Average Error: 58.5 → 0.2
Time: 14.6s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{-2}{5} \cdot {\varepsilon}^{5} + {\varepsilon}^{3} \cdot \frac{-2}{3}\right) + \varepsilon \cdot -2\]

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  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. Using strategy rm

  4. Applied associate-+r+0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019005 
(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: 14.1s)Debug log

sample136.0ms

Algorithm
intervals

simplify9.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.5b

localize19.0ms

Local error

Found 2 expressions with local error:

6.5b
(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))
4.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series132.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

116.0ms
(log (/ (- 1 eps) (+ 1 eps)))
16.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.4s

Counts
54 → 64
Calls

54 calls. Slowest were:

281.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
155.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
115.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune582.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize45.0ms

Local error

Found 3 expressions with local error:

0.2b
(* 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)))

rewrite18.0ms

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
(* 2/5 (pow eps 5))
2.0ms
(* 2/3 (pow eps 3))

series102.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

36.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
34.0ms
(* 2/5 (pow eps 5))
33.0ms
(* 2/3 (pow eps 3))

simplify413.0ms

Counts
19 → 46
Calls

19 calls. Slowest were:

126.0ms
(* (exp (* 2/3 (pow eps 3))) (* (exp (* 2/5 (pow eps 5))) (exp (* 2 eps))))
120.0ms
(* (exp (* 2/3 (pow eps 3))) (exp (+ (* 2/5 (pow eps 5)) (* 2 eps))))
52.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5))))

prune522.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 3 expressions with local error:

0.2b
(* 2/3 (pow eps 3))
0.1b
(* 2/5 (pow eps 5))
0.0b
(+ (* 2/3 (pow eps 3)) (* 2/5 (pow eps 5)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

3.0ms
(+ (* 2/3 (pow eps 3)) (* 2/5 (pow eps 5)))
1.0ms
(* 2/3 (pow eps 3))
1.0ms
(* 2/5 (pow eps 5))

series142.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

89.0ms
(+ (* 2/3 (pow eps 3)) (* 2/5 (pow eps 5)))
33.0ms
(* 2/5 (pow eps 5))
20.0ms
(* 2/3 (pow eps 3))

simplify153.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

34.0ms
(* (exp (* 2/3 (pow eps 3))) (exp (* 2/5 (pow eps 5))))
25.0ms
(+ (* 2/3 (pow eps 3)) (* 2/5 (pow eps 5)))
23.0ms
(+ (* 2/3 (pow eps 3)) (* 2/5 (pow eps 5)))

prune556.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

6.5b
(log (/ (- 1 eps) (+ 1 eps)))
6.5b
(log (/ (- 1 eps) (+ 1 eps)))
6.5b
(log (/ (- 1 eps) (+ 1 eps)))
0.1b
(cbrt (log (/ (- 1 eps) (+ 1 eps))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

2.0ms
(log (/ (- 1 eps) (+ 1 eps)))
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))
1.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series436.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

146.0ms
(cbrt (log (/ (- 1 eps) (+ 1 eps))))
98.0ms
(log (/ (- 1 eps) (+ 1 eps)))
98.0ms
(log (/ (- 1 eps) (+ 1 eps)))
95.0ms
(log (/ (- 1 eps) (+ 1 eps)))

simplify2.7s

Counts
70 → 102
Calls

70 calls. Slowest were:

197.0ms
(+ (* 22/405 (* (pow eps 4) (exp (* 1/3 (+ (log -2) (log eps)))))) (+ (* 1/9 (* (pow eps 2) (exp (* 1/3 (+ (log -2) (log eps)))))) (exp (* 1/3 (+ (log -2) (log eps))))))
188.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
172.0ms
(- (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)))))

prune810.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

regimes39.0ms

Accuracy

0% (0.2b remaining)

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

bsearch12.0ms

end0.0ms

sample5.9s

Algorithm
intervals