Average Error: 61.0 → 0.0
Time: 20.2s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

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. Using strategy rm

  7. Applied add-cbrt-cube0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right) \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}}\]

  8. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \left(\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)} \cdot \frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}\right)}\]

Reproduce

herbie shell --seed 2019004 +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: 19.8s)Debug log

sample289.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune10.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 58.3b

localize10.0ms

Local error

Found 2 expressions with local error:

4.9b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log1p x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 53
Calls

2 calls. Slowest were:

5.0ms
(/ (log (- 1 x)) (log1p x))
1.0ms
(log (- 1 x))

series117.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

75.0ms
(/ (log (- 1 x)) (log1p x))
41.0ms
(log (- 1 x))

simplify2.1s

Counts
47 → 59
Calls

47 calls. Slowest were:

348.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
332.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
296.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune266.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 1 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

7.0ms
(/ (log1p (- x)) (log1p x))

series78.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

77.0ms
(/ (log1p (- x)) (log1p x))

simplify571.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

136.0ms
(- (+ (/ (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)))))))))))
123.0ms
(- (+ (* 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)))))
106.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune223.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

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

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

13.0ms
(cbrt (* (* (/ (log1p (- x)) (log1p x)) (/ (log1p (- x)) (log1p x))) (/ (log1p (- x)) (log1p x))))
4.0ms
(/ (log1p (- x)) (log1p x))
3.0ms
(/ (log1p (- x)) (log1p x))

series276.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(/ (log1p (- x)) (log1p x))
68.0ms
(/ (log1p (- x)) (log1p x))
65.0ms
(/ (log1p (- x)) (log1p x))
63.0ms
(cbrt (* (* (/ (log1p (- x)) (log1p x)) (/ (log1p (- x)) (log1p x))) (/ (log1p (- x)) (log1p x))))

simplify3.3s

Counts
96 → 118
Calls

96 calls. Slowest were:

231.0ms
(- (+ (* 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)))))
223.0ms
(- (+ (/ (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)))))))))))
205.0ms
(- (+ (* 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)))))

prune1.0s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize11.0ms

Local error

Found 2 expressions with local error:

0.2b
(* (log1p (- x)) (/ 1 (log1p x)))
0.0b
(/ 1 (log1p x))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 41
Calls

2 calls. Slowest were:

9.0ms
(* (log1p (- x)) (/ 1 (log1p x)))
2.0ms
(/ 1 (log1p x))

series121.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

80.0ms
(* (log1p (- x)) (/ 1 (log1p x)))
42.0ms
(/ 1 (log1p x))

simplify1.7s

Counts
22 → 47
Calls

22 calls. Slowest were:

522.0ms
(- (+ (/ 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))))
304.0ms
(* (* (* (log1p (- x)) (log1p (- x))) (log1p (- x))) (* (* (/ 1 (log1p x)) (/ 1 (log1p x))) (/ 1 (log1p x))))
230.0ms
(- (+ (* 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)))))

prune365.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes29.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample9.2s

Algorithm
intervals