Average Error: 61.0 → 0.0
Time: 18.6s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\log \left(e^{\frac{\log_* (1 + \left(-x\right))}{\log_* (1 + x)}}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 61.0

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

  2. Simplified60.1

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

  4. Applied sub-neg60.1

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

  5. Applied log1p-def0.0

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

  7. Applied add-log-exp0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 +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: 18.1s)Debug log

sample284.0ms

Algorithm
intervals

simplify8.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(/ (log (- 1 x)) (log (+ 1 x)))

prune12.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 60.3b

localize10.0ms

Local error

Found 2 expressions with local error:

5.2b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log1p x))

rewrite8.0ms

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
6.0ms
(/ (log (- 1 x)) (log1p x))
1.0ms
(log (- 1 x))

series138.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
100.0ms
(/ (log (- 1 x)) (log1p x))
38.0ms
(log (- 1 x))

simplify1.8s

Counts
47 → 59
Calls
47 calls:
Slowest
339.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
291.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
268.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
204.0ms
(- (+ (* 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)))))
164.0ms
(- (+ (/ (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)))))))))))

prune242.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 1 expressions with local error:

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

rewrite7.0ms

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
7.0ms
(/ (log1p (- x)) (log1p x))

series86.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
86.0ms
(/ (log1p (- x)) (log1p x))

simplify766.0ms

Counts
28 → 32
Calls
28 calls:
Slowest
211.0ms
(- (+ (/ (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)))))))))))
164.0ms
(- (+ (* 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)))))
159.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
139.0ms
(/ (* (* (log1p (- x)) (log1p (- x))) (log1p (- x))) (* (* (log1p x) (log1p x)) (log1p x)))
16.0ms
(/ (* (cbrt (log1p (- x))) (cbrt (log1p (- x)))) 1)

prune216.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize14.0ms

Local error

Found 2 expressions with local error:

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

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
18×times-frac
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
13×exp-prod
4×add-exp-log
4×add-cbrt-cube
3×add-log-exp
3×associate-/l*
3×associate-/r*
2×div-inv
2×log1p-expm1-u
2×pow1
2×expm1-log1p-u
1×rem-exp-log
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
2 → 52
Calls
2 calls:
Slowest
7.0ms
(/ (log1p (- x)) (log1p x))
6.0ms
(exp (/ (log1p (- x)) (log1p x)))

series190.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
111.0ms
(exp (/ (log1p (- x)) (log1p x)))
80.0ms
(/ (log1p (- x)) (log1p x))

simplify1.7s

Counts
45 → 58
Calls
45 calls:
Slowest
424.0ms
(- (+ (* 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))))
287.0ms
(- (exp -1) (+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1))))
238.0ms
(- (+ (* 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)))))
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)))))))))))
153.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune619.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize23.0ms

Local error

Found 3 expressions with local error:

0.3b
(* 1/12 x)
0.0b
(fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1))
0.0b
(log (fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1)))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
4×add-cube-cbrt
4×add-exp-log
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
3×add-log-exp
3×log1p-expm1-u
3×add-cbrt-cube
3×log-prod
3×expm1-log1p-u
1×fma-udef
1×*-commutative
1×log-pow
1×rem-log-exp
Counts
3 → 34
Calls
3 calls:
Slowest
1.0ms
(log (fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1)))
0.0ms
(* 1/12 x)
0.0ms
(fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1))

series208.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
125.0ms
(log (fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1)))
76.0ms
(fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1))
7.0ms
(* 1/12 x)

simplify2.0s

Counts
13 → 43
Calls
13 calls:
Slowest
345.0ms
(log (* (cbrt (fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1))) (cbrt (fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1)))))
284.0ms
(log (sqrt (fma (fma (* 1/12 x) (* x x) x) (- (exp -1)) (exp -1))))
280.0ms
(- (exp -1) (+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1))))
270.0ms
(- (exp -1) (+ (* 1/12 (* (pow x 3) (exp -1))) (* x (exp -1))))
232.0ms
(- (+ (log (* 1/12 (exp -1))) (* 12 (/ 1 (pow x 2)))) (+ (* 3 (log (/ -1 x))) (* 12 (/ 1 (pow x 3)))))

prune372.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes30.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample9.4s

Algorithm
intervals