Average Error: 30.7 → 0.0
Time: 10.7s
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

Original30.7
Target0.0
Herbie0.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 30.7

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

  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 2019021 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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

Details

Time bar (total: 10.4s)Debug log

sample296.0ms

Algorithm
intervals
Results
128.0ms48×body2560valid
91.0ms80×body1280valid
46.0ms70×body640valid
15.0ms31×body320valid
6.0ms18×body160valid
2.0ms9×body80valid

simplify15.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
15.0ms
(/ (- 1 (cos x)) (sin x))

prune50.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize9.0ms

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

prune2.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample10.1s

Algorithm
intervals
Results
4.2s1627×body2560valid
3.4s2742×body1280valid
1.6s1810×body640valid
490.0ms938×body320valid
189.0ms503×body160valid
60.0ms376×body80valid
25.0ms4×body5120valid