Average Error: 30.0 → 0.7
Time: 7.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.008959247045566699763075035889414721168578:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{1 - \cos x}{\sin x}\right)}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.008959247045566699763075035889414721168578:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\frac{1 - \cos x}{\sin x}\right)}\\

\end{array}
double f(double x) {
        double r57393 = 1.0;
        double r57394 = x;
        double r57395 = cos(r57394);
        double r57396 = r57393 - r57395;
        double r57397 = sin(r57394);
        double r57398 = r57396 / r57397;
        return r57398;
}
↓
double f(double x) {
        double r57399 = 1.0;
        double r57400 = x;
        double r57401 = cos(r57400);
        double r57402 = r57399 - r57401;
        double r57403 = sin(r57400);
        double r57404 = r57402 / r57403;
        double r57405 = -0.0089592470455667;
        bool r57406 = r57404 <= r57405;
        double r57407 = 3.0;
        double r57408 = pow(r57399, r57407);
        double r57409 = pow(r57401, r57407);
        double r57410 = r57408 - r57409;
        double r57411 = r57401 * r57401;
        double r57412 = r57399 * r57399;
        double r57413 = r57411 - r57412;
        double r57414 = r57401 - r57399;
        double r57415 = r57413 / r57414;
        double r57416 = r57401 * r57415;
        double r57417 = r57416 + r57412;
        double r57418 = r57417 * r57403;
        double r57419 = r57410 / r57418;
        double r57420 = 0.0;
        bool r57421 = r57404 <= r57420;
        double r57422 = 0.041666666666666664;
        double r57423 = pow(r57400, r57407);
        double r57424 = r57422 * r57423;
        double r57425 = 0.004166666666666667;
        double r57426 = 5.0;
        double r57427 = pow(r57400, r57426);
        double r57428 = r57425 * r57427;
        double r57429 = 0.5;
        double r57430 = r57429 * r57400;
        double r57431 = r57428 + r57430;
        double r57432 = r57424 + r57431;
        double r57433 = log(r57404);
        double r57434 = exp(r57433);
        double r57435 = r57421 ? r57432 : r57434;
        double r57436 = r57406 ? r57419 : r57435;
        return r57436;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.0
Herbie0.7
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.0089592470455667

    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. 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

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

    7. Applied add-sqr-sqrt1.0

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

    9. Applied flip-+1.0

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

    10. Applied sqrt-div64.0

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

    11. Applied flip-+64.0

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

    12. Applied sqrt-div64.0

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

    13. Applied frac-times64.0

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

    14. Simplified64.0

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

    15. Simplified1.0

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

    if -0.0089592470455667 < (/ (- 1.0 (cos x)) (sin x)) < 0.0

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.3

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

    if 0.0 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 29.8

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

    3. Applied add-exp-log31.5

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

    4. Applied add-exp-log31.5

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

    5. Applied div-exp31.6

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

    6. Simplified29.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.008959247045566699763075035889414721168578:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{1 - \cos x}{\sin x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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