Average Error: 58.4 → 0.2
Time: 16.7s
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 58.4

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

  2. Simplified58.4

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

  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 2019008 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))

Details

Time bar (total: 16.0s)Debug log

sample240.0ms

Algorithm
intervals

simplify72.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

72.0ms
(* (/ 1 2) (log (/ (+ 1 x) (- 1 x))))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.6b

localize17.0ms

Local error

Found 2 expressions with local error:

5.2b
(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))
2.0ms
(log (/ (+ x 1) (- 1 x)))

series90.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

79.0ms
(log (/ (+ x 1) (- 1 x)))
11.0ms
(/ (+ x 1) (- 1 x))

simplify926.0ms

Counts
53 → 63
Calls

53 calls. Slowest were:

231.0ms
(/ (* (* (+ x 1) (+ x 1)) (+ x 1)) (* (* (- 1 x) (- 1 x)) (- 1 x)))
108.0ms
(- (+ (* 2 (/ 1 (pow x 2))) (+ (* 2 (/ 1 x)) 1)))
102.0ms
(- (+ (* 2 (/ 1 (pow x 2))) (+ (* 2 (/ 1 x)) 1)))

prune472.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize47.0ms

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)))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 35
Calls

3 calls. Slowest were:

6.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
2.0ms
(* 2/3 (pow x 3))
2.0ms
(* 2/5 (pow x 5))

series148.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

85.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
36.0ms
(* 2/5 (pow x 5))
28.0ms
(* 2/3 (pow x 3))

simplify116.0ms

Counts
17 → 44
Calls

17 calls. Slowest were:

23.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
22.0ms
(+ (* 2/3 (pow x 3)) (* 2/5 (pow x 5)))
18.0ms
(* (exp (* 2/3 (pow x 3))) (exp (* 2/5 (pow x 5))))

prune491.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 2 expressions with local error:

5.3b
(log (- 1 x))
5.2b
(log (+ x 1))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

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

series67.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

37.0ms
(log (+ x 1))
30.0ms
(log (- 1 x))

simplify1.3s

Counts
18 → 34
Calls

18 calls. Slowest were:

322.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
246.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
223.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune186.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

5.2b
(log (/ (+ x 1) (- 1 x)))
5.2b
(log (/ (+ x 1) (- 1 x)))
5.2b
(log (/ (+ x 1) (- 1 x)))
0.1b
(cbrt (log (/ (+ x 1) (- 1 x))))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

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

series349.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

143.0ms
(cbrt (log (/ (+ x 1) (- 1 x))))
72.0ms
(log (/ (+ x 1) (- 1 x)))
72.0ms
(log (/ (+ x 1) (- 1 x)))
62.0ms
(log (/ (+ x 1) (- 1 x)))

simplify2.5s

Counts
70 → 102
Calls

70 calls. Slowest were:

480.0ms
(- (+ (* 2/3 (* (/ 1 x) (pow (/ 1 (pow (log -1) 2)) 1/3))) (pow (log -1) 1/3)) (* 4/9 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (log -1) 5)) 1/3))))
441.0ms
(- (+ (* 2/3 (* (/ 1 x) (pow (/ 1 (pow (log -1) 2)) 1/3))) (pow (log -1) 1/3)) (* 4/9 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (log -1) 5)) 1/3))))
191.0ms
(+ (* 22/405 (* (pow x 4) (exp (* 1/3 (+ (log 2) (log x)))))) (+ (exp (* 1/3 (+ (log 2) (log x)))) (* 1/9 (* (pow x 2) (exp (* 1/3 (+ (log 2) (log x))))))))

prune619.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

regimes25.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms

end0.0ms

sample8.2s

Algorithm
intervals