Average Error: 17.7 → 1.2
Time: 16.2s
Precision: 64
Internal Precision: 128
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{t1}{t1 + u} \cdot \left(-v\right)}{t1 + u}\]

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 17.7

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

  3. Applied times-frac1.1

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

  5. Applied associate-*r/1.2

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

  6. Final simplification1.2

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

Details

Time bar (total: 15.5s)Debug log

sample116.0ms

Algorithm
intervals

simplify310.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
309.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 11.7b

localize39.0ms

Local error

Found 2 expressions with local error:

11.7b
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
0.0b
(* (+ t1 u) (+ t1 u))

rewrite30.0ms

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
19.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
7.0ms
(* (+ t1 u) (+ t1 u))

series93.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
78.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
14.0ms
(* (+ t1 u) (+ t1 u))

simplify4.5s

Counts
48 → 66
Calls
48 calls:
Slowest
389.0ms
(* (+ (* t1 t1) (- (* u u) (* t1 u))) (+ (* t1 t1) (- (* u u) (* t1 u))))
381.0ms
(* (- (* t1 t1) (* u u)) (+ (pow t1 3) (pow u 3)))
341.0ms
(/ (* (* (* (- t1) v) (* (- t1) v)) (* (- t1) v)) (* (* (* (+ t1 u) (+ t1 u)) (* (+ t1 u) (+ t1 u))) (* (+ t1 u) (+ t1 u))))
301.0ms
(/ (* (* (* (- t1) v) (* (- t1) v)) (* (- t1) v)) (* (* (* (+ t1 u) (+ t1 u)) (+ t1 u)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u))))
298.0ms
(* (+ (* t1 t1) (- (* u u) (* t1 u))) (+ (* t1 t1) (- (* u u) (* t1 u))))

prune561.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 3 expressions with local error:

2.7b
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
0.0b
(/ v (+ t1 u))
0.0b
(/ (- t1) (+ t1 u))

rewrite21.0ms

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
12.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
6.0ms
(/ (- t1) (+ t1 u))
1.0ms
(/ v (+ t1 u))

series122.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
73.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
26.0ms
(/ (- t1) (+ t1 u))
24.0ms
(/ v (+ t1 u))

simplify1.7s

Counts
82 → 106
Calls
82 calls:
Slowest
339.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
275.0ms
(+ (- (log (- t1)) (log (+ t1 u))) (log (/ v (+ t1 u))))
180.0ms
(* (/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u))) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))
104.0ms
(+ (* t1 t1) (- (* u u) (* t1 u)))
103.0ms
(/ (- t1) (- (* t1 t1) (* u u)))

prune1.0s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 3 expressions with local error:

3.6b
(* (/ (- t1) (+ t1 u)) v)
0.4b
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
0.0b
(/ (- t1) (+ t1 u))

rewrite32.0ms

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
14.0ms
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
10.0ms
(* (/ (- t1) (+ t1 u)) v)
6.0ms
(/ (- t1) (+ t1 u))

series188.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
99.0ms
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
69.0ms
(* (/ (- t1) (+ t1 u)) v)
20.0ms
(/ (- t1) (+ t1 u))

simplify1.5s

Counts
79 → 101
Calls
79 calls:
Slowest
492.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
144.0ms
(/ (* (* (* (/ (- t1) (+ t1 u)) v) (* (/ (- t1) (+ t1 u)) v)) (* (/ (- t1) (+ t1 u)) v)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
133.0ms
(- (log (* (/ (- t1) (+ t1 u)) v)) (log (+ t1 u)))
107.0ms
(+ (* t1 t1) (- (* u u) (* t1 u)))
86.0ms
(/ (- t1) (- (* t1 t1) (* u u)))

prune936.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

localize11.0ms

Local error

Found 3 expressions with local error:

11.8b
(* (- t1) (/ v (+ t1 u)))
0.4b
(/ (* (- t1) (/ v (+ t1 u))) (+ t1 u))
0.0b
(/ v (+ t1 u))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
10×add-exp-log
10×add-cbrt-cube
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×associate-/r/
6×associate-*r*
6×associate-/r*
5×pow1
4×associate-*l*
3×add-log-exp
3×div-inv
3×log1p-expm1-u
3×flip-+
3×times-frac
3×flip3-+
3×expm1-log1p-u
2×cbrt-unprod
2×associate-*r/
2×prod-exp
2×div-exp
2×frac-2neg
2×clear-num
2×cbrt-undiv
2×distribute-lft-neg-out
1×associate-/l/
1×associate-/l*
1×*-commutative
1×distribute-frac-neg
1×pow-prod-down
1×neg-mul-1
Counts
3 → 69
Calls
3 calls:
Slowest
13.0ms
(/ (* (- t1) (/ v (+ t1 u))) (+ t1 u))
8.0ms
(* (- t1) (/ v (+ t1 u)))
1.0ms
(/ v (+ t1 u))

series130.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
59.0ms
(/ (* (- t1) (/ v (+ t1 u))) (+ t1 u))
52.0ms
(* (- t1) (/ v (+ t1 u)))
18.0ms
(/ v (+ t1 u))

simplify1.8s

Counts
45 → 78
Calls
45 calls:
Slowest
385.0ms
(* (* (* (- t1) (- t1)) (- t1)) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))
213.0ms
(- (log (* (- t1) (/ v (+ t1 u)))) (log (+ t1 u)))
191.0ms
(/ (* (* (* (- t1) (- t1)) (- t1)) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u)))) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
185.0ms
(/ (* (* (* (- t1) (/ v (+ t1 u))) (* (- t1) (/ v (+ t1 u)))) (* (- t1) (/ v (+ t1 u)))) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
180.0ms
(- (+ (log (- t1)) (log (/ v (+ t1 u)))) (log (+ t1 u)))

prune732.0ms

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

regimes172.0ms

Accuracy

0% (1.2b remaining)

Error of 1.2b against oracle of 0.0b and baseline of 1.2b

bsearch4.0ms

end0.0ms

sample1.4s

Algorithm
intervals