Average Error: 60.8 → 0.5
Time: 17.1s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[-1 - \left(x + \left(\frac{1}{2} \cdot x\right) \cdot x\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.8
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 60.8

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

  2. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{-\left(x + \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)}\]

  3. Simplified0.5

    \[\leadsto \color{blue}{\left(-1 - x\right) - x \cdot \left(x \cdot \frac{1}{2}\right)}\]
  4. Using strategy rm

  5. Applied associate--l-0.5

    \[\leadsto \color{blue}{-1 - \left(x + x \cdot \left(x \cdot \frac{1}{2}\right)\right)}\]

  6. Final simplification0.5

    \[\leadsto -1 - \left(x + \left(\frac{1}{2} \cdot x\right) \cdot x\right)\]

Reproduce

herbie shell --seed 2019016 
(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: 16.8s)Debug log

sample303.0ms

Algorithm
intervals
Results
256×(pre true 80)
108×(body real 1280)
75×(body real 640)
34×(body real 320)
32×(body real 160)
7×(body real 80)

simplify9.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
9.0ms
(/ (log (- 1 x)) (log (+ 1 x)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.1b

localize42.0ms

Local error

Found 3 expressions with local error:

8.6b
(log (- 1 x))
8.6b
(log (+ 1 x))
0.0b
(/ (log (- 1 x)) (log (+ 1 x)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
16×times-frac
15×add-cube-cbrt
15×*-un-lft-identity
15×pow1
15×add-sqr-sqrt
12×log-pow
7×add-exp-log
6×log-prod
6×log-div
5×add-cbrt-cube
4×associate-/l*
4×associate-/r*
3×add-log-exp
2×flip--
2×rem-log-exp
2×div-sub
2×flip3--
1×div-inv
1×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
3 → 66
Calls
3 calls:
Slowest
5.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
1.0ms
(log (- 1 x))
1.0ms
(log (+ 1 x))

series141.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
84.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
29.0ms
(log (- 1 x))
28.0ms
(log (+ 1 x))

simplify2.2s

Counts
63 → 75
Calls
63 calls:
Slowest
353.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
244.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
223.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
203.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
196.0ms
(- (+ (/ (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)))))))))))

prune653.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 1 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
3×add-log-exp
2×associate--l+
2×sub-neg
1×flip--
1×add-cube-cbrt
1×add-exp-log
1×diff-log
1×add-cbrt-cube
1×flip3--
1×*-un-lft-identity
1×pow1
1×add-sqr-sqrt
1×associate--l-
Counts
1 → 15
Calls
1 calls:
Slowest
11.0ms
(- (- -1 x) (* x (* x 1/2)))

series16.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
16.0ms
(- (- -1 x) (* x (* x 1/2)))

simplify500.0ms

Counts
4 → 18
Calls
4 calls:
Slowest
171.0ms
(/ (exp (- -1 x)) (exp (* x (* x 1/2))))
127.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
107.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
94.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune93.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune5.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

8.6b
(log (- 1 x))
8.6b
(log (+ 1 x))
0.0b
(/ (log (+ 1 x)) (log (- 1 x)))
0.0b
(/ 1 (/ (log (+ 1 x)) (log (- 1 x))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
32×times-frac
25×add-cube-cbrt
25×*-un-lft-identity
25×pow1
25×add-sqr-sqrt
24×associate-/r*
20×log-pow
11×add-exp-log
6×add-cbrt-cube
6×log-prod
6×log-div
4×add-log-exp
4×associate-/l*
3×div-inv
2×flip-+
2×rem-log-exp
2×div-exp
2×div-sub
2×flip3-+
2×frac-2neg
2×clear-num
2×rec-exp
1×inv-pow
1×flip--
1×pow-flip
1×associate-/r/
1×flip3--
1×cbrt-undiv
Counts
4 → 101
Calls
4 calls:
Slowest
9.0ms
(/ (log (+ 1 x)) (log (- 1 x)))
2.0ms
(/ 1 (/ (log (+ 1 x)) (log (- 1 x))))
2.0ms
(log (- 1 x))
1.0ms
(log (+ 1 x))

series215.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
83.0ms
(/ (log (+ 1 x)) (log (- 1 x)))
72.0ms
(/ 1 (/ (log (+ 1 x)) (log (- 1 x))))
36.0ms
(log (+ 1 x))
23.0ms
(log (- 1 x))

simplify2.6s

Counts
88 → 113
Calls
88 calls:
Slowest
264.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
259.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
221.0ms
(- (+ (/ (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)))))))))))
204.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
191.0ms
(- (+ (* 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)))))

prune837.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes33.0ms

Accuracy

0% (0.5b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample9.1s

Algorithm
intervals
Results
8000×(pre true 80)
3368×(body real 1280)
2524×(body real 640)
1208×(body real 320)
669×(body real 160)
231×(body real 80)