Average Error: 12.1 → 0.4
Time: 32.0s
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - (\left(\frac{(-2 \cdot v + 3)_*}{\frac{1 - v}{0.125}}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.1

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

  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 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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019008 +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: 30.3s)Debug log

sample409.0ms

Algorithm
intervals

simplify516.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

515.0ms
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5)

prune19.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.5b

localize74.0ms

Local error

Found 4 expressions with local error:

1.1b
(/ 2 (* r r))
0.4b
(* (* w r) (* w r))
0.3b
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
0.0b
(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))
8.0ms
(* (* w r) (* w r))
3.0ms
(/ 2 (* r r))

series139.0ms

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)
24.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
21.0ms
(/ 2 (* r r))
17.0ms
(* (* w r) (* w r))

simplify4.0s

Counts
50 → 87
Calls

50 calls. Slowest were:

645.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
632.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))
569.0ms
(- 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: 0.1b

localize10.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (* w r) (* w r))
0.3b
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
0.3b
(/ (/ 2 r) r)
0.0b
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

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

series115.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

68.0ms
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)
20.0ms
(* (* w r) (* w r))
15.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
11.0ms
(/ (/ 2 r) r)

simplify3.6s

Counts
52 → 91
Calls

52 calls. Slowest were:

549.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
537.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))
534.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))

prune1.4s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

1.1b
(/ 2 (* r r))
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))

rewrite50.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

33.0ms
(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))))
8.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
7.0ms
(* (* w r) (* w r))

series108.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

56.0ms
(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))))
19.0ms
(/ 2 (* r r))
17.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
16.0ms
(* (* w r) (* w r))

simplify5.7s

Counts
59 → 96
Calls

59 calls. Slowest were:

688.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
565.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))
561.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))

prune1.6s

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

1.1b
(/ 2 (* r r))
0.4b
(expm1 (log1p (/ (fma -2 v 3) (/ (- 1 v) 0.125))))
0.4b
(* (* w r) (* w r))
0.3b
(/ (fma -2 v 3) (/ (- 1 v) 0.125))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

11.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
6.0ms
(* (* w r) (* w r))
0.0ms
(/ 2 (* r r))

series107.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

61.0ms
(expm1 (log1p (/ (fma -2 v 3) (/ (- 1 v) 0.125))))
20.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
15.0ms
(* (* w r) (* w r))
11.0ms
(/ 2 (* r r))

simplify3.4s

Counts
51 → 88
Calls

51 calls. Slowest were:

586.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
542.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))
527.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))

prune1.3s

Pruning

13 alts after pruning (10 fresh and 3 done)

Merged error: 0.0b

regimes375.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample5.9s

Algorithm
intervals