Average Error: 17.4 → 1.3
Time: 17.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\frac{t1}{t1 + u} \cdot v\right) \cdot \frac{-1}{t1 + u}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.4

    \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
  2. Using strategy rm

  3. Applied times-frac1.3

    \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}}\]
  4. Using strategy rm

  5. Applied div-inv1.4

    \[\leadsto \frac{-t1}{t1 + u} \cdot \color{blue}{\left(v \cdot \frac{1}{t1 + u}\right)}\]

  6. Applied associate-*r*1.3

    \[\leadsto \color{blue}{\left(\frac{-t1}{t1 + u} \cdot v\right) \cdot \frac{1}{t1 + u}}\]

  7. Final simplification1.3

    \[\leadsto \left(\frac{t1}{t1 + u} \cdot v\right) \cdot \frac{-1}{t1 + u}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

Details

Time bar (total: 17.2s)Debug log

sample98.0ms

Algorithm
intervals

simplify140.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

139.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.4b

localize18.0ms

Local error

Found 2 expressions with local error:

16.4b
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
0.0b
(* (+ t1 u) (+ t1 u))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
2 → 60
Calls

2 calls. Slowest were:

9.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
4.0ms
(* (+ t1 u) (+ t1 u))

series66.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

56.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
9.0ms
(* (+ t1 u) (+ t1 u))

simplify4.6s

Counts
48 → 66
Calls

48 calls. Slowest were:

874.0ms
(* (- (* t1 t1) (* u u)) (+ (pow t1 3) (pow u 3)))
312.0ms
(* (+ (* t1 t1) (- (* u u) (* t1 u))) (+ (* t1 t1) (- (* u u) (* t1 u))))
289.0ms
(* (+ (* t1 t1) (- (* u u) (* t1 u))) (+ (* t1 t1) (- (* u u) (* t1 u))))

prune649.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 2 expressions with local error:

1.1b
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
0.0b
(/ (- t1) (+ t1 u))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
2 → 80
Calls

2 calls. Slowest were:

6.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
3.0ms
(/ (- t1) (+ t1 u))

series96.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

75.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
21.0ms
(/ (- t1) (+ t1 u))

simplify1.8s

Counts
74 → 86
Calls

74 calls. Slowest were:

523.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
258.0ms
(+ (- (log (- t1)) (log (+ t1 u))) (log (/ v (+ t1 u))))
168.0ms
(* (/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u))) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))

prune794.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize17.0ms

Local error

Found 3 expressions with local error:

1.8b
(* (/ (- t1) (+ t1 u)) v)
0.4b
(* (* (/ (- t1) (+ t1 u)) v) (/ 1 (+ t1 u)))
0.0b
(/ (- t1) (+ t1 u))

rewrite40.0ms

Algorithm
rewrite-expression-head
Counts
3 → 93
Calls

3 calls. Slowest were:

23.0ms
(* (* (/ (- t1) (+ t1 u)) v) (/ 1 (+ t1 u)))
9.0ms
(* (/ (- t1) (+ t1 u)) v)
6.0ms
(/ (- t1) (+ t1 u))

series166.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

73.0ms
(* (/ (- t1) (+ t1 u)) v)
72.0ms
(* (* (/ (- t1) (+ t1 u)) v) (/ 1 (+ t1 u)))
21.0ms
(/ (- t1) (+ t1 u))

simplify2.3s

Counts
80 → 102
Calls

80 calls. Slowest were:

424.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
391.0ms
(* (* (/ (- t1) (+ t1 u)) v) (/ 1 (+ t1 u)))
306.0ms
(* (* (/ t1 (+ t1 u)) v) (/ 1 (+ t1 u)))

prune996.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

localize12.0ms

Local error

Found 3 expressions with local error:

9.4b
(/ t1 (/ (+ u t1) (- v)))
0.3b
(/ (/ t1 (/ (+ u t1) (- v))) (+ t1 u))
0.0b
(/ (+ u t1) (- v))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 104
Calls

3 calls. Slowest were:

6.0ms
(/ (+ u t1) (- v))
4.0ms
(/ (/ t1 (/ (+ u t1) (- v))) (+ t1 u))
1.0ms
(/ t1 (/ (+ u t1) (- v)))

series124.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

62.0ms
(/ (/ t1 (/ (+ u t1) (- v))) (+ t1 u))
41.0ms
(/ t1 (/ (+ u t1) (- v)))
21.0ms
(/ (+ u t1) (- v))

simplify2.8s

Counts
99 → 113
Calls

99 calls. Slowest were:

449.0ms
(/ (* (* (+ u t1) (+ u t1)) (+ u t1)) (* (* (- v) (- v)) (- v)))
254.0ms
(/ (/ 1 (/ (+ u t1) (- v))) (sqrt (+ t1 u)))
236.0ms
(/ (/ 1 (/ (+ u t1) (- v))) (+ t1 u))

prune1.0s

Pruning

4 alts after pruning (0 fresh and 4 done)

Merged error: 0b

regimes241.0ms

Accuracy

0% (1.3b remaining)

Error of 1.3b against oracle of 0.0b and baseline of 1.3b

bsearch13.0ms

end0.0ms

sample1.1s

Algorithm
intervals