Average Error: 58.6 → 0.2
Time: 22.5s
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 1137770271 
(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: 21.8s)Debug log

sample316.0ms

Algorithm
intervals
Results
119×153.0ms(body real 1280)
65×52.0ms(body real 640)
246×30.0ms(body nan 80)
42×23.0ms(body real 320)
502×13.0ms(pre true 80)
22×8.0ms(body real 160)
8×1.0ms(body real 80)

simplify49.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
49.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.6b

localize41.0ms

Local error

Found 2 expressions with local error:

6.7b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite13.0ms

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
7.0ms
(/ (- 1 eps) (+ 1 eps))
4.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series159.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
129.0ms
(log (/ (- 1 eps) (+ 1 eps)))
29.0ms
(/ (- 1 eps) (+ 1 eps))

simplify2.7s

Counts
54 → 64
Calls
54 calls:
Slowest
502.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
239.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
238.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
218.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
167.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune999.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize76.0ms

Local error

Found 2 expressions with local error:

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

rewrite7.0ms

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

series83.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
46.0ms
(* 2/5 (pow eps 5))
37.0ms
(* 2/3 (pow eps 3))

simplify75.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
15.0ms
(* 2/3 (pow eps 3))
13.0ms
(* 2/3 (pow eps 3))
12.0ms
(* 2/3 (pow eps 3))
8.0ms
(* 2/5 (pow eps 5))
5.0ms
(cbrt (pow eps 3))

prune543.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize72.0ms

Local error

Found 4 expressions with local error:

6.7b
(log (/ (sqrt (- 1 eps)) (cbrt (+ 1 eps))))
6.7b
(log (/ (sqrt (- 1 eps)) (* (cbrt (+ 1 eps)) (cbrt (+ 1 eps)))))
0.0b
(cbrt (+ 1 eps))
0.0b
(cbrt (+ 1 eps))

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
60×log-prod
42×times-frac
34×add-cube-cbrt
34×*-un-lft-identity
34×add-sqr-sqrt
24×cbrt-prod
21×sqrt-prod
18×cbrt-div
13×add-exp-log
10×associate-/r/
9×flip-+
9×flip3-+
6×pow1
5×rem-log-exp
4×add-log-exp
4×add-cbrt-cube
4×frac-times
3×div-exp
2×div-inv
2×log-pow
2×associate-*r/
2×associate-*l/
2×pow1/3
2×log-div
1×prod-exp
Counts
4 → 109
Calls
4 calls:
Slowest
19.0ms
(log (/ (sqrt (- 1 eps)) (* (cbrt (+ 1 eps)) (cbrt (+ 1 eps)))))
7.0ms
(log (/ (sqrt (- 1 eps)) (cbrt (+ 1 eps))))
1.0ms
(cbrt (+ 1 eps))
1.0ms
(cbrt (+ 1 eps))

series567.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
210.0ms
(log (/ (sqrt (- 1 eps)) (cbrt (+ 1 eps))))
205.0ms
(log (/ (sqrt (- 1 eps)) (* (cbrt (+ 1 eps)) (cbrt (+ 1 eps)))))
78.0ms
(cbrt (+ 1 eps))
75.0ms
(cbrt (+ 1 eps))

simplify4.6s

Counts
89 → 121
Calls
89 calls:
Slowest
422.0ms
(log (/ (sqrt (* (cbrt (- 1 eps)) (cbrt (- 1 eps)))) 1))
403.0ms
(- (+ (* 1/12 (pow eps 2)) (+ (* 5/18 (pow eps 3)) (* 5/6 eps))))
354.0ms
(- (+ (* 1/3 (/ (exp (* 1/3 (- (log -1) (log (/ -1 eps))))) eps)) (exp (* 1/3 (- (log -1) (log (/ -1 eps)))))) (* 1/9 (/ (exp (* 1/3 (- (log -1) (log (/ -1 eps))))) (pow eps 2))))
352.0ms
(- (* 1/12 (pow eps 2)) (+ (* 7/18 (pow eps 3)) (* 7/6 eps)))
318.0ms
(- (+ (* 1/3 (/ (exp (* 1/3 (- (log -1) (log (/ -1 eps))))) eps)) (exp (* 1/3 (- (log -1) (log (/ -1 eps)))))) (* 1/9 (/ (exp (* 1/3 (- (log -1) (log (/ -1 eps))))) (pow eps 2))))

prune1.7s

Pruning

2 alts after pruning (0 fresh and 2 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

bsearch15.0ms

end0.0ms

sample9.6s

Algorithm
intervals
Results
3443×4.9s(body real 1280)
2435×2.1s(body real 640)
8195×1.0s(body nan 80)
1265×629.0ms(body real 320)
16196×376.0ms(pre true 80)
673×215.0ms(body real 160)
185×27.0ms(body real 80)