Average Error: 30.3 → 0.0
Time: 11.4s
Precision: 64
Internal Precision: 128
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\tan \left(\frac{x}{2}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 30.3

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)}\]

  3. Final simplification0.0

    \[\leadsto \tan \left(\frac{x}{2}\right)\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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

Details

Time bar (total: 10.8s)Debug log

sample328.0ms

Algorithm
intervals
Results
178.0ms59×body2560valid
84.0ms83×body1280valid
29.0ms45×body640valid
17.0ms37×body320valid
6.0ms16×body160valid
3.0ms16×body80valid

simplify17.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
17.0ms
(/ (- 1 (cos x)) (sin x))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize13.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune5.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample10.4s

Algorithm
intervals
Results
4.2s1592×body2560valid
3.9s2777×body1280valid
1.5s1823×body640valid
424.0ms939×body320valid
170.0ms472×body160valid
69.0ms393×body80valid
32.0ms4×body5120valid