Average Error: 60.9 → 0.0
Time: 26.1s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{1}{\frac{\log_* (1 + x)}{\log_* (1 + \left(-x\right))}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.9
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 60.9

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

  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. Using strategy rm

  7. Applied *-un-lft-identity0.0

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

  8. Applied associate-/l*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019005 +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: 25.7s)Debug log

sample261.0ms

Algorithm
intervals

simplify10.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune44.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 59.1b

localize22.0ms

Local error

Found 2 expressions with local error:

6.5b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log1p x))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
2 → 53
Calls

2 calls. Slowest were:

10.0ms
(/ (log (- 1 x)) (log1p x))
2.0ms
(log (- 1 x))

series124.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

85.0ms
(/ (log (- 1 x)) (log1p x))
39.0ms
(log (- 1 x))

simplify1.8s

Counts
47 → 59
Calls

47 calls. Slowest were:

339.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
297.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
279.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune261.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 40
Calls

2 calls. Slowest were:

6.0ms
(/ (log1p (- x)) (log1p x))
1.0ms
(log1p (- x))

series131.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

104.0ms
(/ (log1p (- x)) (log1p x))
27.0ms
(log1p (- x))

simplify1.6s

Counts
32 → 46
Calls

32 calls. Slowest were:

303.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
282.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
266.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune333.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize6.0ms

Local error

Found 3 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 70
Calls

3 calls. Slowest were:

5.0ms
(/ (log1p x) (log1p (- x)))
2.0ms
(/ 1 (/ (log1p x) (log1p (- x))))
0.0ms
(log1p (- x))

series202.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

104.0ms
(/ 1 (/ (log1p x) (log1p (- x))))
74.0ms
(/ (log1p x) (log1p (- x)))
23.0ms
(log1p (- x))

simplify2.2s

Counts
50 → 79
Calls

50 calls. Slowest were:

337.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
302.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
232.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune575.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 124
Calls

4 calls. Slowest were:

11.0ms
(cbrt (* (* (/ 1 (/ (log1p x) (log1p (- x)))) (/ 1 (/ (log1p x) (log1p (- x))))) (/ 1 (/ (log1p x) (log1p (- x))))))
6.0ms
(* (/ 1 (/ (log1p x) (log1p (- x)))) (/ 1 (/ (log1p x) (log1p (- x)))))
5.0ms
(/ (log1p x) (log1p (- x)))

series362.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

115.0ms
(* (/ 1 (/ (log1p x) (log1p (- x)))) (/ 1 (/ (log1p x) (log1p (- x)))))
91.0ms
(cbrt (* (* (/ 1 (/ (log1p x) (log1p (- x)))) (/ 1 (/ (log1p x) (log1p (- x))))) (/ 1 (/ (log1p x) (log1p (- x))))))
84.0ms
(/ (log1p x) (log1p (- x)))
73.0ms
(/ (log1p x) (log1p (- x)))

simplify6.8s

Counts
111 → 136
Calls

111 calls. Slowest were:

493.0ms
(cbrt (* (* (/ 1 (/ (log1p x) (log1p (- x)))) 1) (/ 1 (/ (log1p x) (log1p (- x))))))
417.0ms
(- (+ (* 2 (/ (pow (log -1) 2) (* (pow (log (/ 1 x)) 3) x))) (+ (* 3 (/ (pow (log -1) 2) (* (pow (log (/ 1 x)) 4) (pow x 2)))) (+ (/ (pow (log -1) 2) (pow (log (/ 1 x)) 2)) (+ (* 8 (/ 1 (* (pow (log (/ 1 x)) 2) (pow x 2)))) (+ (* 4 (/ 1 (* (log (/ 1 x)) x))) (+ (/ (log -1) (* (pow (log (/ 1 x)) 2) (pow x 2))) 1)))))) (+ (/ (pow (log -1) 2) (* (pow (log (/ 1 x)) 3) (pow x 2))) (+ (* 10 (/ (log -1) (* (pow (log (/ 1 x)) 3) (pow x 2)))) (+ (* 6 (/ (log -1) (* (pow (log (/ 1 x)) 2) x))) (* 2 (/ (log -1) (log (/ 1 x))))))))
321.0ms
(+ (- (- (log (log1p x)) (log (log1p (- x))))) (log (/ 1 (/ (log1p x) (log1p (- x))))))

prune1.3s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes25.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.5s

Algorithm
intervals