Average Error: 61.0 → 0.4
Time: 20.3s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(-1 - x\right) - x \cdot \left(\frac{1}{2} \cdot x\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.0

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019002 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))

Details

Time bar (total: 19.9s)Debug log

sample305.0ms

Algorithm
intervals

simplify7.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 61.0b

localize22.0ms

Local error

Found 3 expressions with local error:

6.4b
(log (+ 1 x))
6.4b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log (+ 1 x)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
3 → 66
Calls

3 calls. Slowest were:

28.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
1.0ms
(log (+ 1 x))
1.0ms
(log (- 1 x))

series170.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

96.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
38.0ms
(log (+ 1 x))
35.0ms
(log (- 1 x))

simplify1.6s

Counts
63 → 75
Calls

63 calls. Slowest were:

219.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
188.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
167.0ms
(- (+ (/ (log (/ -1 x)) (* x (pow (- (log -1) (log (/ -1 x))) 2))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (+ (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))) (+ (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (/ 1 (* x (- (log -1) (log (/ -1 x))))) (+ (* 1/2 (/ (log (/ -1 x)) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))) (* 1/2 (/ 1 (* (pow x 2) (- (log -1) (log (/ -1 x)))))))))))

prune610.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize12.0ms

Local error

Found 1 expressions with local error:

0.0b
(- (- -1 x) (* x (* x 1/2)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 15
Calls

1 calls. Slowest were:

4.0ms
(- (- -1 x) (* x (* x 1/2)))

series10.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

10.0ms
(- (- -1 x) (* x (* x 1/2)))

simplify479.0ms

Counts
4 → 18
Calls

4 calls. Slowest were:

202.0ms
(/ (exp (- -1 x)) (exp (* x (* x 1/2))))
104.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
101.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune108.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize24.0ms

Local error

Found 4 expressions with local error:

6.4b
(log (+ 1 x))
6.4b
(log (+ 1 x))
3.4b
(log (- 1 (* x x)))
0.0b
(/ (- (log (- 1 (* x x))) (log (+ 1 x))) (log (+ 1 x)))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

26.0ms
(/ (- (log (- 1 (* x x))) (log (+ 1 x))) (log (+ 1 x)))
1.0ms
(log (+ 1 x))
1.0ms
(log (+ 1 x))

series238.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

90.0ms
(/ (- (log (- 1 (* x x))) (log (+ 1 x))) (log (+ 1 x)))
83.0ms
(log (- 1 (* x x)))
38.0ms
(log (+ 1 x))
26.0ms
(log (+ 1 x))

simplify3.2s

Counts
106 → 113
Calls

106 calls. Slowest were:

374.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
286.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
208.0ms
(- (log -1) (+ (* 2 (log (/ 1 x))) (+ (* 1/2 (/ 1 (pow x 4))) (/ 1 (pow x 2)))))

prune664.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize6.0ms

Local error

Found 4 expressions with local error:

6.4b
(log (+ 1 x))
6.4b
(log (- 1 x))
0.0b
(/ 1 (/ (log (+ 1 x)) (log (- 1 x))))
0.0b
(/ (log (+ 1 x)) (log (- 1 x)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 101
Calls

4 calls. Slowest were:

5.0ms
(/ (log (+ 1 x)) (log (- 1 x)))
2.0ms
(/ 1 (/ (log (+ 1 x)) (log (- 1 x))))
1.0ms
(log (+ 1 x))

series220.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(/ 1 (/ (log (+ 1 x)) (log (- 1 x))))
71.0ms
(/ (log (+ 1 x)) (log (- 1 x)))
36.0ms
(log (+ 1 x))
29.0ms
(log (- 1 x))

simplify2.7s

Counts
88 → 113
Calls

88 calls. Slowest were:

280.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
274.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
263.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune716.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes28.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample8.7s

Algorithm
intervals