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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.006199629553278163:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le 0.0051340515948461625:\\ \;\;\;\;\left(\frac{1}{2} \cdot x + \frac{1}{240} \cdot {x}^{5}\right) + {x}^{3} \cdot \frac{1}{24}\\ \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le +\infty:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\left(\cos x \cdot \cos x + \cos x\right) + 1\right) \cdot \sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\left(\cos x \cdot \cos x + \cos x\right) + 1\right) \cdot \sin x}\\ \end{array}\]

Original	30.1
Target	0.0
Herbie	0.7

Derivation

Split input into 3 regimes
if (/ (- 1 (cos x)) (sin x)) < -0.006199629553278163
1. Initial program 0.9
  \[\frac{1 - \cos x}{\sin x}\]
2. Using strategy rm
3. Applied div-sub1.1
  \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]
if -0.006199629553278163 < (/ (- 1 (cos x)) (sin x)) < 0.0051340515948461625
1. Initial program 59.7
  \[\frac{1 - \cos x}{\sin x}\]
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 0.0051340515948461625 < (/ (- 1 (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
  \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\sin x \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]
Recombined 3 regimes into one program.
Applied simplify0.7
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.006199629553278163:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le 0.0051340515948461625:\\ \;\;\;\;\left(\frac{1}{2} \cdot x + \frac{1}{240} \cdot {x}^{5}\right) + {x}^{3} \cdot \frac{1}{24}\\ \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le +\infty:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\left(\cos x \cdot \cos x + \cos x\right) + 1\right) \cdot \sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\left(\cos x \cdot \cos x + \cos x\right) + 1\right) \cdot \sin x}\\ \end{array}}\]

Runtime

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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

Error

Target

Derivation

`if (/ (- 1 (cos x)) (sin x)) < -0.006199629553278163`

`if -0.006199629553278163 < (/ (- 1 (cos x)) (sin x)) < 0.0051340515948461625`

`if 0.0051340515948461625 < (/ (- 1 (cos x)) (sin x))`

Runtime