Average Error: 58.3 → 0.0
Time: 12.1s
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. Simplified50.3

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

  4. Applied sub-neg50.3

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

  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 2019016 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))

Details

Time bar (total: 11.7s)Debug log

sample255.0ms

Algorithm
intervals
Results
524×(pre true 80)
268×(body nan 80)
105×(body real 1280)
82×(body real 640)
41×(body real 320)
17×(body real 160)
11×(body real 80)

simplify109.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
108.0ms
(* (/ 1 2) (log (/ (+ 1 x) (- 1 x))))

prune19.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 50.1b

localize17.0ms

Local error

Found 2 expressions with local error:

6.1b
(log (- 1 x))
0.0b
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
3×add-cube-cbrt
3×add-exp-log
3×*-un-lft-identity
3×log-prod
3×pow1
3×add-sqr-sqrt
2×add-log-exp
2×log1p-expm1-u
2×add-cbrt-cube
2×log1p-def
2×log-div
2×expm1-log1p-u
1×flip--
1×fma-udef
1×log-pow
1×rem-log-exp
1×flip3--
1×sub-neg
Counts
2 → 28
Calls
2 calls:
Slowest
1.0ms
(log (- 1 x))
0.0ms
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))

series190.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
146.0ms
(fma (log (- 1 x)) -1/2 (/ (log1p x) 2))
44.0ms
(log (- 1 x))

simplify1.3s

Counts
14 → 34
Calls
14 calls:
Slowest
349.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
301.0ms
(- (+ x (+ (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))))
270.0ms
(- (+ (log (/ -1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))
116.0ms
(- (+ (* 1/3 (/ 1 (pow x 3))) (/ 1 x)) (* 1/2 (log -1)))
96.0ms
(+ (/ 1 x) (+ (* 1/3 (/ 1 (pow x 3))) (* 1/2 (log -1))))

prune150.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 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
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))

series141.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
141.0ms
(fma (log1p (- x)) -1/2 (/ (log1p x) 2))

simplify257.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
107.0ms
(- (+ (* 1/3 (/ 1 (pow x 3))) (/ 1 x)) (* 1/2 (log -1)))
82.0ms
(+ x (+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5))))
66.0ms
(+ (/ 1 x) (+ (* 1/3 (/ 1 (pow x 3))) (* 1/2 (log -1))))

prune95.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize34.0ms

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)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
8×associate-*r*
5×add-log-exp
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
3×log1p-expm1-u
3×add-exp-log
3×add-cbrt-cube
3×pow1
3×expm1-log1p-u
2×*-commutative
1×flip-+
1×unpow3
1×flip3-+
1×fma-def
1×sum-log
1×+-commutative
1×cube-mult
Counts
3 → 42
Calls
3 calls:
Slowest
6.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
3.0ms
(* 1/3 (pow x 3))
2.0ms
(* 1/5 (pow x 5))

series129.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
67.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
32.0ms
(* 1/5 (pow x 5))
30.0ms
(* 1/3 (pow x 3))

simplify103.0ms

Counts
17 → 51
Calls
17 calls:
Slowest
22.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
19.0ms
(* (exp (* 1/3 (pow x 3))) (exp (* 1/5 (pow x 5))))
17.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
13.0ms
(+ (* 1/3 (pow x 3)) (* 1/5 (pow x 5)))
6.0ms
(* 1/3 (pow x 3))

prune504.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes18.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample8.4s

Algorithm
intervals
Results
15997×(pre true 80)
7996×(body nan 80)
3363×(body real 1280)
2518×(body real 640)
1257×(body real 320)
636×(body real 160)
227×(body real 80)