Average Error: 58.6 → 0.2
Time: 25.7s
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 1042149663 
(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: 24.9s)Debug log

sample293.0ms

Algorithm
intervals
Results
531×(pre true 80)
275×(body nan 80)
110×(body real 1280)
77×(body real 640)
43×(body real 320)
20×(body real 160)
6×(body real 80)

simplify23.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
22.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.3b

localize29.0ms

Local error

Found 2 expressions with local error:

5.9b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite28.0ms

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
23.0ms
(/ (- 1 eps) (+ 1 eps))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series136.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
116.0ms
(log (/ (- 1 eps) (+ 1 eps)))
20.0ms
(/ (- 1 eps) (+ 1 eps))

simplify2.3s

Counts
54 → 64
Calls
54 calls:
Slowest
433.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
228.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
211.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
208.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
143.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune803.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize58.0ms

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

series78.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
40.0ms
(* 2/5 (pow eps 5))
37.0ms
(* 2/3 (pow eps 3))

simplify68.0ms

Counts
13 → 30
Calls
13 calls:
Slowest
12.0ms
(* 2/3 (pow eps 3))
12.0ms
(* 2/3 (pow eps 3))
12.0ms
(* 2/3 (pow eps 3))
4.0ms
(cbrt (pow eps 3))
4.0ms
(sqrt (pow eps 3))

prune482.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 4 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
60×sqrt-prod
40×add-sqr-sqrt
38×times-frac
36×log-prod
34×add-cube-cbrt
34×*-un-lft-identity
8×associate-/r/
6×add-exp-log
6×pow1
4×add-log-exp
4×div-inv
4×flip-+
4×log-pow
4×sqrt-div
4×pow1/2
4×add-cbrt-cube
4×flip3-+
4×rem-sqrt-square
2×rem-log-exp
2×log-div
Counts
4 → 110
Calls
4 calls:
Slowest
3.0ms
(log (sqrt (/ (- 1 eps) (+ 1 eps))))
3.0ms
(log (sqrt (/ (- 1 eps) (+ 1 eps))))
3.0ms
(sqrt (/ (- 1 eps) (+ 1 eps)))
3.0ms
(sqrt (/ (- 1 eps) (+ 1 eps)))

series392.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
149.0ms
(log (sqrt (/ (- 1 eps) (+ 1 eps))))
147.0ms
(log (sqrt (/ (- 1 eps) (+ 1 eps))))
48.0ms
(sqrt (/ (- 1 eps) (+ 1 eps)))
47.0ms
(sqrt (/ (- 1 eps) (+ 1 eps)))

simplify5.5s

Counts
88 → 122
Calls
88 calls:
Slowest
499.0ms
(- (+ (/ 1 (* (sqrt -1) eps)) (sqrt -1)) (+ (* 1/2 (/ 1 (* (pow (sqrt -1) 3) (pow eps 2)))) (/ 1 (* (sqrt -1) (pow eps 2)))))
498.0ms
(- (+ (/ 1 (* (sqrt -1) eps)) (sqrt -1)) (+ (* 1/2 (/ 1 (* (pow (sqrt -1) 3) (pow eps 2)))) (/ 1 (* (sqrt -1) (pow eps 2)))))
498.0ms
(- (+ (/ 1 (* (sqrt -1) eps)) (sqrt -1)) (+ (* 1/2 (/ 1 (* (pow (sqrt -1) 3) (pow eps 2)))) (/ 1 (* (sqrt -1) (pow eps 2)))))
498.0ms
(- (+ (/ 1 (* (sqrt -1) eps)) (sqrt -1)) (+ (* 1/2 (/ 1 (* (pow (sqrt -1) 3) (pow eps 2)))) (/ 1 (* (sqrt -1) (pow eps 2)))))
442.0ms
(- (+ (* 1/2 (pow eps 2)) 1) eps)

prune1.4s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 3 expressions with local error:

5.9b
(log (/ (- 1 eps) (+ 1 eps)))
0.2b
(exp (log (log (/ (- 1 eps) (+ 1 eps)))))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
20×add-cube-cbrt
20×*-un-lft-identity
20×add-sqr-sqrt
19×log-prod
18×times-frac
8×add-exp-log
6×pow1
5×add-cbrt-cube
4×associate-/r/
4×exp-sum
4×exp-prod
3×add-log-exp
3×associate-/l*
3×log-pow
3×associate-/r*
2×associate-/l/
2×div-inv
2×flip-+
2×rem-log-exp
2×div-exp
2×flip3-+
1×rem-exp-log
1×flip--
1×div-sub
1×flip3--
1×frac-2neg
1×clear-num
1×log-div
1×cbrt-undiv
Counts
3 → 74
Calls
3 calls:
Slowest
6.0ms
(/ (- 1 eps) (+ 1 eps))
5.0ms
(exp (log (log (/ (- 1 eps) (+ 1 eps)))))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series252.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
118.0ms
(log (/ (- 1 eps) (+ 1 eps)))
113.0ms
(exp (log (log (/ (- 1 eps) (+ 1 eps)))))
20.0ms
(/ (- 1 eps) (+ 1 eps))

simplify3.1s

Counts
66 → 83
Calls
66 calls:
Slowest
426.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
231.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
230.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
213.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
211.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune851.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.0b

regimes163.0ms

Accuracy

0% (0.2b remaining)

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

bsearch14.0ms

end0.0ms

sample8.8s

Algorithm
intervals
Results
16069×(pre true 80)
8067×(body nan 80)
3475×(body real 1280)
2476×(body real 640)
1242×(body real 320)
602×(body real 160)
207×(body real 80)