Average Error: 61.0 → 0.0
Time: 19.7s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 61.0

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

  2. Simplified60.1

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

  4. Applied sub-neg60.1

    \[\leadsto \frac{\log \color{blue}{\left(1 + \left(-x\right)\right)}}{\log_* (1 + x)}\]

  5. Applied log1p-def0.0

    \[\leadsto \frac{\color{blue}{\log_* (1 + \left(-x\right))}}{\log_* (1 + x)}\]

  6. Final simplification0.0

    \[\leadsto \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(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: 19.3s)Debug log

sample243.0ms

Algorithm
intervals

simplify5.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune10.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 59.9b

localize10.0ms

Local error

Found 2 expressions with local error:

5.5b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log1p x))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
2 → 53
Calls

2 calls. Slowest were:

13.0ms
(log (- 1 x))
6.0ms
(/ (log (- 1 x)) (log1p x))

series112.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

85.0ms
(/ (log (- 1 x)) (log1p x))
27.0ms
(log (- 1 x))

simplify1.9s

Counts
47 → 59
Calls

47 calls. Slowest were:

336.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
305.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
252.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune257.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 1 expressions with local error:

0.0b
(/ (log1p (- x)) (log1p x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

5.0ms
(/ (log1p (- x)) (log1p x))

series63.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

63.0ms
(/ (log1p (- x)) (log1p x))

simplify793.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

247.0ms
(- (+ (/ (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)))))))))))
185.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
183.0ms
(- (+ (* 2 (/ 1 (* (pow (log (/ 1 x)) 2) (pow x 2)))) (+ (* 2 (/ 1 (* (log (/ 1 x)) x))) (+ (* 1/2 (/ (log -1) (* (pow (log (/ 1 x)) 2) (pow x 2)))) 1))) (+ (/ (log -1) (* (pow (log (/ 1 x)) 3) (pow x 2))) (+ (/ (log -1) (log (/ 1 x))) (/ (log -1) (* (pow (log (/ 1 x)) 2) x)))))

prune259.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 2 expressions with local error:

0.0b
(/ 1 (/ (log1p x) (log1p (- x))))
0.0b
(/ (log1p x) (log1p (- x)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 59
Calls

2 calls. Slowest were:

3.0ms
(/ (log1p x) (log1p (- x)))
2.0ms
(/ 1 (/ (log1p x) (log1p (- x))))

series152.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

80.0ms
(/ (log1p x) (log1p (- x)))
72.0ms
(/ 1 (/ (log1p x) (log1p (- x))))

simplify1.2s

Counts
46 → 65
Calls

46 calls. Slowest were:

202.0ms
(- (+ (* 1/2 (/ (log -1) (* (pow x 2) (pow (log (/ -1 x)) 2)))) (+ (* 2 (/ 1 (* (pow x 2) (pow (log (/ -1 x)) 2)))) (+ (/ (log -1) (* x (pow (log (/ -1 x)) 2))) 1))) (+ (/ (log -1) (* (pow x 2) (pow (log (/ -1 x)) 3))) (+ (/ (log -1) (log (/ -1 x))) (* 2 (/ 1 (* x (log (/ -1 x))))))))
193.0ms
(- (+ (* 2 (/ 1 (* (pow (log (/ 1 x)) 2) (pow x 2)))) (+ (* 2 (/ 1 (* (log (/ 1 x)) x))) (+ (* 1/2 (/ (log -1) (* (pow (log (/ 1 x)) 2) (pow x 2)))) 1))) (+ (/ (log -1) (* (pow (log (/ 1 x)) 3) (pow x 2))) (+ (/ (log -1) (log (/ 1 x))) (/ (log -1) (* (pow (log (/ 1 x)) 2) x)))))
193.0ms
(- (+ (/ (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)))))))))))

prune470.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize14.0ms

Local error

Found 3 expressions with local error:

0.0b
(/ 1 (log1p (- x)))
0.0b
(/ 1 (log1p x))
0.0b
(/ (/ 1 (log1p x)) (/ 1 (log1p (- x))))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 79
Calls

3 calls. Slowest were:

4.0ms
(/ (/ 1 (log1p x)) (/ 1 (log1p (- x))))
1.0ms
(/ 1 (log1p (- x)))
1.0ms
(/ 1 (log1p x))

series167.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

91.0ms
(/ (/ 1 (log1p x)) (/ 1 (log1p (- x))))
43.0ms
(/ 1 (log1p (- x)))
32.0ms
(/ 1 (log1p x))

simplify3.9s

Counts
61 → 88
Calls

61 calls. Slowest were:

1.3s
(/ (* (* (/ 1 (log1p x)) (/ 1 (log1p x))) (/ 1 (log1p x))) (* (* (/ 1 (log1p (- x))) (/ 1 (log1p (- x)))) (/ 1 (log1p (- x)))))
549.0ms
(- (+ (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (/ 1 (- (log -1) (log (/ -1 x)))) (* 1/2 (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2)))))) (/ 1 (* x (pow (- (log -1) (log (/ -1 x))) 2))))
283.0ms
(+ (/ 1 (- (log -1) (log (/ 1 x)))) (+ (/ 1 (* x (pow (- (log -1) (log (/ 1 x))) 2))) (+ (* 1/2 (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ 1 x))) 2)))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ 1 x))) 3))))))

prune585.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes39.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.1s

Algorithm
intervals