Average Error: 61.0 → 0.4
Time: 23.5s
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 2019004 
(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: 22.9s)Debug log

sample218.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.8b

localize23.0ms

Local error

Found 3 expressions with local error:

4.9b
(log (- 1 x))
4.9b
(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:

9.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
2.0ms
(log (- 1 x))
1.0ms
(log (+ 1 x))

series162.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

91.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
36.0ms
(log (- 1 x))
33.0ms
(log (+ 1 x))

simplify2.0s

Counts
63 → 75
Calls

63 calls. Slowest were:

235.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
228.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
187.0ms
(- (+ (/ (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)))))))))))

prune697.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

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)))

simplify428.0ms

Counts
4 → 18
Calls

4 calls. Slowest were:

175.0ms
(/ (exp (- -1 x)) (exp (* x (* x 1/2))))
90.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
81.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune134.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 4 expressions with local error:

4.9b
(log (+ 1 x))
4.9b
(log (+ 1 x))
4.9b
(log (+ 1 x))
3.3b
(log (- 1 (* x x)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

2.0ms
(log (- 1 (* x x)))
2.0ms
(log (+ 1 x))
2.0ms
(log (+ 1 x))

series191.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

110.0ms
(log (- 1 (* x x)))
29.0ms
(log (+ 1 x))
27.0ms
(log (+ 1 x))
25.0ms
(log (+ 1 x))

simplify2.3s

Counts
36 → 68
Calls

36 calls. Slowest were:

312.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
301.0ms
(- (log -1) (+ (* 2 (log (/ 1 x))) (+ (* 1/2 (/ 1 (pow x 4))) (/ 1 (pow x 2)))))
285.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))

prune447.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

30.5b
(/ (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x)))
4.9b
(log (+ 1 x))
4.9b
(log (+ 1 x))
4.9b
(log (+ 1 x))

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

27.0ms
(/ (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x)))
1.0ms
(log (+ 1 x))
1.0ms
(log (+ 1 x))

series141.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

49.0ms
(/ (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x)))
38.0ms
(log (+ 1 x))
27.0ms
(log (+ 1 x))
27.0ms
(log (+ 1 x))

simplify5.7s

Counts
65 → 88
Calls

65 calls. Slowest were:

842.0ms
(/ (* (* (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4)))) (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4)))) (* (* (log (+ 1 x)) (log (+ 1 x))) (log (+ 1 x))))
326.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
300.0ms
(- (* 1/3 (/ (pow x 5) (pow (- (log -1) (log (/ -1 x))) 2))) (+ (* 1/3 (/ (pow x 4) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (* 1/2 (/ (pow x 4) (- (log -1) (log (/ -1 x))))) (+ (* 1/6 (/ (pow x 4) (pow (- (log -1) (log (/ -1 x))) 2))) (* 1/3 (/ (pow x 6) (- (log -1) (log (/ -1 x)))))))))

prune988.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes27.0ms

Accuracy

0% (0.4b remaining)

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

bsearch1.0ms

end0.0ms

sample9.3s

Algorithm
intervals