Average Error: 58.7 → 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

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  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 2019010 +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

sample167.0ms

Algorithm
intervals

simplify10.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
10.0ms
(log (/ (- 1 eps) (+ 1 eps)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.5b

localize21.0ms

Local error

Found 2 expressions with local error:

5.4b
(log (/ (- 1 eps) (+ 1 eps)))
0.0b
(/ (- 1 eps) (+ 1 eps))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
15×log-prod
7×add-exp-log
4×associate-/r/
4×add-cbrt-cube
3×associate-/l*
3×associate-/r*
3×pow1
2×associate-/l/
2×add-log-exp
2×div-inv
2×log1p-expm1-u
2×flip-+
2×rem-log-exp
2×div-exp
2×flip3-+
2×expm1-log1p-u
1×flip--
1×log-pow
1×div-sub
1×flip3--
1×frac-2neg
1×clear-num
1×log-div
1×cbrt-undiv
Counts
2 → 62
Calls
2 calls:
Slowest
5.0ms
(/ (- 1 eps) (+ 1 eps))
3.0ms
(log (/ (- 1 eps) (+ 1 eps)))

series116.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
100.0ms
(log (/ (- 1 eps) (+ 1 eps)))
17.0ms
(/ (- 1 eps) (+ 1 eps))

simplify2.0s

Counts
54 → 68
Calls
54 calls:
Slowest
353.0ms
(/ (* (* (- 1 eps) (- 1 eps)) (- 1 eps)) (* (* (+ 1 eps) (+ 1 eps)) (+ 1 eps)))
273.0ms
(- (+ (* 2/3 (pow eps 3)) (+ (* 2/5 (pow eps 5)) (* 2 eps))))
212.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
203.0ms
(- (log -1) (+ (* 2 (/ 1 eps)) (* 2/3 (/ 1 (pow eps 3)))))
164.0ms
(- (* 2 (/ 1 eps)) (+ (* 2 (/ 1 (pow eps 2))) 1))

prune554.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize32.0ms

Local error

Found 1 expressions with local error:

0.1b
(* (pow eps 5) -2/5)

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
3×associate-*l*
2×add-cube-cbrt
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×log1p-expm1-u
1×*-commutative
1×add-exp-log
1×add-cbrt-cube
1×pow1
1×expm1-log1p-u
Counts
1 → 13
Calls
1 calls:
Slowest
4.0ms
(* (pow eps 5) -2/5)

series41.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
41.0ms
(* (pow eps 5) -2/5)

simplify16.0ms

Counts
5 → 16
Calls
5 calls:
Slowest
4.0ms
(* -2/5 (pow eps 5))
4.0ms
(* -2/5 (pow eps 5))
4.0ms
(* -2/5 (pow eps 5))
2.0ms
(* (cbrt (pow eps 5)) (cbrt (pow eps 5)))
1.0ms
(sqrt (pow eps 5))

prune126.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize26.0ms

Local error

Found 4 expressions with local error:

5.5b
(log (- 1 eps))
5.5b
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
0.0b
(- (log (- 1 eps)) (log1p (pow eps 3)))
0.0b
(log1p (pow eps 3))

rewrite20.0ms

Algorithm
rewrite-expression-head
Rules
19×*-un-lft-identity
17×add-sqr-sqrt
15×add-cube-cbrt
12×prod-diff
11×pow1
10×log-prod
7×log-pow
6×add-exp-log
6×log-div
5×add-log-exp
5×expm1-log1p-u
4×log1p-expm1-u
4×fma-neg
4×add-cbrt-cube
3×flip--
3×associate--l+
3×flip3--
2×distribute-lft-out--
2×rem-log-exp
2×diff-log
2×log1p-def
2×sub-neg
2×log1p-udef
2×associate--l-
1×difference-of-squares
1×distribute-lft-out
1×flip-+
1×log1p-expm1
1×flip3-+
Counts
4 → 85
Calls
4 calls:
Slowest
11.0ms
(- (log (- 1 eps)) (log1p (pow eps 3)))
6.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))
1.0ms
(log (- 1 eps))
1.0ms
(log1p (pow eps 3))

series502.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
372.0ms
(log1p (pow eps 3))
55.0ms
(- (log (- 1 eps)) (log1p (pow eps 3)))
40.0ms
(log (- 1 eps))
34.0ms
(log (+ (* 1 1) (- (* eps eps) (* 1 eps))))

simplify4.1s

Counts
65 → 97
Calls
65 calls:
Slowest
490.0ms
(log (- (* (* 1 1) (* 1 1)) (* (- (* eps eps) (* 1 eps)) (- (* eps eps) (* 1 eps)))))
304.0ms
(- (* 2 (log (/ -1 eps))) (+ (/ 1 eps) (+ (* 1/2 (/ 1 (pow eps 2))) (log -1))))
291.0ms
(- (+ (* 2 (log (/ 1 eps))) (log -1)) (+ (/ 1 eps) (* 1/2 (/ 1 (pow eps 2)))))
269.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 4/3 (pow eps 3)) eps)))
255.0ms
(- (+ (* 1/2 (pow eps 2)) (+ (* 1/3 (pow eps 3)) eps)))

prune926.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 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

sample6.3s

Algorithm
intervals