Average Error: 58.6 → 0.2
Time: 9.0s
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

Original58.6
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 2019021 
(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: 8.4s)Debug log

sample216.0ms

Algorithm
intervals
Results
111.0ms111×body1280valid
44.0ms75×body640valid
19.0ms246×body80nan
15.0ms43×body320valid
5.0ms21×body160valid
1.0ms6×body80valid

simplify18.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
18.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.6b

localize21.0ms

Local error

Found 2 expressions with local error:

5.8b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
15×log-prod
7×add-exp-log
4×associate-/r/
4×add-cbrt-cube
3×associate-/l*
3×associate-/r*
3×pow1
2×associate-/l/
2×add-log-exp
2×div-inv
2×flip-+
2×rem-log-exp
2×div-exp
2×flip3-+
1×flip--
1×log-pow
1×div-sub
1×flip3--
1×frac-2neg
1×clear-num
1×log-div
1×cbrt-undiv
Counts
2 → 58
Calls
2 calls:
Slowest
5.0ms
(/ (- 1 eps) (+ 1 eps))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series121.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
111.0ms
(log (/ (- 1 eps) (+ 1 eps)))
10.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.5s

Counts
54 → 64
Calls
54 calls:
Slowest
236.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
155.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
139.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
112.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
102.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune518.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize25.0ms

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
Rules
8×associate-*r*
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
2×add-log-exp
2×*-commutative
2×add-exp-log
2×add-cbrt-cube
2×pow1
1×unpow3
1×cube-mult
Counts
2 → 24
Calls
2 calls:
Slowest
1.0ms
(* 2/5 (pow eps 5))
1.0ms
(* 2/3 (pow eps 3))

series51.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
29.0ms
(* 2/5 (pow eps 5))
22.0ms
(* 2/3 (pow eps 3))

simplify26.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
5.0ms
(* 2/3 (pow eps 3))
4.0ms
(* 2/3 (pow eps 3))
4.0ms
(* 2/3 (pow eps 3))
2.0ms
(cbrt (pow eps 3))
2.0ms
(* 2/5 (pow eps 5))

prune230.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample5.7s

Algorithm
intervals
Results
2.8s3492×body1280valid
1.2s2439×body640valid
834.0ms1235×body320valid
568.0ms7881×body80nan
139.0ms634×body160valid
16.0ms202×body80valid