Average Error: 12.4 → 0.5
Time: 50.3s
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \frac{0.375 - 0.25 \cdot v}{\sqrt[3]{1 - v}}\right) - 4.5\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.4

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

  2. Taylor expanded around inf 21.1

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{0.375 \cdot \left({r}^{2} \cdot {w}^{2}\right) - 0.25 \cdot \left({r}^{2} \cdot \left(v \cdot {w}^{2}\right)\right)}}{1 - v}\right) - 4.5\]

  3. Simplified6.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(v \cdot \left(-0.25\right) + 0.375\right)}}{1 - v}\right) - 4.5\]
  4. Using strategy rm

  5. Applied add-cube-cbrt6.5

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(v \cdot \left(-0.25\right) + 0.375\right)}{\color{blue}{\left(\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}\right) \cdot \sqrt[3]{1 - v}}}\right) - 4.5\]

  6. Applied times-frac0.5

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \frac{v \cdot \left(-0.25\right) + 0.375}{\sqrt[3]{1 - v}}}\right) - 4.5\]

  7. Simplified0.5

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \color{blue}{\frac{0.375 - v \cdot 0.25}{\sqrt[3]{1 - v}}}\right) - 4.5\]

  8. Final simplification0.5

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{\sqrt[3]{1 - v} \cdot \sqrt[3]{1 - v}} \cdot \frac{0.375 - 0.25 \cdot v}{\sqrt[3]{1 - v}}\right) - 4.5\]

Reproduce

herbie shell --seed 2019016 
(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: 49.3s)Debug log

sample1.0s

Algorithm
intervals
Results
555×(pre true 80)
555×(body real 80)

simplify117.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
117.0ms
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5)

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 12.5b

localize52.0ms

Local error

Found 4 expressions with local error:

10.6b
(* (* w w) r)
6.4b
(/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))
5.8b
(* (* (* w w) r) r)
2.0b
(* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r))

rewrite48.0ms

Algorithm
rewrite-expression-head
Rules
11×add-exp-log
11×add-cbrt-cube
6×add-cube-cbrt
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
4×add-log-exp
4×associate-*r/
4×associate-*l/
3×flip--
3×*-commutative
3×associate-*l*
3×flip3--
3×times-frac
3×associate-/r*
2×associate-/l/
2×cbrt-unprod
2×associate-/r/
2×prod-exp
2×div-exp
2×cbrt-undiv
1×associate-/l*
1×div-inv
1×pow-prod-down
1×associate-*r*
1×frac-2neg
1×clear-num
Counts
4 → 58
Calls
4 calls:
Slowest
19.0ms
(/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))
17.0ms
(* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r))
6.0ms
(* (* (* w w) r) r)
4.0ms
(* (* w w) r)

series137.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
53.0ms
(* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r))
49.0ms
(/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))
18.0ms
(* (* w w) r)
17.0ms
(* (* (* w w) r) r)

simplify6.4s

Counts
36 → 70
Calls
36 calls:
Slowest
646.0ms
(* (* 0.125 (- (* 3 3) (* (* 2 v) (* 2 v)))) (* (* (* w w) r) r))
573.0ms
(* (* 0.125 (- (* 3 3) (* (* 2 v) (* 2 v)))) (* (* (* w w) r) r))
435.0ms
(+ (* 0.375 (* (pow r 2) (pow w 2))) (+ (* 0.125 (* (pow r 2) (* v (pow w 2)))) (* 0.125 (* (pow r 2) (* (pow v 2) (pow w 2))))))
414.0ms
(* (* 0.125 (- (pow 3 3) (pow (* 2 v) 3))) (* (* (* w w) r) r))
368.0ms
(+ (log (* 0.125 (- 3 (* 2 v)))) (log (* (* (* w w) r) r)))

prune995.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.2b

localize46.0ms

Local error

Found 4 expressions with local error:

6.4b
(/ (* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375)) (- 1 v))
2.0b
(* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375))
0.5b
(/ 2 (* r r))
0.4b
(* (* r w) (* r w))

rewrite40.0ms

Algorithm
rewrite-expression-head
Rules
20×add-exp-log
20×add-cbrt-cube
14×pow1
7×cbrt-unprod
7×add-cube-cbrt
7×prod-exp
7×*-un-lft-identity
7×add-sqr-sqrt
4×add-log-exp
4×associate-*r/
4×pow-prod-down
4×associate-*r*
4×associate-/r*
3×div-exp
3×times-frac
3×cbrt-undiv
2×associate-/l/
2×div-inv
2×flip-+
2×*-commutative
2×associate-/r/
2×associate-*l*
2×flip3-+
2×frac-2neg
2×clear-num
1×associate-/l*
1×flip--
1×pow-prod-up
1×flip3--
1×pow-plus
1×distribute-lft-in
1×distribute-rgt-in
1×pow2
Counts
4 → 76
Calls
4 calls:
Slowest
22.0ms
(/ (* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375)) (- 1 v))
13.0ms
(* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375))
4.0ms
(* (* r w) (* r w))
0.0ms
(/ 2 (* r r))

series156.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
61.0ms
(/ (* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375)) (- 1 v))
48.0ms
(* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375))
28.0ms
(* (* r w) (* r w))
18.0ms
(/ 2 (* r r))

simplify7.5s

Counts
52 → 88
Calls
52 calls:
Slowest
650.0ms
(* (* (* (* r w) (* r w)) (* r w)) (* (* (* r w) (* r w)) (* r w)))
498.0ms
(* (* (* r w) (* r w)) (- (* (* v (- 0.25)) (* v (- 0.25))) (* 0.375 0.375)))
461.0ms
(+ (* (* v (- 0.25)) (* v (- 0.25))) (- (* 0.375 0.375) (* (* v (- 0.25)) 0.375)))
400.0ms
(/ (* (* r w) (* r w)) (* (cbrt (- 1 v)) (cbrt (- 1 v))))
389.0ms
(+ (* 0.125 (* v (* (pow r 2) (pow w 2)))) (+ (* 0.125 (* (pow v 2) (* (pow r 2) (pow w 2)))) (* 0.375 (* (pow r 2) (pow w 2)))))

prune1.3s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.1b

localize15.0ms

Local error

Found 4 expressions with local error:

