Average Error: 58.5 → 0.2
Time: 18.0s
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

Original58.5
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 2019007 +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: 17.3s)Debug log

sample224.0ms

Algorithm
intervals

simplify9.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

9.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.7b

localize33.0ms

Local error

Found 2 expressions with local error:

7.5b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 62
Calls

2 calls. Slowest were:

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

series76.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

66.0ms
(log (/ (- 1 eps) (+ 1 eps)))
10.0ms
(/ (- 1 eps) (+ 1 eps))

simplify1.8s

Counts
54 → 68
Calls

54 calls. Slowest were:

335.0ms
(/ (* (* (- 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))))
219.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))

prune485.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

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))

series48.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

24.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
23.0ms
(* (pow eps 5) -2/5)

simplify827.0ms

Counts
8 → 29
Calls

8 calls. Slowest were:

322.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
263.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))
228.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))

prune245.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (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))

rewrite23.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

10.0ms
(* (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))))
7.0ms
(* (pow eps 5) -2/5)
4.0ms
(* (pow eps 5) -2/5)

series121.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

40.0ms
(fma eps (fma -2/3 (* eps eps) -2) (* (pow eps 5) -2/5))
36.0ms
(* (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))))
26.0ms
(* (pow eps 5) -2/5)
19.0ms
(* (pow eps 5) -2/5)

simplify3.4s

Counts
38 → 80
Calls

38 calls. Slowest were:

577.0ms
(* (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)))
485.0ms
(* (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)))
269.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2 eps) (* 2/5 (pow eps 5)))))

prune708.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

7.5b
(log (/ 1 (+ 1 eps)))
7.5b
(log (- 1 eps))
0.0b
(/ 1 (+ 1 eps))
0.0b
(+ (log (- 1 eps)) (log (/ 1 (+ 1 eps))))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 95
Calls

4 calls. Slowest were:

6.0ms
(+ (log (- 1 eps)) (log (/ 1 (+ 1 eps))))
1.0ms
(log (/ 1 (+ 1 eps)))
1.0ms
(log (- 1 eps))

series154.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

98.0ms
(+ (log (- 1 eps)) (log (/ 1 (+ 1 eps))))
27.0ms
(log (/ 1 (+ 1 eps)))
23.0ms
(log (- 1 eps))
6.0ms
(/ 1 (+ 1 eps))

simplify2.8s

Counts
58 → 107
Calls

58 calls. Slowest were:

495.0ms
(- (* 1/2 (pow eps 2)) (+ (* 1/3 (pow eps 3)) eps))
337.0ms
(- (log -1) (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ 1 eps)))))
292.0ms
(- (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log (/ -1 eps)))))

prune389.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes27.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms

end0.0ms

sample5.9s

Algorithm
intervals