Average Error: 60.9 → 0.5
Time: 43.0s
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

Original60.9
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 60.9

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

  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. Final simplification0.5

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

Reproduce

herbie shell --seed 1137770271 
(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: 42.1s)Debug log

sample465.0ms

Algorithm
intervals
Results
104×259.0ms(body real 1280)
76×113.0ms(body real 640)
46×40.0ms(body real 320)
256×15.0ms(pre true 80)
19×11.0ms(body real 160)
11×3.0ms(body real 80)

simplify13.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
13.0ms
(/ (log (- 1 x)) (log (+ 1 x)))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.2b

localize41.0ms

Local error

Found 3 expressions with local error:

6.6b
(log (+ 1 x))
6.6b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log (+ 1 x)))

rewrite18.0ms

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
13.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
2.0ms
(log (- 1 x))
2.0ms
(log (+ 1 x))

series241.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
128.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
65.0ms
(log (- 1 x))
48.0ms
(log (+ 1 x))

simplify3.2s

Counts
63 → 75
Calls
63 calls:
Slowest
452.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
363.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
276.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
274.0ms
(- (+ (/ (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)))))))))))
263.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune1.1s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 1 expressions with local error:

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

rewrite10.0ms

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
10.0ms
(- (- -1 x) (* x (* x 1/2)))

series20.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
20.0ms
(- (- -1 x) (* x (* x 1/2)))

simplify657.0ms

Counts
4 → 18
Calls
4 calls:
Slowest
225.0ms
(/ (exp (- -1 x)) (exp (* x (* x 1/2))))
146.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
144.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
139.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune203.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize37.0ms

Local error

Found 4 expressions with local error:

6.6b
(log (+ 1 x))
6.6b
(log (+ 1 x))
2.7b
(log (- 1 (* x x)))
0.0b
(- (log (- 1 (* x x))) (log (+ 1 x)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
15×log-prod
13×*-un-lft-identity
11×pow1
11×add-sqr-sqrt
10×log-div
9×add-cube-cbrt
7×log-pow
7×add-exp-log
4×add-log-exp
4×distribute-lft-out--
4×add-cbrt-cube
3×associate--r+
3×flip--
3×flip-+
3×associate--l+
3×rem-log-exp
3×flip3--
3×flip3-+
2×associate--r-
2×associate--l-
1×difference-of-squares
1×diff-log
1×sub-neg
Counts
4 → 69
Calls
4 calls:
Slowest
13.0ms
(- (log (- 1 (* x x))) (log (+ 1 x)))
2.0ms
(log (+ 1 x))
2.0ms
(log (- 1 (* x x)))
2.0ms
(log (+ 1 x))

series375.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
177.0ms
(log (- 1 (* x x)))
100.0ms
(- (log (- 1 (* x x))) (log (+ 1 x)))
51.0ms
(log (+ 1 x))
47.0ms
(log (+ 1 x))

simplify3.9s

Counts
47 → 81
Calls
47 calls:
Slowest
435.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
430.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
384.0ms
(- (log -1) (+ (* 2 (log (/ 1 x))) (+ (* 1/2 (/ 1 (pow x 4))) (/ 1 (pow x 2)))))
364.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
274.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))

prune874.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize70.0ms

Local error

Found 4 expressions with local error:

28.0b
(/ (- (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x))) (log (+ 1 x)))
6.6b
(log (+ 1 x))
6.6b
(log (+ 1 x))
0.1b
(* -1/3 (pow x 6))

rewrite250.0ms

Algorithm
rewrite-expression-head
Rules
34×*-un-lft-identity
29×add-sqr-sqrt
24×times-frac
19×add-cube-cbrt
18×pow1
14×log-pow
10×distribute-lft-out--
8×add-exp-log
6×associate-/l*
6×add-cbrt-cube
6×log-prod
5×difference-of-squares
4×add-log-exp
4×associate-/r*
4×log-div
3×associate-*r*
2×associate-/l/
2×flip-+
2×rem-log-exp
2×flip3-+
1×flip--
1×div-inv
1×*-commutative
1×div-exp
1×div-sub
1×flip3--
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 88
Calls
4 calls:
Slowest
241.0ms
(/ (- (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x))) (log (+ 1 x)))
3.0ms
(* -1/3 (pow x 6))
2.0ms
(log (+ 1 x))
2.0ms
(log (+ 1 x))

series276.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
135.0ms
(/ (- (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x))) (log (+ 1 x)))
47.0ms
(* -1/3 (pow x 6))
47.0ms
(log (+ 1 x))
46.0ms
(log (+ 1 x))

simplify15.1s

Counts
87 → 100
Calls
87 calls:
Slowest
852.0ms
(/ (* (* (- (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x))) (- (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x)))) (- (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x)))) (* (* (log (+ 1 x)) (log (+ 1 x))) (log (+ 1 x))))
551.0ms
(/ (* (cbrt (- (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x)))) (cbrt (- (- (- (* -1/3 (pow x 6)) (* x x)) (* 1/2 (pow x 4))) (log (+ 1 x))))) (sqrt (log (+ 1 x))))
446.0ms
(- (* 1/3 (/ (pow x 5) (pow (- (log -1) (log (/ -1 x))) 2))) (+ (* 1/3 (/ (pow x 4) (pow (- (log -1) (log (/ -1 x))) 3))) (+ (* 1/2 (/ (pow x 4) (- (log -1) (log (/ -1 x))))) (+ (* 1/6 (/ (pow x 4) (pow (- (log -1) (log (/ -1 x))) 2))) (* 1/3 (/ (pow x 6) (- (log -1) (log (/ -1 x)))))))))
440.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
434.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))

prune1.8s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes116.0ms

Accuracy

0% (0.5b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample13.4s

Algorithm
intervals
Results
3439×7.9s(body real 1280)
2462×3.6s(body real 640)
1215×974.0ms(body real 320)
8000×390.0ms(pre true 80)
658×288.0ms(body real 160)
226×46.0ms(body real 80)