Average Error: 58.3 → 0.3
Time: 15.3s
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

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

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]

  3. Final simplification0.3

    \[\leadsto {\varepsilon}^{3} \cdot \frac{-2}{3} + \left(\frac{-2}{5} \cdot {\varepsilon}^{5} + -2 \cdot \varepsilon\right)\]

Reproduce

herbie shell --seed 2019016 
(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: 14.9s)Debug log

sample163.0ms

Algorithm
intervals
Results
524×(pre true 80)
268×(body nan 80)
104×(body real 1280)
83×(body real 640)
41×(body real 320)
17×(body real 160)
11×(body real 80)

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: 58.3b

localize32.0ms

Local error

Found 2 expressions with local error:

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

rewrite11.0ms

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

series94.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
78.0ms
(log (/ (- 1 eps) (+ 1 eps)))
16.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.5s

Counts
54 → 64
Calls
54 calls:
Slowest
295.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
178.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
159.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
133.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
86.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune546.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 2 expressions with local error:

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

rewrite2.0ms

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
(* 2/3 (pow eps 3))
1.0ms
(* 2/5 (pow eps 5))

series49.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
30.0ms
(* 2/5 (pow eps 5))
19.0ms
(* 2/3 (pow eps 3))

simplify36.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
8.0ms
(* 2/3 (pow eps 3))
5.0ms
(* 2/3 (pow eps 3))
5.0ms
(* 2/3 (pow eps 3))
3.0ms
(* 2/5 (pow eps 5))
3.0ms
(* 2/5 (pow eps 5))

prune340.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize24.0ms

Local error

Found 3 expressions with local error:

2.3b
(log (exp (* 2/5 (pow eps 5))))
0.1b
(* 2/3 (pow eps 3))
0.1b
(* 2/5 (pow eps 5))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*r*
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×pow1
3×add-log-exp
3×add-exp-log
3×add-cbrt-cube
3×log-prod
2×*-commutative
2×log-pow
1×rem-log-exp
1×unpow3
1×exp-prod
1×cube-mult
Counts
3 → 37
Calls
3 calls:
Slowest
3.0ms
(log (exp (* 2/5 (pow eps 5))))
2.0ms
(* 2/3 (pow eps 3))
2.0ms
(* 2/5 (pow eps 5))

series102.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
36.0ms
(log (exp (* 2/5 (pow eps 5))))
36.0ms
(* 2/5 (pow eps 5))
31.0ms
(* 2/3 (pow eps 3))

simplify57.0ms

Counts
20 → 46
Calls
20 calls:
Slowest
8.0ms
(log (* (cbrt (exp (* 2/5 (pow eps 5)))) (cbrt (exp (* 2/5 (pow eps 5))))))
7.0ms
(* 2/3 (pow eps 3))
6.0ms
(log (sqrt (exp (* 2/5 (pow eps 5)))))
5.0ms
(* 2/3 (pow eps 3))
5.0ms
(* 2/3 (pow eps 3))

prune535.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize31.0ms

Local error

Found 4 expressions with local error:

6.2b
(log (* (cbrt (- 1 eps)) (cbrt (- 1 eps))))
6.1b
(log (/ (cbrt (- 1 eps)) (+ 1 eps)))
0.0b
(* (cbrt (- 1 eps)) (cbrt (- 1 eps)))
0.0b
(+ (log (* (cbrt (- 1 eps)) (cbrt (- 1 eps)))) (log (/ (cbrt (- 1 eps)) (+ 1 eps))))

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
50×log-prod
39×*-un-lft-identity
36×times-frac
36×cbrt-div
34×add-cube-cbrt
34×add-sqr-sqrt
25×pow1
24×associate-+r+
24×cbrt-prod
18×flip--
18×flip3--
18×log-div
15×log-pow
12×add-exp-log
12×frac-times
8×associate-+l-
8×pow1/3
6×distribute-lft-out
6×associate-*r/
6×associate-*l/
6×pow-prod-down
6×associate-*l*
6×associate-*r*
4×add-log-exp
4×pow-prod-up
4×associate-/r/
4×rem-log-exp
4×add-cbrt-cube
3×flip-+
3×flip3-+
2×div-inv
2×prod-exp
2×pow-plus
2×pow2
1×cbrt-unprod
1×*-commutative
1×div-exp
1×sum-log
1×associate-+r-
1×associate-+l+
1×+-commutative
Counts
4 → 147
Calls
4 calls:
Slowest
17.0ms
(+ (log (* (cbrt (- 1 eps)) (cbrt (- 1 eps)))) (log (/ (cbrt (- 1 eps)) (+ 1 eps))))
11.0ms
(log (* (cbrt (- 1 eps)) (cbrt (- 1 eps))))
7.0ms
(* (cbrt (- 1 eps)) (cbrt (- 1 eps)))
3.0ms
(log (/ (cbrt (- 1 eps)) (+ 1 eps)))

series603.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
405.0ms
(+ (log (* (cbrt (- 1 eps)) (cbrt (- 1 eps)))) (log (/ (cbrt (- 1 eps)) (+ 1 eps))))
80.0ms
(log (/ (cbrt (- 1 eps)) (+ 1 eps)))
73.0ms
(log (* (cbrt (- 1 eps)) (cbrt (- 1 eps))))
45.0ms
(* (cbrt (- 1 eps)) (cbrt (- 1 eps)))

simplify3.8s

Counts
126 → 159
Calls
126 calls:
Slowest
446.0ms
(- (+ (log (/ -1 eps)) (+ (log (* -1 (pow (* eps -1) 1/3))) (log (pow (/ -1 eps) -2/3)))) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
279.0ms
(- (+ (* 1/3 (/ 1 (pow eps 2))) (+ (* 2/3 (log (/ 1 eps))) (* 1/3 (log -1)))) (* 4/3 (/ 1 eps)))
225.0ms
(- (+ (log (pow (/ 1 eps) -2/3)) (+ (* 2/3 (log (/ 1 eps))) (* 1/3 (log -1)))) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
204.0ms
(- (+ (* 1/3 (/ 1 (pow eps 2))) (+ (log (/ -1 eps)) (log (* -1 (pow (* eps -1) 1/3))))) (* 4/3 (/ 1 eps)))
201.0ms
(- (+ (* 1/3 (pow eps 2)) (+ (* 2/9 (pow eps 3)) (* 2/3 eps))))

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes24.0ms

Accuracy

0% (0.2b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch9.0ms

end0.0ms

sample5.6s

Algorithm
intervals
Results
15997×(pre true 80)
7996×(body nan 80)
3361×(body real 1280)
2520×(body real 640)
1258×(body real 320)
626×(body real 160)
236×(body real 80)