Average Error: 18.0 → 1.4
Time: 21.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\left(\frac{\sqrt[3]{-t1}}{\sqrt[3]{u + t1}} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{u + t1}}\right) \cdot \frac{\frac{\sqrt[3]{-t1}}{\sqrt[3]{u + t1}}}{\frac{u + t1}{v}}\]

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 18.0

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

  2. Initial simplification1.7

    \[\leadsto \frac{\frac{-t1}{t1 + u}}{\frac{t1 + u}{v}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity1.7

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

  5. Applied add-cube-cbrt2.3

    \[\leadsto \frac{\frac{-t1}{\color{blue}{\left(\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}\right) \cdot \sqrt[3]{t1 + u}}}}{1 \cdot \frac{t1 + u}{v}}\]

  6. Applied add-cube-cbrt2.0

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}\right) \cdot \sqrt[3]{-t1}}}{\left(\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}\right) \cdot \sqrt[3]{t1 + u}}}{1 \cdot \frac{t1 + u}{v}}\]

  7. Applied times-frac2.0

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}}}{1 \cdot \frac{t1 + u}{v}}\]

  8. Applied times-frac1.4

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{-t1} \cdot \sqrt[3]{-t1}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}}{1} \cdot \frac{\frac{\sqrt[3]{-t1}}{\sqrt[3]{t1 + u}}}{\frac{t1 + u}{v}}}\]

  9. Simplified1.4

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

  10. Final simplification1.4

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

Reproduce

herbie shell --seed 2018362 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

Details

Time bar (total: 18.3s)Debug log

start106.0ms

Algorithm
intervals

setup145.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.7b

localize31.0ms

Local error

Found 3 expressions with local error:

1.4b
(/ (/ (- t1) (+ t1 u)) (/ (+ t1 u) v))
0.0b
(/ (+ t1 u) v)
0.0b
(/ (- t1) (+ t1 u))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
3 → 162
Calls

3 calls. Slowest were:

8.0ms
(/ (/ (- t1) (+ t1 u)) (/ (+ t1 u) v))
3.0ms
(/ (- t1) (+ t1 u))
2.0ms
(/ (+ t1 u) v)

series137.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

99.0ms
(/ (/ (- t1) (+ t1 u)) (/ (+ t1 u) v))
24.0ms
(/ (- t1) (+ t1 u))
14.0ms
(/ (+ t1 u) v)

simplify8.5s

Counts
220 → 171
Calls

220 calls. Slowest were:

603.0ms
(/ (/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u))) (* (* (/ (+ t1 u) v) (/ (+ t1 u) v)) (/ (+ t1 u) v)))
358.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
352.0ms
(/ (/ (- t1) (- (* t1 t1) (* u u))) (* (cbrt (/ (+ t1 u) v)) (cbrt (/ (+ t1 u) v))))

prune1.6s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize39.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (- t1))
0.6b
(cbrt (- t1))
0.6b
(cbrt (- t1))
0.6b
(cbrt (+ t1 u))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 49
Calls

4 calls. Slowest were:

1.0ms
(cbrt (- t1))
1.0ms
(cbrt (+ t1 u))
1.0ms
(cbrt (- t1))

series1.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

302.0ms
(cbrt (- t1))
291.0ms
(cbrt (- t1))
285.0ms
(cbrt (- t1))
137.0ms
(cbrt (+ t1 u))

simplify339.0ms

Counts
29 → 61
Calls

29 calls. Slowest were:

188.0ms
(- (+ (* 1/3 (* (pow (/ 1 (pow u 2)) 1/3) t1)) (pow u 1/3)) (* 1/9 (* (pow (/ 1 (pow u 5)) 1/3) (pow t1 2))))
52.0ms
(cbrt (- (* t1 t1) (* u u)))
10.0ms
(* (pow (* t1 -1) 1/3) (cbrt -1))

prune877.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

localize8.0ms

Local error

Found 3 expressions with local error:

1.2b
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
0.0b
(/ v (+ t1 u))
0.0b
(/ (- t1) (+ t1 u))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 91
Calls

3 calls. Slowest were:

6.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
3.0ms
(/ (- t1) (+ t1 u))
0.0ms
(/ v (+ t1 u))

series129.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

81.0ms
(* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))
28.0ms
(/ v (+ t1 u))
20.0ms
(/ (- t1) (+ t1 u))

simplify1.8s

Counts
82 → 100
Calls

82 calls. Slowest were:

433.0ms
(/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u)))
273.0ms
(+ (- (log (- t1)) (log (+ t1 u))) (log (/ v (+ t1 u))))
157.0ms
(* (/ (* (* (- t1) (- t1)) (- t1)) (* (* (+ t1 u) (+ t1 u)) (+ t1 u))) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))

prune911.0ms

Pruning

7 alts after pruning (4 fresh and 3 done)

Merged error: 0b

localize11.0ms

Local error

Found 3 expressions with local error:

8.6b
(* (- t1) (/ v (+ t1 u)))
0.1b
(/ (* (- t1) (/ v (+ t1 u))) (+ t1 u))
0.0b
(/ v (+ t1 u))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
3 → 63
Calls

3 calls. Slowest were:

13.0ms
(/ (* (- t1) (/ v (+ t1 u))) (+ t1 u))
7.0ms
(* (- t1) (/ v (+ t1 u)))
1.0ms
(/ v (+ t1 u))

series110.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

48.0ms
(* (- t1) (/ v (+ t1 u)))
44.0ms
(/ (* (- t1) (/ v (+ t1 u))) (+ t1 u))
17.0ms
(/ v (+ t1 u))

simplify1.7s

Counts
45 → 72
Calls

45 calls. Slowest were:

427.0ms
(* (* (* (- t1) (- t1)) (- t1)) (* (* (/ v (+ t1 u)) (/ v (+ t1 u))) (/ v (+ t1 u))))
199.0ms
(- (log (* (- t1) (/ v (+ t1 u)))) (log (+ t1 u)))
189.0ms
(- (+ (log (- t1)) (log (/ v (+ t1 u)))) (log (+ t1 u)))

prune679.0ms

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0b

regimes145.0ms

Accuracy

0% (1.4b remaining)

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

bsearch3.0ms