Average Error: 58.4 → 0.2
Time: 15.3s
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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  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 2019008 
(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: 14.9s)Debug log

sample190.0ms

Algorithm
intervals

simplify9.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

9.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.6b

localize16.0ms

Local error

Found 2 expressions with local error:

5.2b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 58
Calls

2 calls. Slowest were:

5.0ms
(/ (- 1 eps) (+ 1 eps))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series100.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

86.0ms
(log (/ (- 1 eps) (+ 1 eps)))
14.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.4s

Counts
54 → 64
Calls

54 calls. Slowest were:

271.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
132.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
130.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune561.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize48.0ms

Local error

Found 3 expressions with local error:

0.1b
(* 2/3 (pow eps 3))
0.1b
(* 2/5 (pow eps 5))
0.0b
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 37
Calls

3 calls. Slowest were:

13.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
3.0ms
(* 2/3 (pow eps 3))
2.0ms
(* 2/5 (pow eps 5))

series108.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

45.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
34.0ms
(* 2/5 (pow eps 5))
28.0ms
(* 2/3 (pow eps 3))

simplify379.0ms

Counts
19 → 46
Calls

19 calls. Slowest were:

128.0ms
(* (exp (* 2/3 (pow eps 3))) (exp (+ (* 2/5 (pow eps 5)) (* 2 eps))))
87.0ms
(* (exp (* 2/3 (pow eps 3))) (* (exp (* 2/5 (pow eps 5))) (exp (* 2 eps))))
52.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5))))

prune427.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

5.2b
(log (/ (- 1 eps) (+ 1 eps)))
5.2b
(log (/ (- 1 eps) (+ 1 eps)))
5.2b
(log (/ (- 1 eps) (+ 1 eps)))
0.1b
(cbrt (log (/ (- 1 eps) (+ 1 eps))))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

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

series374.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

134.0ms
(cbrt (log (/ (- 1 eps) (+ 1 eps))))
87.0ms
(log (/ (- 1 eps) (+ 1 eps)))
82.0ms
(log (/ (- 1 eps) (+ 1 eps)))
71.0ms
(log (/ (- 1 eps) (+ 1 eps)))

simplify2.8s

Counts
70 → 102
Calls

70 calls. Slowest were:

227.0ms
(- (pow (log -1) 1/3) (+ (* 4/9 (* (/ 1 (pow eps 2)) (pow (/ 1 (pow (log -1) 5)) 1/3))) (* 2/3 (* (/ 1 eps) (pow (/ 1 (pow (log -1) 2)) 1/3)))))
190.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
187.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune729.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize16.0ms

Local error

Found 2 expressions with local error:

5.3b
(log (- 1 eps))
5.2b
(log (+ 1 eps))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

8.0ms
(log (- 1 eps))
2.0ms
(log (+ 1 eps))

series72.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

42.0ms
(log (- 1 eps))
30.0ms
(log (+ 1 eps))

simplify1.3s

Counts
18 → 34
Calls

18 calls. Slowest were:

302.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 1/3 (pow eps 3)) eps)))
249.0ms
(- (log -1) (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ 1 eps)))))
225.0ms
(- (+ (/ 1 eps) (log -1)) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps))))

prune254.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes108.0ms

Accuracy

0% (0.2b remaining)

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

bsearch6.0ms

end0.0ms

sample6.0s

Algorithm
intervals