Average Error: 12.1 → 0.4
Time: 46.1s
Precision: 64
Internal Precision: 128
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{1}{\frac{r}{\frac{2}{r}}}\right) - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{\frac{1 - v}{\left(-0.25\right) \cdot v + 0.375}}\right) - 4.5\]

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 12.1

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

  2. Taylor expanded around -inf 20.9

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

  3. Simplified6.2

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

  5. Applied associate-/l*0.4

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

  7. Applied associate-/r*0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Applied associate-/l*0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019022 
(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: 44.9s)Debug log

sample210.0ms

Algorithm
intervals
Results
141.0ms542×body80valid

simplify105.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
104.0ms
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 12.0b

localize504.0ms

Local error

Found 4 expressions with local error:

9.3b
(* (* w w) r)
6.7b
(* (* (* w w) r) r)
6.4b
(/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))
2.7b
(* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r))

rewrite55.0ms

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
21.0ms
(/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))
15.0ms
(* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r))
11.0ms
(* (* (* w w) r) r)
7.0ms
(* (* w w) r)

series157.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
64.0ms
(/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))
47.0ms
(* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r))
30.0ms
(* (* (* w w) r) r)
17.0ms
(* (* w w) r)

simplify5.7s

Counts
36 → 70
Calls
36 calls:
Slowest
556.0ms
(* (* 0.125 (- (* 3 3) (* (* 2 v) (* 2 v)))) (* (* (* w w) r) r))
476.0ms
(* (* 0.125 (- (* 3 3) (* (* 2 v) (* 2 v)))) (* (* (* w w) r) r))
408.0ms
(+ (log (* 0.125 (- 3 (* 2 v)))) (log (* (* (* w w) r) r)))
337.0ms
(+ (* 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))))))
328.0ms
(* (* 0.125 (- (pow 3 3) (pow (* 2 v) 3))) (* (* (* w w) r) r))

prune813.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.2b

localize35.0ms

Local error

Found 4 expressions with local error:

6.4b
(/ (* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375)) (- 1 v))
2.7b
(* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375))
0.4b
(* (* r w) (* r w))
0.3b
(/ 2 (* r r))

rewrite29.0ms

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
12.0ms
(/ (* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375)) (- 1 v))
10.0ms
(* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375))
5.0ms
(* (* r w) (* r w))
1.0ms
(/ 2 (* r r))

series177.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
70.0ms
(* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375))
65.0ms
(/ (* (* (* r w) (* r w)) (+ (* v (- 0.25)) 0.375)) (- 1 v))
25.0ms
(* (* r w) (* r w))
17.0ms
(/ 2 (* r r))

simplify7.5s

Counts
52 → 88
Calls
52 calls:
Slowest
599.0ms
(* (* (* (* r w) (* r w)) (* r w)) (* (* (* r w) (* r w)) (* r w)))
559.0ms
(* (* (* r w) (* r w)) (- (* (* v (- 0.25)) (* v (- 0.25))) (* 0.375 0.375)))
486.0ms
(/ (* (* r w) (* r w)) (* (cbrt (- 1 v)) (cbrt (- 1 v))))
483.0ms
(+ (* (* v (- 0.25)) (* v (- 0.25))) (- (* 0.375 0.375) (* (* v (- 0.25)) 0.375)))
395.0ms
(+ (* 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)))))

prune1.1s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.1b

localize10.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (* r w) (* r w))
0.3b
(/ 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)

rewrite158.0ms

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
143.0ms
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)
10.0ms
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
3.0ms
(* (* r w) (* r w))
0.0ms
(/ 2 (* r r))

series227.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
118.0ms
(- (- (+ 3 (/ 2 (* r r))) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)
78.0ms
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
20.0ms
(* (* r w) (* r w))
11.0ms
(/ 2 (* r r))

simplify9.9s

Counts
76 → 104
Calls
76 calls:
Slowest
648.0ms
(* (* (* (* r w) (* r w)) (* r w)) (* (* (* r w) (* r w)) (* r w)))
482.0ms
(+ (* (* v (- 0.25)) (* v (- 0.25))) (- (* 0.375 0.375) (* (* v (- 0.25)) 0.375)))
451.0ms
(/ (* r w) (/ (- 1 v) (sqrt (+ (* v (- 0.25)) 0.375))))
437.0ms
(- (log (* (* r w) (* r w))) (log (/ (- 1 v) (+ (* v (- 0.25)) 0.375))))
372.0ms
(+ (* 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)))))

prune1.4s

Pruning

13 alts after pruning (13 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)

rewrite139.0ms

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
109.0ms
(- (- (+ 3 (/ (/ 2 r) r)) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)
18.0ms
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
6.0ms
(* (* r w) (* r w))
5.0ms
(/ (/ 2 r) r)

series196.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
101.0ms
(- (- (+ 3 (/ (/ 2 r) r)) (/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))) 4.5)
60.0ms
(/ (* (* r w) (* r w)) (/ (- 1 v) (+ (* v (- 0.25)) 0.375)))
21.0ms
(* (* r w) (* r w))
14.0ms
(/ (/ 2 r) r)

simplify9.5s

Counts
78 → 108
Calls
78 calls:
Slowest
630.0ms
(* (* (* (* r w) (* r w)) (* r w)) (* (* (* r w) (* r w)) (* r w)))
484.0ms
(+ (* (* v (- 0.25)) (* v (- 0.25))) (- (* 0.375 0.375) (* (* v (- 0.25)) 0.375)))
419.0ms
(- (log (* (* r w) (* r w))) (log (/ (- 1 v) (+ (* v (- 0.25)) 0.375))))
405.0ms
(+ (* 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)))))
357.0ms
(- (+ (log (* r w)) (log (* r w))) (- (log (- 1 v)) (log (+ (* v (- 0.25)) 0.375))))

prune1.4s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.0b

regimes501.0ms

Accuracy

0% (0.3b remaining)

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

bsearch6.0ms

end0.0ms

sample5.0s

Algorithm
intervals
Results
4.6s17147×body80valid