Average Error: 58.3 → 0.3
Time: 23.7s
Precision: 64
Internal Precision: 128
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(x \cdot 2 + \left(\frac{2}{5} \cdot {x}^{5} + {x}^{3} \cdot \frac{2}{3}\right)\right) \cdot \frac{1}{2}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.3

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

  2. Initial simplification58.3

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

  3. Taylor expanded around 0 0.3

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \left(\frac{2}{3} \cdot {x}^{3} + \frac{2}{5} \cdot {x}^{5}\right)\right)}\]

  4. Final simplification0.3

    \[\leadsto \left(x \cdot 2 + \left(\frac{2}{5} \cdot {x}^{5} + {x}^{3} \cdot \frac{2}{3}\right)\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))

Details

Time bar (total: 8.8s)Debug log

start416.0ms

Algorithm
intervals

setup86.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.8b

localize18.0ms

Local error

Found 2 expressions with local error:

7.7b
(log (/ (+ x 1) (- 1 x)))
0.0b
(/ (+ x 1) (- 1 x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 57
Calls

2 calls. Slowest were:

3.0ms
(/ (+ x 1) (- 1 x))
1.0ms
(log (/ (+ x 1) (- 1 x)))

series75.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

64.0ms
(log (/ (+ x 1) (- 1 x)))
11.0ms
(/ (+ x 1) (- 1 x))

simplify1.1s

Counts
53 → 63
Calls

53 calls. Slowest were:

227.0ms
(/ (* (* (+ x 1) (+ x 1)) (+ x 1)) (* (* (- 1 x) (- 1 x)) (- 1 x)))
164.0ms
(- (+ (* 2 (/ 1 (pow x 2))) (+ (* 2 (/ 1 x)) 1)))
136.0ms
(- (+ (* 2 (/ 1 (pow x 2))) (+ (* 2 (/ 1 x)) 1)))

prune576.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 3 expressions with local error:

0.1b
(* 2/3 (pow x 3))
0.1b
(* 2/5 (pow x 5))
0.0b
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

3.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
1.0ms
(* 2/3 (pow x 3))
1.0ms
(* 2/5 (pow x 5))

series108.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

69.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
21.0ms
(* 2/5 (pow x 5))
17.0ms
(* 2/3 (pow x 3))

simplify92.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

27.0ms
(* (exp (* 2/3 (pow x 3))) (exp (* 2/5 (pow x 5))))
12.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
11.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))

prune853.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 3 expressions with local error:

7.8b
(log (+ x 1))
7.8b
(log (- 1 x))
0.0b
(- (log (+ x 1)) (log (- 1 x)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 55
Calls

3 calls. Slowest were:

5.0ms
(- (log (+ x 1)) (log (- 1 x)))
1.0ms
(log (+ x 1))
1.0ms
(log (- 1 x))

series230.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

158.0ms
(- (log (+ x 1)) (log (- 1 x)))
38.0ms
(log (- 1 x))
33.0ms
(log (+ x 1))

simplify1.7s

Counts
38 → 64
Calls

38 calls. Slowest were:

336.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
309.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
213.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune515.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize42.0ms

Local error

Found 4 expressions with local error:

7.9b
(log (sqrt (+ 1 x)))
7.7b
(log (/ (sqrt (+ x 1)) (- 1 x)))
0.0b
(/ (sqrt (+ x 1)) (- 1 x))
0.0b
(sqrt (+ x 1))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 110
Calls

4 calls. Slowest were:

8.0ms
(log (sqrt (+ 1 x)))
7.0ms
(/ (sqrt (+ x 1)) (- 1 x))
3.0ms
(log (/ (sqrt (+ x 1)) (- 1 x)))

series170.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

67.0ms
(log (/ (sqrt (+ x 1)) (- 1 x)))
43.0ms
(/ (sqrt (+ x 1)) (- 1 x))
42.0ms
(log (sqrt (+ 1 x)))
18.0ms
(sqrt (+ x 1))

simplify1.5s

Counts
104 → 122
Calls

104 calls. Slowest were:

168.0ms
(+ (* 3/2 x) (+ (* 1/2 (pow x 3)) (* 1/4 (pow x 2))))
136.0ms
(- (+ (* 1/2 x) (* 1/6 (pow x 3))) (* 1/4 (pow x 2)))
128.0ms
(/ (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (- 1 x) (- 1 x)) (- 1 x)))

prune1.1s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

regimes42.0ms

Accuracy

0% (0.2b remaining)

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

bsearch3.0ms