Average Error: 58.6 → 0.2
Time: 20.9s
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 2019022 
(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: 20.3s)Debug log

sample234.0ms

Algorithm
intervals
Results
116.0ms115×body1280valid
49.0ms83×body640valid
37.0ms246×body80nan
14.0ms37×body320valid
4.0ms16×body160valid
0.0ms5×body80valid

simplify19.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
19.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.0b

localize10.0ms

Local error

Found 2 expressions with local error:

6.4b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite5.0ms

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
3.0ms
(/ (- 1 eps) (+ 1 eps))
1.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series77.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
67.0ms
(log (/ (- 1 eps) (+ 1 eps)))
10.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.4s

Counts
54 → 64
Calls
54 calls:
Slowest
281.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
117.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
117.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
108.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
93.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune556.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize39.0ms

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)))

rewrite17.0ms

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))

series108.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
43.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
39.0ms
(* 2/5 (pow eps 5))
27.0ms
(* 2/3 (pow eps 3))

simplify431.0ms

Counts
19 → 46
Calls
19 calls:
Slowest
134.0ms
(* (exp (* 2/3 (pow eps 3))) (exp (+ (* 2/5 (pow eps 5)) (* 2 eps))))
134.0ms
(* (exp (* 2/3 (pow eps 3))) (* (exp (* 2/5 (pow eps 5))) (exp (* 2 eps))))
41.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5))))
31.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
28.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5))))

prune497.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize39.0ms

Local error

Found 4 expressions with local error:

6.2b
(log (/ (- 1 eps) (+ (pow eps 3) 1)))
6.2b
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
0.0b
(/ (- 1 eps) (+ (pow eps 3) 1))
0.0b
(+ (log (/ (- 1 eps) (+ (pow eps 3) 1))) (log (+ (* 1 1) (- (* eps eps) (* 1 eps)))))

rewrite56.0ms

Algorithm
rewrite-expression-head
Rules
38×log-prod
34×*-un-lft-identity
28×add-cube-cbrt
28×add-sqr-sqrt
27×times-frac
15×associate-+l+
10×add-exp-log
10×pow1
6×distribute-lft-out
6×flip-+
6×log-pow
6×associate-/r/
6×add-cbrt-cube
6×flip3-+
6×log-div
4×add-log-exp
4×associate-+r+
3×associate-/l*
3×div-inv
3×rem-log-exp
3×associate-/r*
2×associate-/l/
2×div-exp
2×associate-+r-
1×flip--
1×associate-+l-
1×div-sub
1×flip3--
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
4 → 110
Calls
4 calls:
Slowest
21.0ms
(+ (log (/ (- 1 eps) (+ (pow eps 3) 1))) (log (+ (* 1 1) (- (* eps eps) (* 1 eps)))))
14.0ms
(log (/ (- 1 eps) (+ (pow eps 3) 1)))
10.0ms
(/ (- 1 eps) (+ (pow eps 3) 1))
7.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))

series297.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
168.0ms
(+ (log (/ (- 1 eps) (+ (pow eps 3) 1))) (log (+ (* 1 1) (- (* eps eps) (* 1 eps)))))
54.0ms
(log (/ (- 1 eps) (+ (pow eps 3) 1)))
46.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
30.0ms
(/ (- 1 eps) (+ (pow eps 3) 1))

simplify5.0s

Counts
94 → 122
Calls
94 calls:
Slowest
611.0ms
(log (+ (pow (* 1 1) 3) (pow (- (* eps eps) (* 1 eps)) 3)))
349.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ (pow eps 3) 1) (+ (pow eps 3) 1)) (+ (pow eps 3) 1)))
348.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 4/3 (pow eps 3)) eps)))
286.0ms
(- (+ (* 2 (log (/ 1 eps))) (log -1)) (+ (/ 1 eps) (* 1/2 (/ 1 (pow eps 2)))))
265.0ms
(+ (* (pow eps 3) (pow eps 3)) (- (* 1 1) (* (pow eps 3) 1)))

prune818.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

6.2b
(log (/ (- 1 eps) (+ (pow eps 3) 1)))
6.2b
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
6.2b
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
6.2b
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
27×log-prod
17×*-un-lft-identity
14×add-cube-cbrt
14×add-sqr-sqrt
10×add-exp-log
9×times-frac
8×pow1
7×log-div
5×rem-log-exp
4×add-log-exp
4×flip-+
4×log-pow
4×add-cbrt-cube
4×flip3-+
3×distribute-lft-out
2×associate-/r/
1×div-inv
1×div-exp
Counts
4 → 71
Calls
4 calls:
Slowest
7.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
7.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
6.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
2.0ms
(log (/ (- 1 eps) (+ (pow eps 3) 1)))

series143.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
45.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
39.0ms
(log (/ (- 1 eps) (+ (pow eps 3) 1)))
33.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
26.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))

simplify4.3s

Counts
51 → 83
Calls
51 calls:
Slowest
554.0ms
(log (+ (pow (* 1 1) 3) (pow (- (* eps eps) (* 1 eps)) 3)))
541.0ms
(log (+ (pow (* 1 1) 3) (pow (- (* eps eps) (* 1 eps)) 3)))
443.0ms
(log (+ (pow (* 1 1) 3) (pow (- (* eps eps) (* 1 eps)) 3)))
291.0ms
(- (+ (* 2 (log (/ 1 eps))) (log -1)) (+ (/ 1 eps) (* 1/2 (/ 1 (pow eps 2)))))
235.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 4/3 (pow eps 3)) eps)))

prune718.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

regimes46.0ms

Accuracy

0% (0.2b remaining)

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

bsearch10.0ms

end0.0ms

sample5.4s

Algorithm
intervals
Results
3.0s3411×body1280valid
1.2s2481×body640valid
514.0ms7987×body80nan
378.0ms1301×body320valid
99.0ms604×body160valid
17.0ms203×body80valid