Average Error: 61.0 → 0.4
Time: 21.7s
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 2019008 
(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: 21.3s)Debug log

sample258.0ms

Algorithm
intervals

simplify10.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

10.0ms
(/ (log (- 1 x)) (log (+ 1 x)))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 61.5b

localize26.0ms

Local error

Found 3 expressions with local error:

5.5b
(log (- 1 x))
5.4b
(log (+ 1 x))
0.0b
(/ (log (- 1 x)) (log (+ 1 x)))

rewrite14.0ms

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))

series167.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

83.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
48.0ms
(log (+ 1 x))
37.0ms
(log (- 1 x))

simplify2.2s

Counts
63 → 75
Calls

63 calls. Slowest were:

374.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
267.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
217.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune645.0ms

Pruning

2 alts after pruning (2 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)))

simplify403.0ms

Counts
4 → 18
Calls

4 calls. Slowest were:

137.0ms
(/ (exp (- -1 x)) (exp (* x (* x 1/2))))
99.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
88.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune94.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 3 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 80
Calls

3 calls. Slowest were:

6.0ms
(/ (exp (- -1 x)) (exp (* x (* x 1/2))))
3.0ms
(log (/ (exp (- -1 x)) (exp (* x (* x 1/2)))))
1.0ms
(exp (- -1 x))

series130.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

70.0ms
(log (/ (exp (- -1 x)) (exp (* x (* x 1/2)))))
45.0ms
(/ (exp (- -1 x)) (exp (* x (* x 1/2))))
15.0ms
(exp (- -1 x))

simplify4.4s

Counts
78 → 89
Calls

78 calls. Slowest were:

396.0ms
(log (/ (exp (- (+ x 1))) (exp (* 1/2 (pow x 2)))))
370.0ms
(/ (* (* (exp (- -1 x)) (exp (- -1 x))) (exp (- -1 x))) (* (* (exp (* x (* x 1/2))) (exp (* x (* x 1/2)))) (exp (* x (* x 1/2)))))
349.0ms
(- (+ (* 1/3 (* (pow x 3) (exp -1))) (exp -1)) (* x (exp -1)))

prune869.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 3 expressions with local error:

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

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
3 → 59
Calls

3 calls. Slowest were:

12.0ms
(* (exp (* x (* x 1/2))) (exp x))
11.0ms
(/ (exp -1) (* (exp (* x (* x 1/2))) (exp x)))
5.0ms
(log (/ (exp -1) (* (exp (* x (* x 1/2))) (exp x))))

series306.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

156.0ms
(log (/ (exp -1) (* (exp (* x (* x 1/2))) (exp x))))
129.0ms
(/ (exp -1) (* (exp (* x (* x 1/2))) (exp x)))
21.0ms
(* (exp (* x (* x 1/2))) (exp x))

simplify1.9s

Counts
42 → 68
Calls

42 calls. Slowest were:

346.0ms
(- (+ (* 1/3 (* (pow x 3) (exp -1))) (exp -1)) (* x (exp -1)))
117.0ms
(/ (exp -1) (* (exp (* x (* x 1/2))) (exp x)))
112.0ms
(- -1 (log (* (exp (* x (* x 1/2))) (exp x))))

prune632.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

regimes42.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample9.1s

Algorithm
intervals