Average Error: 0.0 → 0.0
Time: 25.0s
Precision: 64
Internal Precision: 128
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}{\log \left(e^{\frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}\right) + 2}\]

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied add-log-exp0.0

    \[\leadsto \frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \color{blue}{\log \left(e^{\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\right)}}\]

  4. Final simplification0.0

    \[\leadsto \frac{1 + \frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}{\log \left(e^{\frac{t \cdot 2}{1 + t} \cdot \frac{t \cdot 2}{1 + t}}\right) + 2}\]

Reproduce

herbie shell --seed 2019010 
(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: 24.8s)Debug log

sample49.0ms

Algorithm
intervals

simplify540.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
540.0ms
(/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))))

prune9.0ms

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

rewrite44.0ms

Algorithm
rewrite-expression-head
Rules
32×add-exp-log
32×add-cbrt-cube
18×associate-*l*
18×associate-*r*
18×times-frac
16×add-cube-cbrt
16×*-un-lft-identity
16×add-sqr-sqrt
14×pow1
12×associate-/r/
10×div-exp
10×cbrt-undiv
8×cbrt-unprod
8×prod-exp
6×div-inv
6×flip-+
6×flip3-+
6×associate-/r*
4×add-log-exp
2×associate-/l*
2×pow-prod-up
2×*-commutative
2×associate-*r/
2×associate-*l/
2×pow-prod-down
2×pow-plus
2×frac-2neg
2×clear-num
2×frac-times
2×pow2
Counts
4 → 124
Calls
4 calls:
Slowest
19.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
14.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
6.0ms
(/ (* 2 t) (+ 1 t))
4.0ms
(/ (* 2 t) (+ 1 t))

series53.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
18.0ms
(/ (* 2 t) (+ 1 t))
12.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
12.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
11.0ms
(/ (* 2 t) (+ 1 t))

simplify9.8s

Counts
104 → 136
Calls
104 calls:
Slowest
830.0ms
(* (* (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))) (/ (* 2 t) (+ 1 t))) (* (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))) (/ (* 2 t) (+ 1 t))))
785.0ms
(* (* (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))) (/ (* 2 t) (+ 1 t))) (* (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))) (/ (* 2 t) (+ 1 t))))
486.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
485.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
453.0ms
(- (+ (* 2 t) (* 2 (pow t 3))) (* 2 (pow t 2)))

prune977.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

1.5b
(log (exp (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))))
0.0b
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
0.0b
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
0.0b
(exp (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
28×add-exp-log
28×add-cbrt-cube
18×associate-*l*
18×associate-*r*
15×pow1
13×add-cube-cbrt
13×*-un-lft-identity
13×add-sqr-sqrt
12×times-frac
8×cbrt-unprod
8×associate-/r/
8×prod-exp
8×div-exp
8×cbrt-undiv
7×add-log-exp
4×div-inv
4×flip-+
4×flip3-+
3×log-pow
3×log-prod
2×pow-prod-up
2×*-commutative
2×associate-*r/
2×associate-*l/
2×pow-prod-down
2×exp-to-pow
2×pow-plus
2×exp-prod
2×frac-times
2×pow2
1×rem-exp-log
1×rem-log-exp
Counts
4 → 106
Calls
4 calls:
Slowest
15.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
12.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
6.0ms
(log (exp (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))))
3.0ms
(exp (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))

series67.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
25.0ms
(exp (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))
18.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
12.0ms
(log (exp (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))))
12.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))

simplify10.4s

Counts
87 → 118
Calls
87 calls:
Slowest
1.1s
(* (* (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))) (/ (* 2 t) (+ 1 t))) (* (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))) (/ (* 2 t) (+ 1 t))))
702.0ms
(* (* (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))) (/ (* 2 t) (+ 1 t))) (* (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))) (/ (* 2 t) (+ 1 t))))
547.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))
535.0ms
(log (sqrt (exp (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))
523.0ms
(* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))

prune1.4s

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample1.2s

Algorithm
intervals