0.5b
(/ 2 (* r r))
0.4b
(* (* r w) (* r w))
0.2b
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
0.0b
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)

rewrite123.0ms

Algorithm
rewrite-expression-head
Rules
33×times-frac
18×add-cube-cbrt
18×add-exp-log
18×add-cbrt-cube
18×*-un-lft-identity
18×add-sqr-sqrt
16×associate-/r*
9×pow1
6×div-exp
6×cbrt-undiv
5×associate-/r/
4×add-log-exp
4×div-inv
4×associate--l+
3×cbrt-unprod
3×prod-exp
2×flip-+
2×flip3-+
2×frac-2neg
2×sub-neg
2×clear-num
1×associate-/l*
1×flip--
1×pow-prod-up
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×associate-*r*
1×flip3--
1×pow-plus
1×pow2
1×associate--l-
Counts
4 → 92
Calls
4 calls:
Slowest
105.0ms
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)
13.0ms
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
4.0ms
(* (* r w) (* r w))
0.0ms
(/ 2 (* r r))

series202.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
109.0ms
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)
65.0ms
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
17.0ms
(* (* r w) (* r w))
11.0ms
(/ 2 (* r r))

simplify10.6s

Counts
76 → 104
Calls
76 calls:
Slowest
571.0ms
(* (* (* (* r w) (* r w)) (* r w)) (* (* (* r w) (* r w)) (* r w)))
486.0ms
(+ (* (* v (- 0.25)) (* v (- 0.25))) (- (* 0.375 0.375) (* (* v (- 0.25)) 0.375)))
481.0ms
(- (log (* (* r w) (* r w))) (log (/ (- 1 v) (+ (* v (- 0.25)) 0.375))))
459.0ms
(/ (* r w) (* (cbrt (/ (- 1 v) (+ (* v (- 0.25)) 0.375))) (cbrt (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))))
424.0ms
(+ (* 0.125 (* v (* (pow r 2) (pow w 2)))) (+ (* 0.125 (* (pow v 2) (* (pow r 2) (pow w 2)))) (* 0.375 (* (pow r 2) (pow w 2)))))

prune1.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize28.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (* r w) (* r w))
0.2b
(/ (/ 2 r) r)
0.2b
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
0.0b
(- (- (+ 3 (/ (/ 2 r) r)) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)

rewrite156.0ms

Algorithm
rewrite-expression-head
Rules
33×times-frac
19×add-cube-cbrt
19×*-un-lft-identity
19×add-sqr-sqrt
18×add-exp-log
18×add-cbrt-cube
15×associate-/r*
9×pow1
6×div-exp
6×cbrt-undiv
5×associate-/l*
5×div-inv
5×associate-/r/
4×add-log-exp
4×associate--l+
3×cbrt-unprod
3×prod-exp
2×flip-+
2×flip3-+
2×frac-2neg
2×sub-neg
2×clear-num
1×associate-/l/
1×flip--
1×pow-prod-up
1×*-commutative
1×pow-prod-down
1×associate-*l*
1×associate-*r*
1×flip3--
1×pow-plus
1×pow2
1×associate--l-
Counts
4 → 96
Calls
4 calls:
Slowest
129.0ms
(- (- (+ 3 (/ (/ 2 r) r)) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)
14.0ms
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
7.0ms
(* (* r w) (* r w))
5.0ms
(/ (/ 2 r) r)

series191.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
113.0ms
(- (- (+ 3 (/ (/ 2 r) r)) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)
50.0ms
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
17.0ms
(* (* r w) (* r w))
11.0ms
(/ (/ 2 r) r)

simplify11.0s

Counts
78 → 108
Calls
78 calls:
Slowest
702.0ms
(* (* (* (* r w) (* r w)) (* r w)) (* (* (* r w) (* r w)) (* r w)))
521.0ms
(+ (* (* v (- 0.25)) (* v (- 0.25))) (- (* 0.375 0.375) (* (* v (- 0.25)) 0.375)))
478.0ms
(+ (* 0.125 (* v (* (pow r 2) (pow w 2)))) (+ (* 0.125 (* (pow v 2) (* (pow r 2) (pow w 2)))) (* 0.375 (* (pow r 2) (pow w 2)))))
418.0ms
(- (+ (log (* r w)) (log (* r w))) (- (log (- 1 v)) (log (+ (* v (- 0.25)) 0.375))))
413.0ms
(- (log (* (* r w) (* r w))) (log (/ (- 1 v) (+ (* v (- 0.25)) 0.375))))

prune1.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

regimes427.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 0.5b

bsearch6.0ms

end0.0ms

sample5.7s

Algorithm
intervals
Results
17261×(pre true 80)
17261×(body real 80)