Average Error: 61.2 → 0.4
Time: 21.8s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(-1 - x\right) - x \cdot \left(\frac{1}{2} \cdot x\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.2
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.2

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019018 
(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: 21.3s)Debug log

sample306.0ms

Algorithm
intervals
Results
203.0ms120×body1280valid
61.0ms74×body640valid
17.0ms34×body320valid
8.0ms256×pre80true
6.0ms21×body160valid
1.0ms7×body80valid

simplify5.0ms

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

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.8b

localize22.0ms

Local error

Found 3 expressions with local error:

5.6b
(log (- 1 x))
5.6b
(log (+ 1 x))
0.0b
(/ (log (- 1 x)) (log (+ 1 x)))

rewrite14.0ms

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
9.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
2.0ms
(log (- 1 x))
1.0ms
(log (+ 1 x))

series145.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
71.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
52.0ms
(log (- 1 x))
22.0ms
(log (+ 1 x))

simplify2.0s

Counts
63 → 75
Calls
63 calls:
Slowest
289.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
240.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
214.0ms
(- (+ (/ (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)))))))))))
173.0ms
(- (/ 1 x) (+ (log (/ 1 x)) (* 1/2 (/ 1 (pow x 2)))))
165.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune865.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

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

prune20.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

5.6b
(log (- 1 x))
5.6b
(log (+ 1 x))
0.0b
(/ 1 (/ (log (+ 1 x)) (log (- 1 x))))
0.0b
(/ (log (+ 1 x)) (log (- 1 x)))

rewrite16.0ms

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))
2.0ms
(log (+ 1 x))

series214.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
91.0ms
(/ 1 (/ (log (+ 1 x)) (log (- 1 x))))
74.0ms
(/ (log (+ 1 x)) (log (- 1 x)))
24.0ms
(log (- 1 x))
24.0ms
(log (+ 1 x))

simplify2.7s

Counts
88 → 113
Calls
88 calls:
Slowest
355.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
236.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
202.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
188.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
180.0ms
(- (+ (/ 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)))))))))

prune846.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

5.6b
(log (- 1 x))
5.6b
(log (+ 1 x))
0.0b
(/ (/ 1 (log (+ 1 x))) (/ 1 (log (- 1 x))))
0.0b
(/ 1 (log (- 1 x)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
16×times-frac
15×add-exp-log
12×div-inv
8×associate-/r*
8×pow1
6×add-cbrt-cube
6×log-prod
5×rec-exp
4×add-log-exp
4×associate-/l*
4×div-exp
4×log-div
3×log-pow
2×rem-log-exp
2×frac-2neg
2×clear-num
1×associate-/l/
1×inv-pow
1×flip--
1×pow-flip
1×flip-+
1×associate-/r/
1×flip3--
1×flip3-+
1×cbrt-undiv
Counts
4 → 86
Calls
4 calls:
Slowest
4.0ms
(/ (/ 1 (log (+ 1 x))) (/ 1 (log (- 1 x))))
1.0ms
(log (- 1 x))
1.0ms
(log (+ 1 x))
1.0ms
(/ 1 (log (- 1 x)))

series182.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
95.0ms
(/ (/ 1 (log (+ 1 x))) (/ 1 (log (- 1 x))))
32.0ms
(/ 1 (log (- 1 x)))
27.0ms
(log (- 1 x))
27.0ms
(log (+ 1 x))

simplify4.4s

Counts
73 → 98
Calls
73 calls:
Slowest
1.2s
(/ (* (* (/ 1 (log (+ 1 x))) (/ 1 (log (+ 1 x)))) (/ 1 (log (+ 1 x)))) (* (* (/ 1 (log (- 1 x))) (/ 1 (log (- 1 x)))) (/ 1 (log (- 1 x)))))
331.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
308.0ms
(+ (/ 1 (- (log -1) (log (/ 1 x)))) (+ (/ 1 (* x (pow (- (log -1) (log (/ 1 x))) 2))) (+ (* 1/2 (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ 1 x))) 2)))) (/ 1 (* (pow x 2) (pow (- (log -1) (log (/ 1 x))) 3))))))
208.0ms
(- (+ (/ (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)))))))))))
207.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune749.0ms

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.0b

regimes29.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch1.0ms

end0.0ms

sample8.7s

Algorithm
intervals
Results
5.4s3447×body1280valid
2.0s2483×body640valid
580.0ms8000×pre80true
555.0ms1258×body320valid
153.0ms621×body160valid
21.0ms191×body80valid