Average Error: 60.8 → 0.0
Time: 17.4s
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

Original60.8
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 60.8

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

  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 2019016 +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: 17.0s)Debug log

sample241.0ms

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)

simplify5.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
5.0ms
(/ (log (- 1 x)) (log (+ 1 x)))

prune10.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 58.6b

localize10.0ms

Local error

Found 2 expressions with local error:

8.6b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log1p x))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
12×add-cube-cbrt
12×times-frac
12×*-un-lft-identity
12×add-sqr-sqrt
7×pow1
5×log-pow
5×add-exp-log
4×associate-/l*
4×add-cbrt-cube
4×log-div
3×associate-/r*
3×log-prod
2×add-log-exp
2×flip--
2×log1p-expm1-u
2×div-sub
2×flip3--
2×log1p-def
2×expm1-log1p-u
1×div-inv
1×rem-log-exp
1×div-exp
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
2 → 53
Calls
2 calls:
Slowest
15.0ms
(log (- 1 x))
6.0ms
(/ (log (- 1 x)) (log1p x))

series91.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
66.0ms
(/ (log (- 1 x)) (log1p x))
24.0ms
(log (- 1 x))

simplify1.7s

Counts
47 → 59
Calls
47 calls:
Slowest
283.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
255.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
247.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
183.0ms
(- (+ (/ (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)))))))))))
162.0ms
(- (+ (* 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)))))

prune293.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize7.0ms

Local error

Found 1 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×times-frac
9×*-un-lft-identity
9×add-sqr-sqrt
3×associate-/l*
3×add-exp-log
3×add-cbrt-cube
3×associate-/r*
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow1
1×expm1-log1p-u
Counts
1 → 29
Calls
1 calls:
Slowest
5.0ms
(/ (log1p (- x)) (log1p x))

series77.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
77.0ms
(/ (log1p (- x)) (log1p x))

simplify772.0ms

Counts
28 → 32
Calls
28 calls:
Slowest
222.0ms
(- (+ (/ (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)))))))))))
183.0ms
(- (+ (* 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)))))
145.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
125.0ms
(/ (* (* (log1p (- x)) (log1p (- x))) (log1p (- x))) (* (* (log1p x) (log1p x)) (log1p x)))
17.0ms
(/ (* (cbrt (log1p (- x))) (cbrt (log1p (- x)))) 1)

prune235.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 1 expressions with local error:

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

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma x (* -1/2 x) (- -1 x))

series17.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
17.0ms
(fma x (* -1/2 x) (- -1 x))

simplify452.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
172.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
162.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
117.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune93.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize8.0ms

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)))

rewrite8.0ms

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
3×log1p-expm1-u
3×log1p-def
3×expm1-log1p-u
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×sub-neg
1×clear-num
1×cbrt-undiv
Counts
3 → 75
Calls
3 calls:
Slowest
5.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
1.0ms
(log (- 1 x))
1.0ms
(log (+ 1 x))

series138.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
86.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
30.0ms
(log (- 1 x))
22.0ms
(log (+ 1 x))

simplify2.8s

Counts
65 → 84
Calls
65 calls:
Slowest
375.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
305.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
298.0ms
(- (+ x (* 1/3 (pow x 3))) (* 1/2 (pow x 2)))
269.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
261.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune441.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes43.0ms

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
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)