Average Error: 17.8 → 1.5
Time: 15.0s
Precision: 64
Internal Precision: 128
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\left(\sqrt[3]{\frac{-t1}{t1 + u}} \cdot \left(\sqrt[3]{\frac{-t1}{t1 + u}} \cdot \sqrt[3]{\frac{-t1}{t1 + u}}\right)\right) \cdot v}{t1 + u}\]

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 17.8

    \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
  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 associate-*r/1.2

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

  7. Applied add-cube-cbrt1.5

    \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\frac{-t1}{t1 + u}} \cdot \sqrt[3]{\frac{-t1}{t1 + u}}\right) \cdot \sqrt[3]{\frac{-t1}{t1 + u}}\right)} \cdot v}{t1 + u}\]

  8. Final simplification1.5

    \[\leadsto \frac{\left(\sqrt[3]{\frac{-t1}{t1 + u}} \cdot \left(\sqrt[3]{\frac{-t1}{t1 + u}} \cdot \sqrt[3]{\frac{-t1}{t1 + u}}\right)\right) \cdot v}{t1 + u}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

Details

Time bar (total: 14.3s)Debug log

sample148.0ms

Algorithm
intervals
Results
34.0ms265×body80valid

simplify231.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
231.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 14.5b

localize34.0ms

Local error

Found 2 expressions with local error:

14.5b
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
0.0b
(* (+ t1 u) (+ t1 u))

rewrite197.0ms

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×cbrt-unprod
2×prod-exp
2×div-exp
2×cbrt-undiv
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 → 56
Calls
2 calls:
Slowest
106.0ms
(* (+ t1 u) (+ t1 u))
61.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))

series186.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
162.0ms
(/ (* (- t1) v) (* (+ t1 u) (+ t1 u)))
24.0ms
(* (+ t1 u) (+ t1 u))

simplify3.9s

Counts
48 → 62
Calls
48 calls:
Slowest
357.0ms
(* (- (* t1 t1) (* u u)) (+ t1 u))
319.0ms
(* (+ (* t1 t1) (- (* u u) (* t1 u))) (- t1 u))
305.0ms
(* (+ (* t1 t1) (- (* u u) (* t1 u))) (- t1 u))
302.0ms
(- (log (* (- t1) v)) (+ (log (+ t1 u)) (log (+ t1 u))))
297.0ms
(* (- (* t1 t1) (* u u)) (+ (pow t1 3) (pow u 3)))

prune596.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 3 expressions with local error:

1.6b
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
0.0b
(/ v (+ t1 u))
0.0b
(/ (- t1) (+ t1 u))

rewrite10.0ms

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
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 → 91
Calls
3 calls:
Slowest
6.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
3.0ms
(/ (- t1) (+ t1 u))
0.0ms
(/ v (+ t1 u))

series122.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
65.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
33.0ms
(/ (- t1) (+ t1 u))
24.0ms
(/ v (+ t1 u))

simplify1.6s

Counts
82 → 100
Calls
82 calls:
Slowest
341.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
315.0ms
(+ (- (log (- t1)) (log (+ t1 u))) (log (/ v (+ t1 u))))
173.0ms
(* (/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u))) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))
82.0ms
(* (* (* (/ (- t1) (+ t1 u)) (/ (- t1) (+ t1 u))) (/ (- t1) (+ t1 u))) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))
75.0ms
(+ (log (/ (- t1) (+ t1 u))) (log (/ v (+ t1 u))))

prune829.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 3 expressions with local error:

4.3b
(* (/ (- t1) (+ t1 u)) v)
0.7b
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
0.0b
(/ (- t1) (+ t1 u))

rewrite14.0ms

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×flip-+
3×flip3-+
3×pow1
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 → 86
Calls
3 calls:
Slowest
6.0ms
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
4.0ms
(* (/ (- t1) (+ t1 u)) v)
3.0ms
(/ (- t1) (+ t1 u))

series168.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
87.0ms
(/ (* (/ (- t1) (+ t1 u)) v) (+ t1 u))
51.0ms
(* (/ (- t1) (+ t1 u)) v)
30.0ms
(/ (- t1) (+ t1 u))

simplify1.1s

Counts
79 → 95
Calls
79 calls:
Slowest
311.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
141.0ms
(- (log (* (/ (- t1) (+ t1 u)) v)) (log (+ t1 u)))
132.0ms
(/ (* (* (* (/ (- t1) (+ t1 u)) v) (* (/ (- t1) (+ t1 u)) v)) (* (/ (- t1) (+ t1 u)) v)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
85.0ms
(/ (- t1) (- (* t1 t1) (* u u)))
78.0ms
(/ (* (/ t1 (+ t1 u)) v) (+ t1 u))

prune823.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

9.8b
(cbrt (/ (- t1) (+ t1 u)))
9.8b
(cbrt (/ (- t1) (+ t1 u)))
9.8b
(cbrt (/ (- t1) (+ t1 u)))
4.3b
(* (* (* (cbrt (/ (- t1) (+ t1 u))) (cbrt (/ (- t1) (+ t1 u)))) (cbrt (/ (- t1) (+ t1 u)))) v)

rewrite40.0ms

Algorithm
rewrite-expression-head
Rules
54×cbrt-prod
36×times-frac
28×add-cube-cbrt
28×*-un-lft-identity
28×add-sqr-sqrt
15×cbrt-div
12×associate-*l/
9×neg-mul-1
6×associate-/r/
5×frac-times
4×add-log-exp
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×div-inv
3×flip-+
3×associate-*r/
3×pow1/3
3×flip3-+
1×*-commutative
1×associate-*l*
Counts
4 → 97
Calls
4 calls:
Slowest
35.0ms
(* (* (* (cbrt (/ (- t1) (+ t1 u))) (cbrt (/ (- t1) (+ t1 u)))) (cbrt (/ (- t1) (+ t1 u)))) v)
1.0ms
(cbrt (/ (- t1) (+ t1 u)))
1.0ms
(cbrt (/ (- t1) (+ t1 u)))
1.0ms
(cbrt (/ (- t1) (+ t1 u)))

series804.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
230.0ms
(cbrt (/ (- t1) (+ t1 u)))
220.0ms
(cbrt (/ (- t1) (+ t1 u)))
218.0ms
(cbrt (/ (- t1) (+ t1 u)))
136.0ms
(* (* (* (cbrt (/ (- t1) (+ t1 u))) (cbrt (/ (- t1) (+ t1 u)))) (cbrt (/ (- t1) (+ t1 u)))) v)

simplify890.0ms

Counts
77 → 109
Calls
77 calls:
Slowest
84.0ms
(cbrt (/ (- t1) (- (* t1 t1) (* u u))))
76.0ms
(cbrt (/ (- t1) (- (* t1 t1) (* u u))))
68.0ms
(cbrt (/ (- t1) (- (* t1 t1) (* u u))))
48.0ms
(* (cbrt -1) (exp (* 1/3 (- (log t1) (log u)))))
43.0ms
(* (cbrt -1) (exp (* 1/3 (- (log t1) (log u)))))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

regimes191.0ms

Accuracy

0% (1.5b remaining)

Error of 1.5b against oracle of 0.0b and baseline of 1.5b

bsearch3.0ms

end0.0ms

sample1.1s

Algorithm
intervals
Results
916.0ms8257×body80valid