Average Error: 12.2 → 0.5
Time: 41.9s
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \sqrt{\frac{\frac{2}{r}}{r}} \cdot \sqrt{\frac{\frac{2}{r}}{r}}\right) - (\left(\frac{(-2 \cdot v + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.2

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]

  2. Initial simplification0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - (\left(\frac{(-2 \cdot v + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) + 4.5)_*\]
  3. Using strategy rm

  4. Applied associate-/r*0.4

    \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - (\left(\frac{(-2 \cdot v + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) + 4.5)_*\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.5

    \[\leadsto \left(3 + \color{blue}{\sqrt{\frac{\frac{2}{r}}{r}} \cdot \sqrt{\frac{\frac{2}{r}}{r}}}\right) - (\left(\frac{(-2 \cdot v + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) + 4.5)_*\]

  7. Final simplification0.5

    \[\leadsto \left(3 + \sqrt{\frac{\frac{2}{r}}{r}} \cdot \sqrt{\frac{\frac{2}{r}}{r}}\right) - (\left(\frac{(-2 \cdot v + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))

Details

Time bar (total: 31.2s)Debug log

start413.0ms

Algorithm
intervals

setup487.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.3b

localize34.0ms

Local error

Found 4 expressions with local error:

0.5b
(/ 2 (* r r))
0.4b
(* (* w r) (* w r))
0.3b
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
0.1b
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 75
Calls

4 calls. Slowest were:

4.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
4.0ms
(* (* w r) (* w r))
2.0ms
(/ 2 (* r r))

series164.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

76.0ms
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)
30.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
29.0ms
(/ 2 (* r r))
28.0ms
(* (* w r) (* w r))

simplify4.3s

Counts
50 → 87
Calls

50 calls. Slowest were:

754.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
721.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))
644.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))

prune2.0s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (* w r) (* w r))
0.3b
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
0.2b
(/ (/ 2 r) r)
0.1b
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

4.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
3.0ms
(* (* w r) (* w r))
2.0ms
(/ (/ 2 r) r)

series142.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

74.0ms
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)
30.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
19.0ms
(* (* w r) (* w r))
19.0ms
(/ (/ 2 r) r)

simplify4.0s

Counts
52 → 91
Calls

52 calls. Slowest were:

767.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))
628.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
569.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))

prune2.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 4 expressions with local error:

0.5b
(cbrt (* (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (/ (fma -2 v 3) (/ (- 1 v) 0.125))) (/ (fma -2 v 3) (/ (- 1 v) 0.125))))
0.4b
(* (* w r) (* w r))
0.3b
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
0.3b
(/ (fma -2 v 3) (/ (- 1 v) 0.125))

rewrite51.0ms

Algorithm
rewrite-expression-head
Counts
4 → 105
Calls

4 calls. Slowest were:

28.0ms
(cbrt (* (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (/ (fma -2 v 3) (/ (- 1 v) 0.125))) (/ (fma -2 v 3) (/ (- 1 v) 0.125))))
7.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
7.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))

series136.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

74.0ms
(cbrt (* (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (/ (fma -2 v 3) (/ (- 1 v) 0.125))) (/ (fma -2 v 3) (/ (- 1 v) 0.125))))
27.0ms
(* (* w r) (* w r))
19.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
16.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))

simplify9.1s

Counts
91 → 117
Calls

91 calls. Slowest were:

771.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))
727.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))
668.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))

prune3.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

12.0b
(sqrt (/ (/ 2 r) r))
12.0b
(sqrt (/ (/ 2 r) r))
0.4b
(* (sqrt (/ (/ 2 r) r)) (sqrt (/ (/ 2 r) r)))
0.4b
(* (* w r) (* w r))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

5.0ms
(* (sqrt (/ (/ 2 r) r)) (sqrt (/ (/ 2 r) r)))
3.0ms
(* (* w r) (* w r))
1.0ms
(sqrt (/ (/ 2 r) r))

series143.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

64.0ms
(* (sqrt (/ (/ 2 r) r)) (sqrt (/ (/ 2 r) r)))
33.0ms
(sqrt (/ (/ 2 r) r))
30.0ms
(sqrt (/ (/ 2 r) r))
16.0ms
(* (* w r) (* w r))

simplify1.5s

Counts
58 → 99
Calls

58 calls. Slowest were:

634.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
505.0ms
(* (* (* (sqrt (/ (/ 2 r) r)) (sqrt (/ (/ 2 r) r))) (sqrt (/ (/ 2 r) r))) (* (* (sqrt (/ (/ 2 r) r)) (sqrt (/ (/ 2 r) r))) (sqrt (/ (/ 2 r) r))))
55.0ms
(* (/ (/ 2 r) r) (/ (/ 2 r) r))

prune2.7s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

regimes447.0ms

Accuracy

0% (0.3b remaining)

Error of 0.5b against oracle of 0.2b and baseline of 0.5b

bsearch5.0ms