Average Error: 17.7 → 1.4
Time: 15.9s
Precision: 64
Internal Precision: 128
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{v}{t1 + u} \cdot \frac{-1}{\frac{t1 + u}{t1}}\]

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 17.7

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

  3. Applied times-frac1.2

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

  5. Applied neg-mul-11.2

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

  6. Applied associate-/l*1.4

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

  7. Final simplification1.4

    \[\leadsto \frac{v}{t1 + u} \cdot \frac{-1}{\frac{t1 + u}{t1}}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

Details

Time bar (total: 14.9s)Debug log

sample100.0ms

Algorithm
intervals
Results
262×(pre true 80)
262×(body real 80)

simplify124.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
124.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 16.0b

localize17.0ms

Local error

Found 2 expressions with local error:

16.0b
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
0.0b
(* (+ t1 u) (+ t1 u))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
12×flip-+
12×flip3-+
9×add-exp-log
9×add-cbrt-cube
8×associate-/r/
8×frac-times
7×pow1
4×add-cube-cbrt
4×associate-*r/
4×associate-*l/
4×*-un-lft-identity
4×add-sqr-sqrt
3×associate-*l*
3×associate-*r*
2×add-log-exp
2×log1p-expm1-u
2×cbrt-unprod
2×prod-exp
2×div-exp
2×cbrt-undiv
2×expm1-log1p-u
1×associate-/l*
1×div-inv
1×pow-prod-up
1×*-commutative
1×distribute-frac-neg
1×pow-prod-down
1×pow-plus
1×times-frac
1×distribute-lft-in
1×frac-2neg
1×associate-/r*
1×clear-num
1×distribute-rgt-in
1×pow2
1×distribute-lft-neg-out
Counts
2 → 60
Calls
2 calls:
Slowest
10.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
4.0ms
(* (+ t1 u) (+ t1 u))

series68.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
51.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
17.0ms
(* (+ t1 u) (+ t1 u))

simplify4.2s

Counts
48 → 66
Calls
48 calls:
Slowest
390.0ms
(* (- (* t1 t1) (* u u)) (+ (pow t1 3) (pow u 3)))
334.0ms
(* (+ (* t1 t1) (- (* u u) (* t1 u))) (+ (* t1 t1) (- (* u u) (* t1 u))))
302.0ms
(/ (* (* (* (- t1) v) (* (- t1) v)) (* (- t1) v)) (* (* (* (+ t1 u) (+ t1 u)) (+ t1 u)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u))))
278.0ms
(/ (* (* (* (- t1) v) (* (- t1) v)) (* (- t1) v)) (* (* (* (+ t1 u) (+ t1 u)) (* (+ t1 u) (+ t1 u))) (* (+ t1 u) (+ t1 u))))
270.0ms
(* (+ (pow t1 3) (pow u 3)) (+ (pow t1 3) (pow u 3)))

prune587.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 3 expressions with local error:

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

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
18×associate-*l*
10×add-exp-log
10×add-cbrt-cube
8×associate-/r/
7×neg-mul-1
6×associate-*r*
6×associate-/r*
5×pow1
4×associate-/l*
4×div-inv
4×flip-+
4×flip3-+
3×add-log-exp
3×log1p-expm1-u
3×expm1-log1p-u
2×cbrt-unprod
2×distribute-frac-neg
2×prod-exp
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×div-sub
1×frac-times
1×neg-sub0
1×distribute-lft-neg-out
Counts
3 → 97
Calls
3 calls:
Slowest
13.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
7.0ms
(/ (- t1) (+ t1 u))
1.0ms
(/ v (+ t1 u))

series152.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
94.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
30.0ms
(/ v (+ t1 u))
28.0ms
(/ (- t1) (+ t1 u))

simplify2.2s

Counts
82 → 106
Calls
82 calls:
Slowest
483.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
328.0ms
(+ (- (log (- t1)) (log (+ t1 u))) (log (/ v (+ t1 u))))
227.0ms
(* (/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u))) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))
125.0ms
(/ (- t1) (- (* t1 t1) (* u u)))
113.0ms
(+ (* t1 t1) (- (* u u) (* t1 u)))

