Average Error: 12.0 → 0.4
Time: 41.0s
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{2}{r \cdot r}\right) - (\left(\sqrt[3]{{\left(\frac{1}{\frac{1 - v}{(v \cdot -2 + 3)_*}}\right)}^{3}} \cdot 0.125\right) \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) + 4.5)_*\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.0

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

  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 add-cbrt-cube7.3

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

  5. Applied add-cbrt-cube21.3

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

  6. Applied cbrt-undiv21.3

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

  7. Simplified0.4

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

  9. Applied div-inv0.4

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

  10. Applied cube-prod0.4

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

  11. Applied cbrt-prod0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2018362 +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: 30.6s)Debug log

start1.3s

Algorithm
intervals

setup203.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.5b

localize61.0ms

Local error

Found 4 expressions with local error:

0.4b
(/ 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)

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 75
Calls

4 calls. Slowest were:

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

series150.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

75.0ms
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)
31.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
25.0ms
(* (* w r) (* w r))
18.0ms
(/ 2 (* r r))

simplify4.0s

Counts
50 → 87
Calls

50 calls. Slowest were:

661.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))
630.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))
630.0ms
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))

prune2.0s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.1b

localize36.0ms

Local error

Found 4 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

7.0ms
(* (* w r) (* w r))
2.0ms
(cbrt (pow (/ 0.125 (/ (- 1 v) (fma v -2 3))) 3))
1.0ms
(/ 2 (* r r))

series132.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(fma (cbrt (pow (/ 0.125 (/ (- 1 v) (fma v -2 3))) 3)) (* (* w r) (* w r)) 4.5)
30.0ms
(cbrt (pow (/ 0.125 (/ (- 1 v) (fma v -2 3))) 3))
16.0ms
(* (* w r) (* w r))
14.0ms
(/ 2 (* r r))

simplify5.3s

Counts
34 → 80
Calls

34 calls. Slowest were:

695.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
553.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))
548.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))

prune1.8s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize32.0ms

Local error

Found 4 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

4.0ms
(* (* w r) (* w r))
1.0ms
(cbrt (pow (/ 1 (/ (- 1 v) (fma v -2 3))) 3))
0.0ms
(/ 2 (* r r))

series122.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

66.0ms
(fma (* 0.125 (cbrt (pow (/ 1 (/ (- 1 v) (fma v -2 3))) 3))) (* (* w r) (* w r)) 4.5)
27.0ms
(* (* w r) (* w r))
16.0ms
(/ 2 (* r r))
13.0ms
(cbrt (pow (/ 1 (/ (- 1 v) (fma v -2 3))) 3))

simplify5.5s

Counts
34 → 80
Calls

34 calls. Slowest were:

768.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
659.0ms
(cbrt (pow (sqrt (/ 1 (/ (- 1 v) (fma v -2 3)))) 3))
595.0ms
(cbrt (pow (* (cbrt (/ 1 (/ (- 1 v) (fma v -2 3)))) (cbrt (/ 1 (/ (- 1 v) (fma v -2 3))))) 3))

prune2.2s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

localize21.0ms

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)

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

7.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
7.0ms
(* (* w r) (* w r))
5.0ms
(/ (/ 2 r) r)

series129.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

77.0ms
(fma (/ (fma -2 v 3) (/ (- 1 v) 0.125)) (* (* w r) (* w r)) 4.5)
25.0ms
(* (* w r) (* w r))
16.0ms
(/ (fma -2 v 3) (/ (- 1 v) 0.125))
11.0ms
(/ (/ 2 r) r)

simplify4.6s

Counts
52 → 91
Calls

52 calls. Slowest were:

1.1s
(/ (* (* (fma -2 v 3) (fma -2 v 3)) (fma -2 v 3)) (* (* (/ (- 1 v) 0.125) (/ (- 1 v) 0.125)) (/ (- 1 v) 0.125)))
665.0ms
(* (* (* (* w r) (* w r)) (* w r)) (* (* (* w r) (* w r)) (* w r)))
630.0ms
(- 0.25 (+ (* 0.125 (/ 1 v)) (* 0.125 (/ 1 (pow v 2)))))

prune1.9s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.0b

regimes1.0s

Accuracy

0% (0.3b remaining)

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

bsearch5.0ms