Average Error: 60.9 → 0.4
Time: 36.9s
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
Target0.3
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 60.9

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

  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 1042149663 
(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: 36.2s)Debug log

sample411.0ms

Algorithm
intervals
Results
256×(pre true 80)
111×(body real 1280)
86×(body real 640)
40×(body real 320)
14×(body real 160)
5×(body real 80)

simplify11.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
10.0ms
(/ (log (- 1 x)) (log (+ 1 x)))

prune12.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 61.2b

localize35.0ms

Local error

Found 3 expressions with local error:

3.4b
(log (+ 1 x))
3.2b
(log (- 1 x))
0.0b
(/ (log (- 1 x)) (log (+ 1 x)))

rewrite17.0ms

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

series220.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
118.0ms
(/ (log (- 1 x)) (log (+ 1 x)))
56.0ms
(log (- 1 x))
46.0ms
(log (+ 1 x))

simplify3.2s

Counts
63 → 75
Calls
63 calls:
Slowest
437.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
370.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
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)))))))))))
273.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
272.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune1.0s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize28.0ms

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
9.0ms
(- (- -1 x) (* x (* x 1/2)))

series19.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
19.0ms
(- (- -1 x) (* x (* x 1/2)))

simplify670.0ms

Counts
4 → 18
Calls
4 calls:
Slowest
237.0ms
(/ (exp (- -1 x)) (exp (* x (* x 1/2))))
144.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
144.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))
144.0ms
(- (+ x (+ (* 1/2 (pow x 2)) 1)))

prune193.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize77.0ms

Local error

Found 4 expressions with local error:

3.2b
(log (- 1 x))
3.2b
(log (+ (* 1 1) (- (* x x) (* 1 x))))
1.6b
(log (+ 1 (pow x 3)))
0.0b
(/ (log (- 1 x)) (- (log (+ 1 (pow x 3))) (log (+ (* 1 1) (- (* x x) (* 1 x))))))

rewrite44.0ms

Algorithm
rewrite-expression-head
Rules
42×*-un-lft-identity
36×pow1
32×log-pow
32×times-frac
31×add-sqr-sqrt
21×add-cube-cbrt
20×distribute-lft-out--
10×log-prod
9×add-exp-log
8×associate-/r*
8×log-div
6×add-cbrt-cube
5×difference-of-squares
4×add-log-exp
4×associate-/l*
3×flip--
3×rem-log-exp
3×flip3--
2×flip-+
2×associate-/r/
2×div-sub
2×flip3-+
1×distribute-lft-out
1×div-inv
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 103
Calls
4 calls:
Slowest
22.0ms
(/ (log (- 1 x)) (- (log (+ 1 (pow x 3))) (log (+ (* 1 1) (- (* x x) (* 1 x))))))
14.0ms
(log (+ (* 1 1) (- (* x x) (* 1 x))))
2.0ms
(log (+ 1 (pow x 3)))
2.0ms
(log (- 1 x))

series789.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
506.0ms
(log (+ 1 (pow x 3)))
180.0ms
(/ (log (- 1 x)) (- (log (+ 1 (pow x 3))) (log (+ (* 1 1) (- (* x x) (* 1 x))))))
55.0ms
(log (+ (* 1 1) (- (* x x) (* 1 x))))
48.0ms
(log (- 1 x))

simplify9.6s

Counts
111 → 115
Calls
111 calls:
Slowest
904.0ms
(log (+ (pow (* 1 1) 3) (pow (- (* x x) (* 1 x)) 3)))
368.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
284.0ms
(log (- (* (* 1 1) (* 1 1)) (* (- (* x x) (* 1 x)) (- (* x x) (* 1 x)))))
277.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
273.0ms
(- (+ (/ (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)))))))))))

prune1.7s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize35.0ms

Local error

Found 4 expressions with local error:

3.2b
(log (- 1 x))
3.2b
(log (+ (* 1 1) (- (* x x) (* 1 x))))
1.7b
(log (sqrt (+ 1 (pow x 3))))
1.7b
(log (sqrt (+ 1 (pow x 3))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
19×log-prod
11×*-un-lft-identity
10×add-cube-cbrt
10×add-sqr-sqrt
8×add-exp-log
8×log-div
8×pow1
6×sqrt-prod
6×log-pow
4×add-log-exp
4×rem-log-exp
4×sqrt-div
4×add-cbrt-cube
3×flip-+
3×flip3-+
2×pow1/2
1×distribute-lft-out
1×flip--
1×flip3--
Counts
4 → 65
Calls
4 calls:
Slowest
9.0ms
(log (+ (* 1 1) (- (* x x) (* 1 x))))
7.0ms
(log (sqrt (+ 1 (pow x 3))))
3.0ms
(log (sqrt (+ 1 (pow x 3))))
2.0ms
(log (- 1 x))

series826.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
363.0ms
(log (sqrt (+ 1 (pow x 3))))
362.0ms
(log (sqrt (+ 1 (pow x 3))))
54.0ms
(log (+ (* 1 1) (- (* x x) (* 1 x))))
46.0ms
(log (- 1 x))

simplify3.5s

Counts
43 → 77
Calls
43 calls:
Slowest
905.0ms
(log (+ (pow (* 1 1) 3) (pow (- (* x x) (* 1 x)) 3)))
372.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
284.0ms
(log (- (* (* 1 1) (* 1 1)) (* (- (* x x) (* 1 x)) (- (* x x) (* 1 x)))))
282.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
273.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune1.1s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

regimes53.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample12.6s

Algorithm
intervals
Results
8000×(pre true 80)
3457×(body real 1280)
2538×(body real 640)
1182×(body real 320)
605×(body real 160)
218×(body real 80)