Average Error: 58.6 → 0.2
Time: 15.5s
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

Original58.6
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. 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 2019002 +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: 15.0s)Debug log

sample156.0ms

Algorithm
intervals

simplify43.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

43.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.3b

localize22.0ms

Local error

Found 2 expressions with local error:

5.3b
(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)))

series118.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

95.0ms
(log (/ (- 1 eps) (+ 1 eps)))
22.0ms
(/ (- 1 eps) (+ 1 eps))

simplify2.1s

Counts
54 → 68
Calls

54 calls. Slowest were:

366.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
290.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
225.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune482.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

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))

series51.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

31.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
20.0ms
(* (pow eps 5) -2/5)

simplify854.0ms

Counts
8 → 29
Calls

8 calls. Slowest were:

342.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
273.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
225.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune318.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 3 expressions with local error:

3.0b
(exp (log (fma eps (fma -2/3 (* eps eps) -2) (* (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))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 40
Calls

3 calls. Slowest were:

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

series111.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

40.0ms
(exp (log (fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))))
36.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
35.0ms
(* (pow eps 5) -2/5)

simplify1.9s

Counts
18 → 49
Calls

18 calls. Slowest were:

341.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
297.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
284.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune540.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 2 expressions with local error:

5.3b
(log (+ 1 eps))
5.3b
(log (- 1 eps))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 35
Calls

2 calls. Slowest were:

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

series55.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

29.0ms
(log (- 1 eps))
26.0ms
(log (+ 1 eps))

simplify1.9s

Counts
20 → 41
Calls

20 calls. Slowest were:

367.0ms
(- (+ (/ 1 eps) (log -1)) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps))))
358.0ms
(- (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps)))))
309.0ms
(- (+ (* 1/3 (pow eps 3)) eps) (* 1/2 (pow eps 2)))

prune264.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes22.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms

end0.0ms

sample6.0s

Algorithm
intervals