0.0
Herbie1.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.044495808914492015

    1. Initial program 0.7

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

    3. Applied flip3--0.8

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

    4. Applied associate-/l/0.8

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

    5. Simplified0.8

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

    7. Applied difference-cubes0.8

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

    8. Simplified0.8

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

    if -0.044495808914492015 < (/ (- 1.0 (cos x)) (sin x)) < 0.003332639573286841

    1. Initial program 59.0

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

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

    if 0.003332639573286841 < (/ (- 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. Simplified1.0

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

    7. Applied log1p-expm1-u1.0

      \[\leadsto \frac{{1}^{3} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{3}\right)\right)}}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.0444958089144920146:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, 1, \cos x \cdot \left(\cos x + 1\right)\right) \cdot \left(1 - \cos x\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0033326395732868408:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{24}, {x}^{3}, \mathsf{fma}\left(\frac{1}{240}, {x}^{5}, \frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{3}\right)\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(\cos x, \cos x, 1 \cdot \cos x\right)\right) \cdot \sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))