Average Error: 58.6 → 0.2
Time: 30.0s
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 323378853 
(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: 29.1s)Debug log

sample270.0ms

Algorithm
intervals
Results
118.0ms108×body1280valid
55.0ms79×body640valid
31.0ms286×body80nan
19.0ms43×body320valid
12.0ms22×body160valid
1.0ms5×body80valid

simplify21.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
21.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.7b

localize29.0ms

Local error

Found 2 expressions with local error:

5.1b
(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
6.0ms
(/ (- 1 eps) (+ 1 eps))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series154.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
134.0ms
(log (/ (- 1 eps) (+ 1 eps)))
20.0ms
(/ (- 1 eps) (+ 1 eps))

simplify2.2s

Counts
54 → 64
Calls
54 calls:
Slowest
415.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
220.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
203.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
199.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
137.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune814.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize58.0ms

Local error

Found 3 expressions with local error:

0.2b
(* 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)))

rewrite26.0ms

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
19.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
3.0ms
(* 2/3 (pow eps 3))
3.0ms
(* 2/5 (pow eps 5))

series125.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
50.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
41.0ms
(* 2/5 (pow eps 5))
34.0ms
(* 2/3 (pow eps 3))

simplify625.0ms

Counts
19 → 46
Calls
19 calls:
Slowest
189.0ms
(* (exp (* 2/3 (pow eps 3))) (exp (+ (* 2/5 (pow eps 5)) (* 2 eps))))
186.0ms
(* (exp (* 2/3 (pow eps 3))) (* (exp (* 2/5 (pow eps 5))) (exp (* 2 eps))))
62.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps)))
61.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5))))
60.0ms
(+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5))))

prune711.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

5.1b
(log (/ (- 1 eps) (+ 1 eps)))
5.1b
(log (/ (- 1 eps) (+ 1 eps)))
5.1b
(log (/ (- 1 eps) (+ 1 eps)))
0.1b
(* (cbrt (log (/ (- 1 eps) (+ 1 eps)))) (cbrt (log (/ (- 1 eps) (+ 1 eps)))))

rewrite28.0ms

Algorithm
rewrite-expression-head
Rules
45×log-prod
29×add-cube-cbrt
29×*-un-lft-identity
29×add-sqr-sqrt
27×times-frac
15×add-exp-log
14×pow1
8×cbrt-prod
7×associate-*l*
7×associate-*r*
6×associate-/r/
6×rem-log-exp
5×log-pow
4×add-log-exp
4×pow1/3
4×add-cbrt-cube
3×div-inv
3×flip-+
3×div-exp
3×flip3-+
3×log-div
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 108
Calls
4 calls:
Slowest
14.0ms
(* (cbrt (log (/ (- 1 eps) (+ 1 eps)))) (cbrt (log (/ (- 1 eps) (+ 1 eps)))))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series610.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
261.0ms
(* (cbrt (log (/ (- 1 eps) (+ 1 eps)))) (cbrt (log (/ (- 1 eps) (+ 1 eps)))))
117.0ms
(log (/ (- 1 eps) (+ 1 eps)))
117.0ms
(log (/ (- 1 eps) (+ 1 eps)))
115.0ms
(log (/ (- 1 eps) (+ 1 eps)))

simplify4.7s

Counts
88 → 120
Calls
88 calls:
Slowest
361.0ms
(+ (exp (* 1/3 (+ (* 2 (log eps)) (log 4)))) (+ (* 2/9 (* (exp (* 1/3 (+ (* 2 (log eps)) (log 4)))) (pow eps 2))) (* 49/405 (* (exp (* 1/3 (+ (* 2 (log eps)) (log 4)))) (pow eps 4)))))
239.0ms
(- (pow (pow (log -1) 2) 1/3) (+ (* 4/3 (* (/ 1 eps) (pow (/ 1 (log -1)) 1/3))) (* 4/9 (* (/ 1 (pow eps 2)) (pow (/ 1 (pow (log -1) 4)) 1/3)))))
239.0ms
(- (pow (pow (log -1) 2) 1/3) (+ (* 4/3 (* (/ 1 eps) (pow (/ 1 (log -1)) 1/3))) (* 4/9 (* (/ 1 (pow eps 2)) (pow (/ 1 (pow (log -1) 4)) 1/3)))))
219.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
218.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune1.6s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize97.0ms

Local error

Found 4 expressions with local error:

5.1b
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
5.1b
(log (/ (- 1 eps) (+ (pow eps 3) 1)))
5.1b
(log (/ (- 1 eps) (+ 1 eps)))
5.1b
(log (/ (- 1 eps) (+ 1 eps)))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
49×log-prod
27×times-frac
27×*-un-lft-identity
26×add-cube-cbrt
26×add-sqr-sqrt
14×add-exp-log
8×pow1
7×rem-log-exp
6×associate-/r/
5×log-div
4×add-log-exp
4×flip-+
4×log-pow
4×add-cbrt-cube
4×flip3-+
3×div-inv
3×div-exp
1×distribute-lft-out
Counts
4 → 93
Calls
4 calls:
Slowest
9.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
6.0ms
(log (/ (- 1 eps) (+ (pow eps 3) 1)))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series362.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
123.0ms
(log (/ (- 1 eps) (+ 1 eps)))
118.0ms
(log (/ (- 1 eps) (+ 1 eps)))
64.0ms
(log (/ (- 1 eps) (+ (pow eps 3) 1)))
56.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))

simplify5.9s

Counts
73 → 105
Calls
73 calls:
Slowest
885.0ms
(log (+ (pow (* 1 1) 3) (pow (- (* eps eps) (* 1 eps)) 3)))
475.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 4/3 (pow eps 3)) eps)))
472.0ms
(- (+ (* 2 (log (/ 1 eps))) (log -1)) (+ (/ 1 eps) (* 1/2 (/ 1 (pow eps 2)))))
273.0ms
(log (- (* (* 1 1) (* 1 1)) (* (- (* eps eps) (* 1 eps)) (- (* eps eps) (* 1 eps)))))
269.0ms
(- (+ (* 2 (log (/ -1 eps))) (log -1)) (+ (/ 1 eps) (* 1/2 (/ 1 (pow eps 2)))))

prune2.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

regimes58.0ms

Accuracy

0% (0.2b remaining)

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

bsearch13.0ms

end0.0ms

sample8.3s

Algorithm
intervals
Results
3.9s3434×body1280valid
1.9s2553×body640valid
1.5s8077×body80nan
565.0ms1195×body320valid
246.0ms626×body160valid
26.0ms194×body80valid