Average Error: 58.3 → 0.0
Time: 23.4s
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 58.3

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

  2. Initial simplification50.3

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

  4. Applied sub-neg50.3

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

  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 2018365 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))

Details

Time bar (total: 9.2s)Debug log

start451.0ms

Algorithm
intervals

setup145.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 50.0b

localize14.0ms

Local error

Found 2 expressions with local error:

7.8b
(log (- 1 x))
0.0b
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
2 → 28
Calls

2 calls. Slowest were:

1.0ms
(log (- 1 x))
0.0ms
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))

series172.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

138.0ms
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))
34.0ms
(log (- 1 x))

simplify1.2s

Counts
14 → 34
Calls

14 calls. Slowest were:

349.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
303.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
293.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune283.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 1 expressions with local error:

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

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

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

series129.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

129.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))

simplify573.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

404.0ms
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))
96.0ms
(- (+ (* 1/3 (/ 1 (pow x 3))) (/ 1 x)) (* 1/2 (log -1)))
72.0ms
(+ (/ 1 x) (+ (* 1/3 (/ 1 (pow x 3))) (* 1/2 (log -1))))

prune128.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize23.0ms

Local error

Found 3 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)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
3 → 42
Calls

3 calls. Slowest were:

11.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
3.0ms
(* 1/3 (pow x 3))
2.0ms
(* 1/5 (pow x 5))

series145.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

84.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
34.0ms
(* 1/5 (pow x 5))
27.0ms
(* 1/3 (pow x 3))

simplify139.0ms

Counts
17 → 51
Calls

17 calls. Slowest were:

33.0ms
(* (exp (* 1/3 (pow x 3))) (exp (* 1/5 (pow x 5))))
20.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
20.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))

prune690.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))
0.6b
(cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))
0.6b
(cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))
0.5b
(* (cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))) (cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))) (cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))))
1.0ms
(cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))
1.0ms
(cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))

series945.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

290.0ms
(* (cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))) (cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2))))
225.0ms
(cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))
223.0ms
(cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))
208.0ms
(cbrt (fma (log1p (- x)) -1/2 (/ (log1p x) 2)))

simplify2.8s

Counts
41 → 81
Calls

41 calls. Slowest were:

311.0ms
(- (+ (* 2/3 (* (/ 1 x) (pow (/ 1/2 (pow (log -1) 2)) 1/3))) (pow (* 1/2 (log -1)) 1/3)) (* 4/9 (* (/ 1 (pow x 2)) (pow (/ 1/2 (pow (log -1) 5)) 1/3))))
307.0ms
(- (pow (* 1/4 (pow (log -1) 2)) 1/3) (+ (* 4/9 (* (pow (/ 1/4 (pow (log -1) 4)) 1/3) (/ 1 (pow x 2)))) (* 4/3 (* (pow (/ 1/4 (log -1)) 1/3) (/ 1 x)))))
281.0ms
(- (+ (* 4/3 (* (pow (/ 1/4 (log -1)) 1/3) (/ 1 x))) (pow (* 1/4 (pow (log -1) 2)) 1/3)) (* 4/9 (* (pow (/ 1/4 (pow (log -1) 4)) 1/3) (/ 1 (pow x 2)))))

prune1.2s

Pruning

4 alts after pruning (0 fresh and 4 done)

Merged error: 0b

regimes57.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms