Average Error: 58.6 → 0.2
Time: 12.7s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{3} \cdot \frac{-2}{3} + \left(\frac{-2}{5} \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right)\]

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]

  3. Final simplification0.2

    \[\leadsto {\varepsilon}^{3} \cdot \frac{-2}{3} + \left(\frac{-2}{5} \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right)\]

Reproduce

herbie shell --seed 2019002 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))

Details

Time bar (total: 12.3s)Debug log

sample187.0ms

Algorithm
intervals

simplify17.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

17.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.3b

localize16.0ms

Local error

Found 2 expressions with local error:

5.3b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 58
Calls

2 calls. Slowest were:

3.0ms
(/ (- 1 eps) (+ 1 eps))
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series93.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

82.0ms
(log (/ (- 1 eps) (+ 1 eps)))
10.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.3s

Counts
54 → 64
Calls

54 calls. Slowest were:

253.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
136.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
112.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune504.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 2 expressions with local error:

0.1b
(* 2/3 (pow eps 3))
0.1b
(* 2/5 (pow eps 5))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
2 → 24
Calls

2 calls. Slowest were:

1.0ms
(* 2/3 (pow eps 3))
1.0ms
(* 2/5 (pow eps 5))

series41.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

23.0ms
(* 2/5 (pow eps 5))
18.0ms
(* 2/3 (pow eps 3))

simplify28.0ms

Counts
13 → 30
Calls

13 calls. Slowest were:

5.0ms
(* 2/3 (pow eps 3))
5.0ms
(* 2/3 (pow eps 3))
5.0ms
(* 2/3 (pow eps 3))

prune280.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 2 expressions with local error:

5.3b
(log (+ 1 eps))
5.3b
(log (- 1 eps))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

1.0ms
(log (+ 1 eps))
1.0ms
(log (- 1 eps))

series69.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

36.0ms
(log (- 1 eps))
33.0ms
(log (+ 1 eps))

simplify1.4s

Counts
18 → 34
Calls

18 calls. Slowest were:

385.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 1/3 (pow eps 3)) eps)))
281.0ms
(- (+ (/ 1 eps) (log -1)) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps))))
202.0ms
(- (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps)))))

prune221.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 3 expressions with local error:

5.3b
(log (sqrt (+ 1 eps)))
5.3b
(log (sqrt (+ 1 eps)))
5.3b
(log (- 1 eps))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
3 → 50
Calls

3 calls. Slowest were:

1.0ms
(log (sqrt (+ 1 eps)))
1.0ms
(log (sqrt (+ 1 eps)))
1.0ms
(log (- 1 eps))

series92.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

33.0ms
(log (- 1 eps))
31.0ms
(log (sqrt (+ 1 eps)))
27.0ms
(log (sqrt (+ 1 eps)))

simplify1.2s

Counts
33 → 59
Calls

33 calls. Slowest were:

361.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 1/3 (pow eps 3)) eps)))
219.0ms
(- (log -1) (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ 1 eps)))))
209.0ms
(- (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps)))))

prune446.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

regimes30.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch8.0ms

end0.0ms

sample6.2s

Algorithm
intervals