Average Error: 30.7 → 0.6
Time: 7.2s
Precision: binary64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.013822936471103725:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0.00023223782787987925:\\ \;\;\;\;\left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right) + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.013822936471103725:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right)}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0.00023223782787987925:\\
\;\;\;\;\left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right) + x \cdot 0.5\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= (/ (- 1.0 (cos x)) (sin x)) -0.013822936471103725)
   (/
    (- (pow 1.0 3.0) (pow (cos x) 3.0))
    (* (sin x) (+ (* 1.0 1.0) (* (cos x) (+ 1.0 (cos x))))))
   (if (<= (/ (- 1.0 (cos x)) (sin x)) 0.00023223782787987925)
     (+
      (+
       (* 0.041666666666666664 (pow x 3.0))
       (* 0.004166666666666667 (pow x 5.0)))
      (* x 0.5))
     (log (exp (/ (- 1.0 (cos x)) (sin x)))))))
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))) <= -0.013822936471103725)) {
		tmp = (((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))) / ((double) (((double) sin(x)) * ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x)))))))))));
	} else {
		double tmp_1;
		if (((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= 0.00023223782787987925)) {
			tmp_1 = ((double) (((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (0.004166666666666667 * ((double) pow(x, 5.0)))))) + ((double) (x * 0.5))));
		} else {
			tmp_1 = ((double) log(((double) exp((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x)))))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.7
Target0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 1.0 (cos.f64 x)) (sin.f64 x)) < -0.0138229364711037247

    1. Initial program 0.9

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

    3. Applied flip3--_binary641.0

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

    4. Applied associate-/l/_binary641.0

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

    5. Simplified1.0

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

    if -0.0138229364711037247 < (/.f64 (-.f64 1.0 (cos.f64 x)) (sin.f64 x)) < 2.32237827879879253e-4

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied associate-+r+_binary640.1

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

    if 2.32237827879879253e-4 < (/.f64 (-.f64 1.0 (cos.f64 x)) (sin.f64 x))

    1. Initial program 1.0

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

    3. Applied add-log-exp_binary641.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.013822936471103725:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)\right)}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0.00023223782787987925:\\ \;\;\;\;\left(0.041666666666666664 \cdot {x}^{3} + 0.004166666666666667 \cdot {x}^{5}\right) + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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