Average Error: 58.6 → 0.2
Time: 23.8s
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 58.6

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

  2. Initial simplification58.6

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

  3. Taylor expanded around 0 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2018362 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))

Details

Time bar (total: 9.4s)Debug log

start794.0ms

Algorithm
intervals

setup72.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.9b

localize13.0ms

Local error

Found 2 expressions with local error:

7.2b
(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)))

series92.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

81.0ms
(log (/ (+ x 1) (- 1 x)))
10.0ms
(/ (+ x 1) (- 1 x))

simplify1.4s

Counts
53 → 63
Calls

53 calls. Slowest were:

310.0ms
(/ (* (* (+ x 1) (+ x 1)) (+ x 1)) (* (* (- 1 x) (- 1 x)) (- 1 x)))
187.0ms
(- (+ (* 2 (/ 1 (pow x 2))) (+ (* 2 (/ 1 x)) 1)))
185.0ms
(- (+ (* 2 (/ 1 (pow x 2))) (+ (* 2 (/ 1 x)) 1)))

prune697.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize33.0ms

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)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

7.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
2.0ms
(* 2/3 (pow x 3))
2.0ms
(* 2/5 (pow x 5))

series160.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

97.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
35.0ms
(* 2/5 (pow x 5))
27.0ms
(* 2/3 (pow x 3))

simplify139.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

35.0ms
(* (exp (* 2/3 (pow x 3))) (exp (* 2/5 (pow x 5))))
23.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
16.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))

prune706.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 2 expressions with local error:

7.2b
(log (- 1 x))
7.0b
(log (+ x 1))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

2.0ms
(log (- 1 x))
1.0ms
(log (+ x 1))

series81.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

42.0ms
(log (+ x 1))
39.0ms
(log (- 1 x))

simplify1.4s

Counts
18 → 34
Calls

18 calls. Slowest were:

421.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
289.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
220.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune241.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

7.2b
(log (- 1 x))
7.2b
(log (- 1 x))
7.2b
(log (- 1 x))
7.0b
(log (+ x 1))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

2.0ms
(log (- 1 x))
2.0ms
(log (+ x 1))
1.0ms
(log (- 1 x))

series160.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

42.0ms
(log (+ x 1))
40.0ms
(log (- 1 x))
39.0ms
(log (- 1 x))
39.0ms
(log (- 1 x))

simplify2.7s

Counts
36 → 68
Calls

36 calls. Slowest were:

317.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
299.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
260.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune625.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes45.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch4.0ms