Average Error: 58.5 → 0.0
Time: 14.8s
Precision: 64
Internal Precision: 128
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[(\left(\log_* (1 + \left(-x\right))\right) \cdot \frac{-1}{2} + \left(\frac{\log_* (1 + x)}{2}\right))_*\]

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  2. Simplified50.4

    \[\leadsto \color{blue}{(\left(\log \left(1 - x\right)\right) \cdot \frac{-1}{2} + \left(\frac{\log_* (1 + x)}{2}\right))_*}\]
  3. Using strategy rm

  4. Applied sub-neg50.4

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

  5. Applied log1p-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))

Details

Time bar (total: 14.2s)Debug log

sample242.0ms

Algorithm
intervals

simplify101.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

101.0ms
(* (/ 1 2) (log (/ (+ 1 x) (- 1 x))))

prune11.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 50.2b

localize391.0ms

Local error

Found 2 expressions with local error:

5.1b
(log (- 1 x))
0.0b
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

2.0ms
(log (- 1 x))
0.0ms
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))

series223.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

181.0ms
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))
41.0ms
(log (- 1 x))

simplify1.2s

Counts
14 → 34
Calls

14 calls. Slowest were:

312.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
296.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
273.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune325.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))

series122.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

122.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))

simplify242.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

95.0ms
(+ (/ 1 x) (+ (* 1/3 (/ 1 (pow x 3))) (* 1/2 (log -1))))
86.0ms
(- (+ (* 1/3 (/ 1 (pow x 3))) (/ 1 x)) (* 1/2 (log -1)))
59.0ms
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))

prune140.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 1/3 (pow x 3))
0.1b
(* 1/5 (pow x 5))
0.0b
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
0.0b
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

6.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
4.0ms
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))
3.0ms
(* 1/3 (pow x 3))

series185.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

92.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
35.0ms
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))
34.0ms
(* 1/5 (pow x 5))
24.0ms
(* 1/3 (pow x 3))

simplify285.0ms

Counts
21 → 67
Calls

21 calls. Slowest were:

51.0ms
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))
49.0ms
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))
46.0ms
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))

prune607.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 3 expressions with local error:

0.2b
(* (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))) (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))))
0.0b
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))
0.0b
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
3 → 52
Calls

3 calls. Slowest were:

7.0ms
(* (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))) (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))))
0.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))
0.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))

series433.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

155.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))
150.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))
127.0ms
(* (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))) (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))))

simplify1.3s

Counts
31 → 61
Calls

31 calls. Slowest were:

178.0ms
(* (fma (log1p (- x)) -1/2 (/ (log1p x) 2)) (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))
144.0ms
(* (fma (log1p (- x)) -1/2 (/ (log1p x) 2)) (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))
143.0ms
(* (* (* (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))) (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))) (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))) (* (* (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))) (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))) (sqrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))))

prune626.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes35.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample7.7s

Algorithm
intervals