Average Error: 30.5 → 0.8
Time: 7.8s
Precision: binary64
\[\frac{1 - \cos x}{\sin x}\]
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.019856612798463034:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{1 - \cos x}{\sin x}\right)}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.019856612798463034:\\
\;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{\sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0:\\
\;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\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)) -0.019856612798463034)
   (/
    (/
     (log (exp (- (pow 1.0 3.0) (pow (cos x) 3.0))))
     (+ (* 1.0 1.0) (* (cos x) (+ 1.0 (cos x)))))
    (sin x))
   (if (<= (/ (- 1.0 (cos x)) (sin x)) 0.0)
     (+
      (* 0.041666666666666664 (pow x 3.0))
      (+ (* 0.004166666666666667 (pow x 5.0)) (* x 0.5)))
     (exp (log (/ (- 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))) <= -0.019856612798463034)) {
		tmp = ((((double) log(((double) exp(((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) cos(x)), 3.0)))))))) / ((double) (((double) (1.0 * 1.0)) + ((double) (((double) cos(x)) * ((double) (1.0 + ((double) cos(x))))))))) / ((double) sin(x)));
	} else {
		double tmp_1;
		if (((((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))) <= 0.0)) {
			tmp_1 = ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (x * 0.5))))));
		} else {
			tmp_1 = ((double) exp(((double) log((((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

Original30.5
Target0.0
Herbie0.8
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes
  2. if (/.f64 (-.f64 1.0 (cos.f64 x)) (sin.f64 x)) < -0.019856612798463034

    1. Initial program 0.8

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm
    3. Applied flip3--_binary640.9

      \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{\sin x}\]
    4. Simplified0.9

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}}{\sin x}\]
    5. Using strategy rm
    6. Applied add-log-exp_binary640.9

      \[\leadsto \frac{\frac{{1}^{3} - \color{blue}{\log \left(e^{{\left(\cos x\right)}^{3}}\right)}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{\sin x}\]
    7. Applied add-log-exp_binary640.9

      \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{{1}^{3}}\right)} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{\sin x}\]
    8. Applied diff-log_binary641.0

      \[\leadsto \frac{\frac{\color{blue}{\log \left(\frac{e^{{1}^{3}}}{e^{{\left(\cos x\right)}^{3}}}\right)}}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{\sin x}\]
    9. Simplified0.9

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

    if -0.019856612798463034 < (/.f64 (-.f64 1.0 (cos.f64 x)) (sin.f64 x)) < 0.0

    1. Initial program 59.9

      \[\frac{1 - \cos x}{\sin x}\]
    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + 0.5 \cdot x\right)}\]
    3. Simplified0.4

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

    if 0.0 < (/.f64 (-.f64 1.0 (cos.f64 x)) (sin.f64 x))

    1. Initial program 1.5

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm
    3. Applied add-exp-log_binary641.5

      \[\leadsto \frac{1 - \cos x}{\color{blue}{e^{\log \left(\sin x\right)}}}\]
    4. Applied add-exp-log_binary641.5

      \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{e^{\log \left(\sin x\right)}}\]
    5. Applied div-exp_binary641.6

      \[\leadsto \color{blue}{e^{\log \left(1 - \cos x\right) - \log \left(\sin x\right)}}\]
    6. Simplified1.5

      \[\leadsto e^{\color{blue}{\log \left(\frac{1 - \cos x}{\sin x}\right)}}\]
  3. Recombined 3 regimes into one program.
  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \leq -0.019856612798463034:\\ \;\;\;\;\frac{\frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{1 \cdot 1 + \cos x \cdot \left(1 + \cos x\right)}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \leq 0:\\ \;\;\;\;0.041666666666666664 \cdot {x}^{3} + \left(0.004166666666666667 \cdot {x}^{5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{1 - \cos x}{\sin x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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