Average Error: 0.0 → 0.0
Time: 12.6s
Precision: 64
Internal Precision: 128
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\frac{1}{x} - 1\right) \cdot \frac{1}{2}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[-\log \left(\frac{1}{x} - 1\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

    \[\leadsto -\log \color{blue}{\left(\sqrt{\frac{1}{x} - 1} \cdot \sqrt{\frac{1}{x} - 1}\right)}\]

  4. Applied log-prod0.0

    \[\leadsto -\color{blue}{\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)}\]
  5. Using strategy rm

  6. Applied pow1/20.0

    \[\leadsto -\left(\log \color{blue}{\left({\left(\frac{1}{x} - 1\right)}^{\frac{1}{2}}\right)} + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)\]

  7. Applied log-pow0.0

    \[\leadsto -\left(\color{blue}{\frac{1}{2} \cdot \log \left(\frac{1}{x} - 1\right)} + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)\]

  8. Final simplification0.0

    \[\leadsto -\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\frac{1}{x} - 1\right) \cdot \frac{1}{2}\right)\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x)
  :name "neg log"
  (- (log (- (/ 1 x) 1))))

Details

Time bar (total: 12.2s)Debug log

sample212.0ms

Algorithm
intervals

simplify17.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

16.0ms
(- (log (- (/ 1 x) 1)))

prune39.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 2 expressions with local error:

0.0b
(log (- (/ 1 x) 1))
0.0b
(- (/ 1 x) 1)

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
2 → 36
Calls

2 calls. Slowest were:

7.0ms
(log (- (/ 1 x) 1))
6.0ms
(- (/ 1 x) 1)

series86.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

50.0ms
(log (- (/ 1 x) 1))
36.0ms
(- (/ 1 x) 1)

simplify395.0ms

Counts
17 → 42
Calls

17 calls. Slowest were:

122.0ms
(- (+ x (+ (* 1/2 (pow x 2)) (log x))))
71.0ms
(- (log -1) (+ (/ 1 x) (* 1/2 (/ 1 (pow x 2)))))
68.0ms
(- (log -1) (+ (/ 1 x) (* 1/2 (/ 1 (pow x 2)))))

prune251.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize1.0ms

Local error

Found 2 expressions with local error:

0.0b
(log (- (/ 1 x) 1))
0.0b
(- (/ 1 x) 1)

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
2 → 36
Calls

2 calls. Slowest were:

7.0ms
(log (- (/ 1 x) 1))
6.0ms
(- (/ 1 x) 1)

series38.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

30.0ms
(log (- (/ 1 x) 1))
8.0ms
(- (/ 1 x) 1)

simplify550.0ms

Counts
17 → 42
Calls

17 calls. Slowest were:

154.0ms
(- (+ x (+ (* 1/2 (pow x 2)) (log x))))
133.0ms
(- (log -1) (+ (/ 1 x) (* 1/2 (/ 1 (pow x 2)))))
116.0ms
(- (log -1) (+ (/ 1 x) (* 1/2 (/ 1 (pow x 2)))))

prune295.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 4 expressions with local error:

0.2b
(sqrt (- (/ 1 x) 1))
0.2b
(sqrt (- (/ 1 x) 1))
0.0b
(log (sqrt (- (/ 1 x) 1)))
0.0b
(log (sqrt (- (/ 1 x) 1)))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

6.0ms
(sqrt (- (/ 1 x) 1))
4.0ms
(sqrt (- (/ 1 x) 1))
2.0ms
(log (sqrt (- (/ 1 x) 1)))

series121.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

40.0ms
(log (sqrt (- (/ 1 x) 1)))
38.0ms
(log (sqrt (- (/ 1 x) 1)))
28.0ms
(sqrt (- (/ 1 x) 1))
15.0ms
(sqrt (- (/ 1 x) 1))

simplify2.7s

Counts
46 → 88
Calls

46 calls. Slowest were:

325.0ms
(- (+ (log (sqrt -1)) (* 1/2 (/ 1 (* x (pow (sqrt -1) 2))))) (* 1/4 (/ 1 (* (pow x 2) (pow (sqrt -1) 4)))))
318.0ms
(- (+ (log (sqrt -1)) (* 1/2 (/ 1 (* x (pow (sqrt -1) 2))))) (* 1/4 (/ 1 (* (pow x 2) (pow (sqrt -1) 4)))))
317.0ms
(- (+ (log (sqrt -1)) (* 1/2 (/ 1 (* x (pow (sqrt -1) 2))))) (* 1/4 (/ 1 (* (pow x 2) (pow (sqrt -1) 4)))))

prune813.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

0.2b
(sqrt (- (/ 1 x) 1))
0.0b
(log (sqrt (- (/ 1 x) 1)))
0.0b
(log (- (/ 1 x) 1))
0.0b
(- (/ 1 x) 1)

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
Calls

4 calls. Slowest were:

8.0ms
(log (sqrt (- (/ 1 x) 1)))
6.0ms
(sqrt (- (/ 1 x) 1))
5.0ms
(- (/ 1 x) 1)

series123.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

56.0ms
(log (sqrt (- (/ 1 x) 1)))
31.0ms
(log (- (/ 1 x) 1))
21.0ms
(sqrt (- (/ 1 x) 1))
14.0ms
(- (/ 1 x) 1)

simplify1.9s

Counts
40 → 86
Calls

40 calls. Slowest were:

360.0ms
(- (+ (log (sqrt -1)) (* 1/2 (/ 1 (* x (pow (sqrt -1) 2))))) (* 1/4 (/ 1 (* (pow x 2) (pow (sqrt -1) 4)))))
308.0ms
(- (+ (log (sqrt -1)) (* 1/2 (/ 1 (* x (pow (sqrt -1) 2))))) (* 1/4 (/ 1 (* (pow x 2) (pow (sqrt -1) 4)))))
242.0ms
(- (+ (* 1/2 (/ 1 (* x (sqrt -1)))) (sqrt -1)) (* 1/8 (/ 1 (* (pow x 2) (pow (sqrt -1) 3)))))

prune938.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes145.0ms

Accuracy

0% (0.0b remaining)

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

bsearch7.0ms

end0.0ms

sample3.4s

Algorithm
intervals