Average Error: 58.5 → 0.2
Time: 20.8s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\varepsilon \cdot \left((\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2)_*\right) + \left(\frac{-2}{5} \cdot {\varepsilon}^{5}\right))_*\]

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. 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. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019005 +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: 20.4s)Debug log

sample182.0ms

Algorithm
intervals

simplify19.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

19.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.5b

localize19.0ms

Local error

Found 2 expressions with local error:

6.5b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
2 → 62
Calls

2 calls. Slowest were:

5.0ms
(/ (- 1 eps) (+ 1 eps))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series102.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

92.0ms
(log (/ (- 1 eps) (+ 1 eps)))
10.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.8s

Counts
54 → 68
Calls

54 calls. Slowest were:

241.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
227.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
182.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune508.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 2 expressions with local error:

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

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
2 → 23
Calls

2 calls. Slowest were:

2.0ms
(* (pow eps 5) -2/5)
0.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))

series46.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

26.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
19.0ms
(* (pow eps 5) -2/5)

simplify774.0ms

Counts
8 → 29
Calls

8 calls. Slowest were:

277.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
250.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
239.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune274.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

6.5b
(log (/ (- 1 eps) (+ 1 eps)))
6.5b
(log (/ (- 1 eps) (+ 1 eps)))
6.5b
(log (/ (- 1 eps) (+ 1 eps)))
0.1b
(cbrt (log (/ (- 1 eps) (+ 1 eps))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

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

series424.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

152.0ms
(cbrt (log (/ (- 1 eps) (+ 1 eps))))
104.0ms
(log (/ (- 1 eps) (+ 1 eps)))
92.0ms
(log (/ (- 1 eps) (+ 1 eps)))
76.0ms
(log (/ (- 1 eps) (+ 1 eps)))

simplify3.6s

Counts
70 → 110
Calls

70 calls. Slowest were:

338.0ms
(+ (* 22/405 (* (pow eps 4) (exp (* 1/3 (+ (log -2) (log eps)))))) (+ (* 1/9 (* (pow eps 2) (exp (* 1/3 (+ (log -2) (log eps)))))) (exp (* 1/3 (+ (log -2) (log eps))))))
284.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
259.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune973.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

6.5b
(log (+ 1 eps))
6.5b
(log (/ (- 1 eps) (+ 1 eps)))
6.5b
(log (/ (- 1 eps) (+ 1 eps)))
6.3b
(log (- 1 eps))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

2.0ms
(log (/ (- 1 eps) (+ 1 eps)))
2.0ms
(log (/ (- 1 eps) (+ 1 eps)))
1.0ms
(log (+ 1 eps))

series216.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

77.0ms
(log (/ (- 1 eps) (+ 1 eps)))
76.0ms
(log (/ (- 1 eps) (+ 1 eps)))
37.0ms
(log (+ 1 eps))
25.0ms
(log (- 1 eps))

simplify3.8s

Counts
62 → 103
Calls

62 calls. Slowest were:

334.0ms
(- (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps)))))
333.0ms
(- (+ (/ 1 eps) (log -1)) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps))))
320.0ms
(- (log -1) (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ 1 eps)))))

prune1.3s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes37.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample6.3s

Algorithm
intervals