Average Error: 58.7 → 0.2
Time: 16.4s
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

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  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 2019010 
(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: 16.1s)Debug log

sample241.0ms

Algorithm
intervals

simplify20.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
20.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.5b

localize22.0ms

Local error

Found 2 expressions with local error:

5.4b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite10.0ms

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)))

series118.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
108.0ms
(log (/ (- 1 eps) (+ 1 eps)))
10.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.6s

Counts
54 → 64
Calls
54 calls:
Slowest
343.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
194.0ms
(- (+ (* 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)))))
125.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))
124.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune594.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize42.0ms

Local error

Found 2 expressions with local error:

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

rewrite6.0ms

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

series41.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
24.0ms
(* 2/5 (pow eps 5))
17.0ms
(* 2/3 (pow eps 3))

simplify31.0ms

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

prune233.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 4 expressions with local error:

5.4b
(log (/ (- 1 eps) (+ 1 eps)))
5.4b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(* (sqrt (log (/ (- 1 eps) (+ 1 eps)))) (sqrt (log (/ (- 1 eps) (+ 1 eps)))))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
30×add-cube-cbrt
30×*-un-lft-identity
30×log-prod
30×add-sqr-sqrt
27×times-frac
14×add-exp-log
13×pow1
8×sqrt-prod
8×add-cbrt-cube
7×associate-*l*
7×associate-*r*
6×associate-/r/
4×add-log-exp
4×log-pow
4×rem-log-exp
4×pow1/2
3×associate-/l*
3×div-inv
3×flip-+
3×div-exp
3×flip3-+
3×associate-/r*
2×associate-/l/
2×pow-prod-up
2×pow-prod-down
2×log-div
1×flip--
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×div-sub
1×sqrt-unprod
1×flip3--
1×pow-plus
1×frac-2neg
1×rem-square-sqrt
1×clear-num
1×cbrt-undiv
1×pow2
Counts
4 → 116
Calls
4 calls:
Slowest
6.0ms
(* (sqrt (log (/ (- 1 eps) (+ 1 eps)))) (sqrt (log (/ (- 1 eps) (+ 1 eps)))))
3.0ms
(/ (- 1 eps) (+ 1 eps))
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series215.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
72.0ms
(log (/ (- 1 eps) (+ 1 eps)))
68.0ms
(* (sqrt (log (/ (- 1 eps) (+ 1 eps)))) (sqrt (log (/ (- 1 eps) (+ 1 eps)))))
64.0ms
(log (/ (- 1 eps) (+ 1 eps)))
10.0ms
(/ (- 1 eps) (+ 1 eps))

simplify3.3s

Counts
102 → 128
Calls
102 calls:
Slowest
263.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
192.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
164.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
164.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
157.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune646.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

5.4b
(log (/ (- 1 eps) (+ 1 eps)))
5.4b
(log (/ (- 1 eps) (+ 1 eps)))
5.4b
(log (/ (- 1 eps) (+ 1 eps)))
5.4b
(log (/ (- 1 eps) (+ 1 eps)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
60×log-prod
36×times-frac
32×add-cube-cbrt
32×*-un-lft-identity
32×add-sqr-sqrt
16×add-exp-log
8×associate-/r/
8×rem-log-exp
8×pow1
4×add-log-exp
4×div-inv
4×flip-+
4×log-pow
4×div-exp
4×add-cbrt-cube
4×flip3-+
4×log-div
Counts
4 → 104
Calls
4 calls:
Slowest
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))
1.0ms
(log (/ (- 1 eps) (+ 1 eps)))
1.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series325.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
96.0ms
(log (/ (- 1 eps) (+ 1 eps)))
94.0ms
(log (/ (- 1 eps) (+ 1 eps)))
69.0ms
(log (/ (- 1 eps) (+ 1 eps)))
66.0ms
(log (/ (- 1 eps) (+ 1 eps)))

simplify3.2s

Counts
84 → 116
Calls
84 calls:
Slowest
179.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
171.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
170.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
167.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
166.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune842.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

regimes22.0ms

Accuracy

0% (0.1b remaining)

Error of 0.2b against oracle of 0.1b and baseline of 0.2b

bsearch6.0ms

end0.0ms

sample4.6s

Algorithm
intervals