Average Error: 58.5 → 0.2
Time: 21.3s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\left((\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon + -2)_*\right) \cdot \varepsilon + \left(\frac{-2}{5} \cdot {\varepsilon}^{5}\right))_*\]

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  2. Initial simplification58.5

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(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: 11.2s)Debug log

start305.0ms

Algorithm
intervals

setup35.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.1b

localize20.0ms

Local error

Found 2 expressions with local error:

5.7b
(log (/ (- 1 eps) (+ eps 1)))
0.0b
(/ (- 1 eps) (+ eps 1))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 62
Calls

2 calls. Slowest were:

3.0ms
(/ (- 1 eps) (+ eps 1))
2.0ms
(log (/ (- 1 eps) (+ eps 1)))

series97.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

80.0ms
(log (/ (- 1 eps) (+ eps 1)))
17.0ms
(/ (- 1 eps) (+ eps 1))

simplify2.0s

Counts
54 → 68
Calls

54 calls. Slowest were:

427.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ eps 1) (+ eps 1)) (+ eps 1)))
314.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
165.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune606.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize32.0ms

Local error

Found 3 expressions with local error:

0.3b
(* -2/3 eps)
0.1b
(* (pow eps 5) -2/5)
0.0b
(fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 33
Calls

3 calls. Slowest were:

4.0ms
(* (pow eps 5) -2/5)
0.0ms
(* -2/3 eps)
0.0ms
(fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))

series89.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

41.0ms
(fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))
36.0ms
(* (pow eps 5) -2/5)
12.0ms
(* -2/3 eps)

simplify708.0ms

Counts
11 → 42
Calls

11 calls. Slowest were:

271.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
207.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
201.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune594.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

39.8b
(cbrt (* (* (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5)) (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))) (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))))
0.3b
(* -2/3 eps)
0.3b
(* -2/3 eps)
0.3b
(* -2/3 eps)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 42
Calls

4 calls. Slowest were:

6.0ms
(cbrt (* (* (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5)) (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))) (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))))
0.0ms
(* -2/3 eps)
0.0ms
(* -2/3 eps)

series49.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

26.0ms
(cbrt (* (* (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5)) (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))) (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))))
10.0ms
(* -2/3 eps)
7.0ms
(* -2/3 eps)
7.0ms
(* -2/3 eps)

simplify1.1s

Counts
14 → 54
Calls

14 calls. Slowest were:

302.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
253.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
245.0ms
(cbrt (* (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5)) (fma (fma (* -2/3 eps) eps -2) eps (* (pow eps 5) -2/5))))

prune981.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 3 expressions with local error:

5.8b
(log (- 1 eps))
5.8b
(log (+ eps 1))
0.0b
(- (log (- 1 eps)) (log (+ eps 1)))

rewrite18.0ms

Algorithm
rewrite-expression-head
Counts
3 → 83
Calls

3 calls. Slowest were:

13.0ms
(- (log (- 1 eps)) (log (+ eps 1)))
2.0ms
(log (- 1 eps))
2.0ms
(log (+ eps 1))

series189.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

132.0ms
(- (log (- 1 eps)) (log (+ eps 1)))
35.0ms
(log (- 1 eps))
22.0ms
(log (+ eps 1))

simplify3.7s

Counts
74 → 92
Calls

74 calls. Slowest were:

733.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 1/3 (pow eps 3)) eps)))
543.0ms
(- (log -1) (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ 1 eps)))))
386.0ms
(- (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps)))))

prune534.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

regimes48.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms