Average Error: 58.4 → 0.2
Time: 21.0s
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

Original58.4
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  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 2019008 +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

sample188.0ms

Algorithm
intervals

simplify18.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

18.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune4.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.6b

localize10.0ms

Local error

Found 2 expressions with local error:

5.2b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 62
Calls

2 calls. Slowest were:

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

series120.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

104.0ms
(log (/ (- 1 eps) (+ 1 eps)))
17.0ms
(/ (- 1 eps) (+ 1 eps))

simplify2.0s

Counts
54 → 68
Calls

54 calls. Slowest were:

361.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
244.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
217.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune546.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize26.0ms

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))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 23
Calls

2 calls. Slowest were:

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

series70.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

37.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
33.0ms
(* (pow eps 5) -2/5)

simplify833.0ms

Counts
8 → 29
Calls

8 calls. Slowest were:

297.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
269.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
246.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune259.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
0.1b
(* (pow eps 5) -2/5)
0.1b
(* (pow eps 5) -2/5)
0.0b
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

5.0ms
(* (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
2.0ms
(* (pow eps 5) -2/5)
2.0ms
(* (pow eps 5) -2/5)

series97.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

29.0ms
(* (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))) (sqrt (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
23.0ms
(* (pow eps 5) -2/5)
23.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
23.0ms
(* (pow eps 5) -2/5)

simplify3.8s

Counts
38 → 80
Calls

38 calls. Slowest were:

571.0ms
(* (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
450.0ms
(* (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)) (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5)))
348.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune891.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

5.2b
(log (/ (- 1 eps) (+ 1 eps)))
5.2b
(log (/ (- 1 eps) (+ 1 eps)))
5.2b
(log (/ (- 1 eps) (+ 1 eps)))
0.1b
(cbrt (log (/ (- 1 eps) (+ 1 eps))))

rewrite13.0ms

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)))

series345.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

145.0ms
(cbrt (log (/ (- 1 eps) (+ 1 eps))))
77.0ms
(log (/ (- 1 eps) (+ 1 eps)))
63.0ms
(log (/ (- 1 eps) (+ 1 eps)))
59.0ms
(log (/ (- 1 eps) (+ 1 eps)))

simplify3.8s

Counts
70 → 110
Calls

70 calls. Slowest were:

287.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
286.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
282.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune914.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

regimes93.0ms

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