prune1.1s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

1.1b
(* (/ -1 (/ (+ t1 u) t1)) (/ v (+ t1 u)))
0.2b
(/ -1 (/ (+ t1 u) t1))
0.0b
(/ (+ t1 u) t1)
0.0b
(/ v (+ t1 u))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
7×associate-/r*
6×div-inv
6×associate-/r/
6×add-exp-log
6×associate-*r*
6×add-cbrt-cube
6×pow1
5×associate-*l*
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-/l*
3×flip-+
3×flip3-+
3×frac-2neg
3×clear-num
2×associate-/l/
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×frac-times
Counts
4 → 78
Calls
4 calls:
Slowest
4.0ms
(* (/ -1 (/ (+ t1 u) t1)) (/ v (+ t1 u)))
3.0ms
(/ (+ t1 u) t1)
0.0ms
(/ -1 (/ (+ t1 u) t1))
0.0ms
(/ v (+ t1 u))

series172.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
87.0ms
(* (/ -1 (/ (+ t1 u) t1)) (/ v (+ t1 u)))
34.0ms
(/ -1 (/ (+ t1 u) t1))
30.0ms
(/ v (+ t1 u))
21.0ms
(/ (+ t1 u) t1)

simplify1.2s

Counts
41 → 90
Calls
41 calls:
Slowest
257.0ms
(* (/ -1 (/ (+ t1 u) t1)) (/ v (+ t1 u)))
256.0ms
(* (/ (+ t1 u) t1) (+ t1 u))
156.0ms
(+ (log (/ -1 (/ (+ t1 u) t1))) (log (/ v (+ t1 u))))
150.0ms
(* (* (* (/ -1 (/ (+ t1 u) t1)) (/ -1 (/ (+ t1 u) t1))) (/ -1 (/ (+ t1 u) t1))) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))
117.0ms
(+ (* t1 t1) (- (* u u) (* t1 u)))

prune892.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 3 expressions with local error:

3.2b
(* (/ (- t1) (+ t1 u)) v)
0.7b
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
0.0b
(/ (- t1) (+ t1 u))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
27×times-frac
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
18×associate-*l*
7×add-exp-log
7×add-cbrt-cube
7×neg-mul-1
6×associate-/r/
6×associate-/r*
5×associate-/l*
4×distribute-frac-neg
3×add-log-exp
3×div-inv
3×log1p-expm1-u
3×flip-+
3×flip3-+
3×pow1
3×expm1-log1p-u
2×associate-*l/
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
2×distribute-lft-neg-out
1×associate-/l/
1×*-commutative
1×div-sub
1×neg-sub0
Counts
3 → 92
Calls
3 calls:
Slowest
8.0ms
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
6.0ms
(* (/ (- t1) (+ t1 u)) v)
4.0ms
(/ (- t1) (+ t1 u))

series179.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
105.0ms
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
53.0ms
(* (/ (- t1) (+ t1 u)) v)
21.0ms
(/ (- t1) (+ t1 u))

simplify1.5s

Counts
79 → 101
Calls
79 calls:
Slowest
456.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
156.0ms
(/ (* (* (* (/ (- t1) (+ t1 u)) v) (* (/ (- t1) (+ t1 u)) v)) (* (/ (- t1) (+ t1 u)) v)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
123.0ms
(/ (- t1) (- (* t1 t1) (* u u)))
114.0ms
(+ (* t1 t1) (- (* u u) (* t1 u)))
107.0ms
(- (log (* (/ (- t1) (+ t1 u)) v)) (log (+ t1 u)))

prune994.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes233.0ms

Accuracy

0% (1.4b remaining)

Error of 1.4b against oracle of 0.0b and baseline of 1.4b

bsearch2.0ms

end0.0ms

sample1.1s

Algorithm
intervals
Results
8243×(pre true 80)
8243×(body real 80)