Average Error: 30.3 → 0.6
Time: 8.0s
Precision: binary64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.0024734165641486813 \lor \neg \left(\frac{1 - \cos x}{\sin x} \leq 3.038235675963212 \cdot 10^{-06}\right):\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.0024734165641486813 \lor \neg \left(\frac{1 - \cos x}{\sin x} \leq 3.038235675963212 \cdot 10^{-06}\right):\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\mathbf{else}:\\
\;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\

\end{array}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= (/ (- 1.0 (cos x)) (sin x)) -0.0024734165641486813)
         (not (<= (/ (- 1.0 (cos x)) (sin x)) 3.038235675963212e-06)))
   (log (exp (/ (- 1.0 (cos x)) (sin x))))
   (+
    (* 0.041666666666666664 (pow x 3.0))
    (+ (* 0.004166666666666667 (pow x 5.0)) (* x 0.5)))))
double code(double x) {
	return (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)));
}
↓
double code(double x) {
	double tmp;
	if ((((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= -0.0024734165641486813) || !((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= 3.038235675963212e-06))) {
		tmp = ((double) log(((double) exp((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))))));
	} else {
		tmp = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (x * 0.5))))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.00247341656414868131 or 3.0382356759632119e-6 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program Error: 1.1 bits

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

    3. Applied add-log-expError: 1.2 bits

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

    if -0.00247341656414868131 < (/ (- 1.0 (cos x)) (sin x)) < 3.0382356759632119e-6

    1. Initial program Error: 60.1 bits

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

    2. Taylor expanded around 0 Error: 0.1 bits

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + 0.5 \cdot x\right)}\]

    3. SimplifiedError: 0.1 bits

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 0.6 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.0024734165641486813 \lor \neg \left(\frac{1 - \cos x}{\sin x} \leq 3.038235675963212 \cdot 10^{-06}\right):\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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