Average Error: 61.0 → 0.3
Time: 20.6s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log \left(e^{-1} - \left(\left({x}^{3} \cdot e^{-1}\right) \cdot \frac{1}{12} + x \cdot e^{-1}\right)\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 61.0

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
  2. Using strategy rm

  3. Applied add-log-exp61.0

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

  4. Taylor expanded around 0 0.3

    \[\leadsto \log \color{blue}{\left(e^{-1} - \left(\frac{1}{12} \cdot \left({x}^{3} \cdot e^{-1}\right) + x \cdot e^{-1}\right)\right)}\]

  5. Final simplification0.3

    \[\leadsto \log \left(e^{-1} - \left(\left({x}^{3} \cdot e^{-1}\right) \cdot \frac{1}{12} + x \cdot e^{-1}\right)\right)\]

Reproduce

herbie shell --seed 2019010 
(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: 20.3s)Debug log

sample319.0ms

Algorithm
intervals

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: 61.9b

localize28.0ms

Local error

Found 3 expressions with local error:

5.2b
(log (- 1 x))
5.2b
(log (+ 1 x))
0.0b
(/ (log (- 1 x)) (log (+ 1 x)))

rewrite46.0ms

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
31.0ms
(log (- 1 x))
12.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
2.0ms
(log (+ 1 x))

series184.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
106.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
39.0ms
(log (- 1 x))
38.0ms
(log (+ 1 x))

simplify2.3s

Counts
63 → 75
Calls
63 calls:
Slowest
339.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
258.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
216.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
206.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
182.0ms
(- (+ (/ (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)))))))))))

prune668.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

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

prune12.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

5.2b
(log (- 1 x))
5.2b
(log (+ 1 x))
0.0b
(/ (log (- 1 x)) (log (+ 1 x)))
0.0b
(exp (/ (log (- 1 x)) (log (+ 1 x))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
32×times-frac
25×add-cube-cbrt
25×*-un-lft-identity
25×add-sqr-sqrt
24×pow1
20×log-pow
20×exp-prod
8×add-exp-log
8×log-div
6×add-cbrt-cube
6×log-prod
5×add-log-exp
4×associate-/l*
4×div-sub
4×associate-/r*
3×flip--
3×div-inv
3×flip3--
2×rem-log-exp
2×exp-diff
1×rem-exp-log
1×flip-+
1×div-exp
1×exp-to-pow
1×flip3-+
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 97
Calls
4 calls:
Slowest
10.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
8.0ms
(exp (/ (log (- 1 x)) (log (+ 1 x))))
2.0ms
(log (- 1 x))
2.0ms
(log (+ 1 x))

series285.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
149.0ms
(exp (/ (log (- 1 x)) (log (+ 1 x))))
83.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
29.0ms
(log (- 1 x))
23.0ms
(log (+ 1 x))

simplify3.8s

Counts
90 → 109
Calls
90 calls:
Slowest
551.0ms
(- (+ (exp (* -1 (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))))) (+ (* 1/2 (/ (* (pow (log (/ -1 x)) 2) (exp (* -1 (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x))))))) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 4)))) (+ (/ (* (log (/ -1 x)) (exp (* -1 (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x))))))) (* x (pow (- (log -1) (log (/ -1 x))) 2))) (* 3/2 (/ (exp (* -1 (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))))) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2))))))) (+ (* 2 (/ (* (log (/ -1 x)) (exp (* -1 (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x))))))) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 3)))) (+ (/ (exp (* -1 (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))))) (* x (- (log -1) (log (/ -1 x))))) (+ (* 1/2 (/ (exp (* -1 (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x)))))) (* (pow x 2) (- (log -1) (log (/ -1 x)))))) (* 1/2 (/ (* (log (/ -1 x)) (exp (* -1 (/ (log (/ -1 x)) (- (log -1) (log (/ -1 x))))))) (* (pow x 2) (pow (- (log -1) (log (/ -1 x))) 2))))))))
535.0ms
(- (+ (* 1/2 (/ (* (log -1) (exp (* -1 (/ (- (log -1) (log (/ 1 x))) (log (/ 1 x)))))) (* (pow (log (/ 1 x)) 2) (pow x 2)))) (+ (exp (* -1 (/ (- (log -1) (log (/ 1 x))) (log (/ 1 x))))) (+ (* 4 (/ (exp (* -1 (/ (- (log -1) (log (/ 1 x))) (log (/ 1 x))))) (* (pow (log (/ 1 x)) 2) (pow x 2)))) (+ (* 2 (/ (exp (* -1 (/ (- (log -1) (log (/ 1 x))) (log (/ 1 x))))) (* (log (/ 1 x)) x))) (* 1/2 (/ (* (pow (log -1) 2) (exp (* -1 (/ (- (log -1) (log (/ 1 x))) (log (/ 1 x)))))) (* (pow (log (/ 1 x)) 4) (pow x 2)))))))) (+ (* 3 (/ (* (log -1) (exp (* -1 (/ (- (log -1) (log (/ 1 x))) (log (/ 1 x)))))) (* (pow (log (/ 1 x)) 3) (pow x 2)))) (/ (* (log -1) (exp (* -1 (/ (- (log -1) (log (/ 1 x))) (log (/ 1 x)))))) (* (pow (log (/ 1 x)) 2) x))))
396.0ms
(- (exp -1) (+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1))))
310.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
230.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune828.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize39.0ms

Local error

Found 4 expressions with local error:

0.2b
(* x (exp -1))
0.1b
(* 1/12 (* (pow x 3) (exp -1)))
0.1b
(* (pow x 3) (exp -1))
0.0b
(+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×associate-*r*
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-log-exp
6×add-cbrt-cube
5×add-exp-log
5×associate-*l*
4×pow1
3×*-commutative
2×prod-exp
1×flip-+
1×cbrt-unprod
1×unpow3
1×flip3-+
1×sum-log
1×+-commutative
1×pow-to-exp
1×cube-mult
Counts
4 → 50
Calls
4 calls:
Slowest
14.0ms
(+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1)))
8.0ms
(* 1/12 (* (pow x 3) (exp -1)))
6.0ms
(* (pow x 3) (exp -1))
2.0ms
(* x (exp -1))

series688.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
398.0ms
(+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1)))
104.0ms
(* x (exp -1))
95.0ms
(* 1/12 (* (pow x 3) (exp -1)))
91.0ms
(* (pow x 3) (exp -1))

simplify1.1s

Counts
26 → 62
Calls
26 calls:
Slowest
464.0ms
(* (* (* (pow x 3) (pow x 3)) (pow x 3)) (* (* (exp -1) (exp -1)) (exp -1)))
133.0ms
(* (exp (* 1/12 (* (pow x 3) (exp -1)))) (exp (* x (exp -1))))
108.0ms
(+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1)))
102.0ms
(+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1)))
100.0ms
(+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1)))

prune711.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes54.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch4.0ms

end0.0ms

sample9.0s

Algorithm
intervals