Average Error: 0.0 → 0.0
Time: 37.1s
Precision: 64
Internal Precision: 128
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{(\left(\frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t}\right) + 1)_*}{(\left(\frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t}\right) + 2)_*}\]

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))

Details

Time bar (total: 36.6s)Debug log

sample44.0ms

Algorithm
intervals

simplify593.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

593.0ms
(/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))))

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize38.0ms

Local error

Found 4 expressions with local error:

0.0b
(/ (* t 2) (+ 1 t))
0.0b
(/ (* t 2) (+ 1 t))
0.0b
(/ (* t 2) (+ 1 t))
0.0b
(/ (* t 2) (+ 1 t))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 92
Calls

4 calls. Slowest were:

6.0ms
(/ (* t 2) (+ 1 t))
6.0ms
(/ (* t 2) (+ 1 t))
6.0ms
(/ (* t 2) (+ 1 t))

series75.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

23.0ms
(/ (* t 2) (+ 1 t))
18.0ms
(/ (* t 2) (+ 1 t))
17.0ms
(/ (* t 2) (+ 1 t))
16.0ms
(/ (* t 2) (+ 1 t))

simplify5.8s

Counts
64 → 104
Calls

64 calls. Slowest were:

743.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
736.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
643.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune969.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize43.0ms

Local error

Found 4 expressions with local error:

0.1b
(/ (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1) (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2))
0.0b
(/ (* t 2) (+ 1 t))
0.0b
(/ (* t 2) (+ 1 t))
0.0b
(/ (* t 2) (+ 1 t))

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

6.0ms
(/ (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1) (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2))
6.0ms
(/ (* t 2) (+ 1 t))
6.0ms
(/ (* t 2) (+ 1 t))

series137.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(/ (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1) (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2))
18.0ms
(/ (* t 2) (+ 1 t))
17.0ms
(/ (* t 2) (+ 1 t))
17.0ms
(/ (* t 2) (+ 1 t))

simplify9.9s

Counts
76 → 110
Calls

76 calls. Slowest were:

749.0ms
(/ (* (* (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1) (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1)) (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1)) (* (* (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2) (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2)) (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2)))
679.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))
666.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune1.3s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize27.0ms

Local error

Found 4 expressions with local error:

20.7b
(cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1))
20.7b
(cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1))
20.7b
(cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1))
0.1b
(/ (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2) (cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

4.0ms
(/ (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2) (cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1)))
1.0ms
(cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1))
1.0ms
(cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1))

series412.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

136.0ms
(/ (fma (/ (* t 2) (+ 1 t)) (/ (* t 2) (+ 1 t)) 2) (cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1)))
95.0ms
(cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1))
93.0ms
(cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1))
87.0ms
(cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1))

simplify14.3s

Counts
67 → 92
Calls

67 calls. Slowest were:

597.0ms
(sqrt (cbrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1)))
541.0ms
(cbrt (sqrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1)))
511.0ms
(cbrt (sqrt (fma (/ (* t 2) (+ 1 t)) (fma (* t t) (fma 2 t -2) (* 2 t)) 1)))

prune1.2s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0.0b

regimes26.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.6s

Algorithm
intervals