Average Error: 12.5 → 0.4
Time: 39.4s
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \sqrt{\frac{(-2 \cdot v + 3)_*}{\frac{1 - v}{0.125}} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot \sqrt{\frac{(-2 \cdot v + 3)_*}{\frac{1 - v}{0.125}} \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}\right)\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.5

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

  2. Simplified0.4

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

  4. Applied fma-udef0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019005 +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: 38.0s)Debug log

sample302.0ms

Algorithm
intervals

simplify117.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

116.0ms
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5)

prune37.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize84.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ 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)

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 75
Calls

4 calls. Slowest were:

8.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
7.0ms
(* (* w r) (* w r))
3.0ms
(/ 2 (* r r))

series142.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

80.0ms
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)
23.0ms
(/ 2 (* r r))
20.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
19.0ms
(* (* w r) (* w r))

simplify4.1s

Counts
50 → 87
Calls

50 calls. Slowest were:

763.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))
594.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
551.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))

prune1.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

0.7b
(/ 2 (* r r))
0.4b
(* (* w r) (* w r))
0.3b
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
0.2b
(* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 104
Calls

4 calls. Slowest were:

25.0ms
(* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))
7.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
7.0ms
(* (* w r) (* w r))

series110.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

54.0ms
(* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))
20.0ms
(* (* w r) (* w r))
20.0ms
(/ 2 (* r r))
15.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))

simplify6.8s

Counts
78 → 116
Calls

78 calls. Slowest were:

607.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))
591.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))
583.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))

prune1.6s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

4.8b
(* (fma -2 v 3) (* (/ 1 (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
0.7b
(/ 2 (* r r))
0.4b
(* (* w r) (* w r))
0.2b
(/ 1 (/ (- 1 v) 0.125))

rewrite51.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

42.0ms
(* (fma -2 v 3) (* (/ 1 (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
6.0ms
(* (* w r) (* w r))
2.0ms
(/ 1 (/ (- 1 v) 0.125))

series114.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

68.0ms
(* (fma -2 v 3) (* (/ 1 (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
18.0ms
(* (* w r) (* w r))
17.0ms
(/ 2 (* r r))
11.0ms
(/ 1 (/ (- 1 v) 0.125))

simplify7.0s

Counts
38 → 89
Calls

38 calls. Slowest were:

1.1s
(* (fma -2 v 3) (* (/ 1 (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
762.0ms
(* (fma -2 v 3) (* (/ 1 (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
666.0ms
(* (fma -2 v 3) (* (/ 1 (/ (- 1 v) 0.125)) (* (* w r) (* w r))))

prune1.3s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

9.7b
(sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
9.7b
(sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
0.7b
(/ 2 (* r r))
0.4b
(* (sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))) (sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))))

rewrite63.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

50.0ms
(* (sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))) (sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))))
6.0ms
(sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
6.0ms
(sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r))))

series799.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

308.0ms
(sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
286.0ms
(sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r))))
187.0ms
(* (sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))) (sqrt (* (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)))))
19.0ms
(/ 2 (* r r))

simplify6.5s

Counts
39 → 80
Calls

39 calls. Slowest were:

382.0ms
(* (sqrt (* (fma -2 v 3) (* (* w r) (* w r)))) (sqrt (* (fma -2 v 3) (* (* w r) (* w r)))))
362.0ms
(sqrt (* (fma -2 v 3) (* (* w r) (* w r))))
352.0ms
(- (* (sqrt 2) (* r (* (sqrt 0.125) w))) (+ (* 1/2 (/ (* r (* (sqrt 0.125) w)) (* (sqrt 2) (pow v 2)))) (+ (* 1/2 (/ (* r (* (sqrt 0.125) w)) (* (sqrt 2) v))) (* 1/8 (/ (* r (* (sqrt 0.125) w)) (* (pow (sqrt 2) 3) (pow v 2)))))))

prune1.3s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

regimes674.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.2b and baseline of 0.4b

bsearch6.0ms

end0.0ms

sample5.4s

Algorithm
intervals