Average Error: 39.0 → 0.0

Time: 3.4s

Precision: 64

Internal Precision: 128

\[\log \left(1 + x\right)\]

↓

\[\log_* (1 + x)\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	39.0
Target	0.2
Herbie	0.0

\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

Initial program 39.0
\[\log \left(1 + x\right)\]
Simplified0.0
\[\leadsto \color{blue}{\log_* (1 + x)}\]
Final simplification0.0
\[\leadsto \log_* (1 + x)\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))

Details

sample126.0ms

Algorithm

intervals

Results

76.0ms	80×	body	1280	valid
23.0ms	47×	body	640	valid
6.0ms	24×	body	320	valid
6.0ms	93×	body	80	valid
3.0ms	76×	body	80	nan
2.0ms	12×	body	160	valid

simplify3.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

3.0ms

(log (+ 1 x))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize6.0ms

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

prune4.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample3.0s

Algorithm

intervals

Results

1.6s	2239×	body	1280	valid
756.0ms	1677×	body	640	valid
179.0ms	879×	body	320	valid
172.0ms	2826×	body	80	valid
117.0ms	2590×	body	80	nan
46.0ms	379×	body	160	valid