Average Error: 58.6 → 0.2
Time: 15.9s
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 2019018 
(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: 15.4s)Debug log

sample123.0ms

Algorithm
intervals
Results
59.0ms100×body1280valid
25.0ms78×body640valid
12.0ms257×body80nan
10.0ms49×body320valid
3.0ms19×body160valid
1.0ms10×body80valid

simplify10.0ms

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

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.2b

localize11.0ms

Local error

Found 2 expressions with local error:

6.0b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite19.0ms

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
16.0ms
(/ (- 1 eps) (+ 1 eps))
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series82.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
66.0ms
(log (/ (- 1 eps) (+ 1 eps)))
16.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.6s

Counts
54 → 64
Calls
54 calls:
Slowest
337.0ms
(/ (* (* (- 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))))
138.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
108.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
100.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune507.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize42.0ms

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)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
8×add-log-exp
8×associate-*r*
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
3×add-exp-log
3×add-cbrt-cube
3×sum-log
3×pow1
2×*-commutative
1×flip-+
1×unpow3
1×flip3-+
1×associate-+r+
1×+-commutative
1×cube-mult
Counts
3 → 37
Calls
3 calls:
Slowest
12.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
2.0ms
(* 2/3 (pow eps 3))
2.0ms
(* 2/5 (pow eps 5))

series106.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
42.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
35.0ms
(* 2/5 (pow eps 5))
29.0ms
(* 2/3 (pow eps 3))

simplify426.0ms

Counts
19 → 46
Calls
19 calls:
Slowest
136.0ms
(* (exp (* 2/3 (pow eps 3))) (* (exp (* 2/5 (pow eps 5))) (exp (* 2 eps))))
123.0ms
(* (exp (* 2/3 (pow eps 3))) (exp (+ (* 2/5 (pow eps 5)) (* 2 eps))))
57.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
37.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5))))
36.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5))))

prune404.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 4 expressions with local error:

6.0b
(log (- (/ 1 (+ 1 eps)) (/ eps (+ 1 eps))))
0.0b
(- (/ 1 (+ 1 eps)) (/ eps (+ 1 eps)))
0.0b
(/ 1 (+ 1 eps))
0.0b
(/ eps (+ 1 eps))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
15×*-un-lft-identity
12×associate-/r/
11×add-sqr-sqrt
10×div-inv
10×log-prod
8×distribute-rgt-out--
7×add-cube-cbrt
6×add-log-exp
6×flip-+
6×add-exp-log
6×flip3-+
6×associate-/r*
6×pow1
4×distribute-lft-out--
4×add-cbrt-cube
4×log-div
2×difference-of-squares
2×sub-div
2×flip--
2×frac-sub
2×flip3--
2×frac-2neg
2×clear-num
1×inv-pow
1×pow-flip
1×log-pow
1×rem-log-exp
1×diff-log
1×sub-neg
1×rec-exp
Counts
4 → 77
Calls
4 calls:
Slowest
5.0ms
(- (/ 1 (+ 1 eps)) (/ eps (+ 1 eps)))
4.0ms
(log (- (/ 1 (+ 1 eps)) (/ eps (+ 1 eps))))
2.0ms
(/ 1 (+ 1 eps))
1.0ms
(/ eps (+ 1 eps))

series142.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
108.0ms
(log (- (/ 1 (+ 1 eps)) (/ eps (+ 1 eps))))
15.0ms
(- (/ 1 (+ 1 eps)) (/ eps (+ 1 eps)))
13.0ms
(/ eps (+ 1 eps))
6.0ms
(/ 1 (+ 1 eps))

simplify2.7s

Counts
55 → 89
Calls
55 calls:
Slowest
519.0ms
(log (- (* 1 (+ 1 eps)) (* (+ 1 eps) eps)))
386.0ms
(log (- (* (/ 1 (+ 1 eps)) (/ 1 (+ 1 eps))) (* (/ eps (+ 1 eps)) (/ eps (+ 1 eps)))))
275.0ms
(- (* 1 (+ 1 eps)) (* (+ 1 eps) eps))
243.0ms
(log (- (pow (/ 1 (+ 1 eps)) 3) (pow (/ eps (+ 1 eps)) 3)))
171.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune665.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

6.1b
(log (- 1 eps))
6.1b
(log (/ 1 (+ 1 eps)))
0.0b
(/ 1 (+ 1 eps))
0.0b
(+ (log (- 1 eps)) (log (/ 1 (+ 1 eps))))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
18×log-prod
13×*-un-lft-identity
12×pow1
9×add-cube-cbrt
9×add-sqr-sqrt
8×log-pow
8×add-exp-log
6×associate-/r/
6×associate-+r+
6×log-div
4×add-log-exp
4×distribute-lft-out
4×flip-+
4×add-cbrt-cube
4×flip3-+
3×div-inv
3×log-rec
3×rem-log-exp
3×associate-/r*
3×associate-+l+
2×inv-pow
2×flip--
2×pow-flip
2×associate-+l-
2×flip3--
2×rec-exp
1×unsub-neg
1×frac-2neg
1×sum-log
1×associate-+r-
1×clear-num
1×+-commutative
Counts
4 → 81
Calls
4 calls:
Slowest
5.0ms
(+ (log (- 1 eps)) (log (/ 1 (+ 1 eps))))
1.0ms
(/ 1 (+ 1 eps))
1.0ms
(log (/ 1 (+ 1 eps)))
1.0ms
(log (- 1 eps))

series197.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
128.0ms
(+ (log (- 1 eps)) (log (/ 1 (+ 1 eps))))
33.0ms
(log (/ 1 (+ 1 eps)))
27.0ms
(log (- 1 eps))
9.0ms
(/ 1 (+ 1 eps))

simplify2.1s

Counts
54 → 93
Calls
54 calls:
Slowest
327.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 1/3 (pow eps 3)) eps)))
258.0ms
(- (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps)))))
250.0ms
(- (+ (* 1/2 (/ 1 (pow eps 2))) (+ (log (/ -1 eps)) (log -1))) (/ 1 eps))
210.0ms
(- (* 1/2 (pow eps 2)) (+ (* 1/3 (pow eps 3)) eps))
188.0ms
(- (log -1) (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ 1 eps)))))

prune708.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

regimes45.0ms

Accuracy

0% (0.2b remaining)

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

bsearch11.0ms

end0.0ms

sample5.5s

Algorithm
intervals
Results
3.2s3428×body1280valid
1.2s2533×body640valid
553.0ms7852×body80nan
332.0ms1217×body320valid
129.0ms632×body160valid
37.0ms191×body80valid