Average Error: 60.7 → 0.0
Time: 27.1s
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

Original60.7
Target0.4
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 60.7

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

  2. Simplified59.8

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

  4. Applied sub-neg59.8

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

  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 2019007 +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: 26.7s)Debug log

sample274.0ms

Algorithm
intervals

simplify5.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune17.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 59.1b

localize21.0ms

Local error

Found 2 expressions with local error:

7.9b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log1p x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 53
Calls

2 calls. Slowest were:

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

series336.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

313.0ms
(/ (log (- 1 x)) (log1p x))
23.0ms
(log (- 1 x))

simplify1.9s

Counts
47 → 59
Calls

47 calls. Slowest were:

301.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
274.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
246.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune310.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 1 expressions with local error:

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

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
1 → 29
Calls

1 calls. Slowest were:

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

series88.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

88.0ms
(/ (log1p (- x)) (log1p x))

simplify964.0ms

Counts
28 → 32
Calls

28 calls. Slowest were:

256.0ms
(- (+ (/ (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)))))))))))
228.0ms
(- (+ (* 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)))))
189.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune290.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize6.0ms

Local error

Found 3 expressions with local error:

0.0b
(cbrt (pow (/ (log1p (- x)) (log1p x)) 3))
0.0b
(pow (/ (log1p (- x)) (log1p x)) 3)
0.0b
(/ (log1p (- x)) (log1p x))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 116
Calls

3 calls. Slowest were:

7.0ms
(/ (log1p (- x)) (log1p x))
3.0ms
(pow (/ (log1p (- x)) (log1p x)) 3)
2.0ms
(cbrt (pow (/ (log1p (- x)) (log1p x)) 3))

series328.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

131.0ms
(pow (/ (log1p (- x)) (log1p x)) 3)
101.0ms
(/ (log1p (- x)) (log1p x))
96.0ms
(cbrt (pow (/ (log1p (- x)) (log1p x)) 3))

simplify10.7s

Counts
99 → 125
Calls

99 calls. Slowest were:

1.0s
(pow (/ 1 (* (cbrt (log1p x)) (cbrt (log1p x)))) 3)
988.0ms
(pow (/ 1 (* (cbrt (log1p x)) (cbrt (log1p x)))) 3)
409.0ms
(cbrt (pow (/ (* (cbrt (log1p (- x))) (cbrt (log1p (- x)))) (* (cbrt (log1p x)) (cbrt (log1p x)))) 3))

prune1.2s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize9.0ms

Local error

Found 2 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
2 → 59
Calls

2 calls. Slowest were:

5.0ms
(/ (log1p x) (log1p (- x)))
3.0ms
(/ 1 (/ (log1p x) (log1p (- x))))

series122.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

63.0ms
(/ (log1p x) (log1p (- x)))
58.0ms
(/ 1 (/ (log1p x) (log1p (- x))))

simplify1.1s

Counts
46 → 65
Calls

46 calls. Slowest were:

207.0ms
(- (+ (/ (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)))))))))))
186.0ms
(- (+ (* 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)))))
184.0ms
(- (+ (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ 1 x))) 2))) (/ 1 (* x (- (log -1) (log (/ 1 x)))))) (+ (/ (log (/ 1 x)) (- (log -1) (log (/ 1 x)))) (+ (* 1/2 (/ 1 (* (pow x 2) (- (log -1) (log (/ 1 x)))))) (+ (/ (log (/ 1 x)) (* (pow (- (log -1) (log (/ 1 x))) 2) x)) (+ (/ (log (/ 1 x)) (* (pow (- (log -1) (log (/ 1 x))) 3) (pow x 2))) (* 1/2 (/ (log (/ 1 x)) (* (pow (- (log -1) (log (/ 1 x))) 2) (pow x 2)))))))))

prune495.0ms

Pruning

3 alts after pruning (0 fresh and 3 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

bsearch0.0ms

end0.0ms

sample8.5s

Algorithm
intervals