Average Error: 15.3 → 0.9
Time: 12.4s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{\frac{g}{2 \cdot a}}\]
↓
\[\frac{1}{\sqrt[3]{\frac{1}{g}} \cdot \sqrt[3]{a \cdot 2}}\]

Error

Bits error versus g

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

    \[\sqrt[3]{\frac{g}{2 \cdot a}}\]
  2. Using strategy rm

  3. Applied cbrt-div0.9

    \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{2 \cdot a}}}\]
  4. Using strategy rm

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

    \[\leadsto \frac{\color{blue}{1 \cdot \sqrt[3]{g}}}{\sqrt[3]{2 \cdot a}}\]

  6. Applied associate-/l*0.9

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{2 \cdot a}}{\sqrt[3]{g}}}}\]
  7. Using strategy rm

  8. Applied div-inv0.9

    \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{2 \cdot a} \cdot \frac{1}{\sqrt[3]{g}}}}\]

  9. Taylor expanded around inf 33.9

    \[\leadsto \frac{1}{\sqrt[3]{2 \cdot a} \cdot \color{blue}{{\left(\frac{1}{g}\right)}^{\frac{1}{3}}}}\]

  10. Simplified0.9

    \[\leadsto \frac{1}{\sqrt[3]{2 \cdot a} \cdot \color{blue}{\sqrt[3]{\frac{1}{g}}}}\]

  11. Final simplification0.9

    \[\leadsto \frac{1}{\sqrt[3]{\frac{1}{g}} \cdot \sqrt[3]{a \cdot 2}}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  (cbrt (/ g (* 2 a))))

Details

Time bar (total: 10.5s)Debug log

sample37.0ms

Algorithm
intervals
Results
14.0ms256×body80valid
13.0ms248×body80nan

simplify3.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
3.0ms
(cbrt (/ g (* 2 a)))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 18.6b

localize17.0ms

Local error

Found 1 expressions with local error:

18.6b
(cbrt (/ g (* 2 a)))

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
4×cbrt-prod
2×add-cube-cbrt
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×div-inv
1×log1p-expm1-u
1×add-exp-log
1×pow1/3
1×add-cbrt-cube
1×cbrt-div
1×pow1
1×expm1-log1p-u
Counts
1 → 15
Calls
1 calls:
Slowest
1.0ms
(cbrt (/ g (* 2 a)))

series128.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
128.0ms
(cbrt (/ g (* 2 a)))

simplify388.0ms

Counts
8 → 18
Calls
8 calls:
Slowest
171.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log (/ 1 a)) (log (/ 1 g))))))
153.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log (/ -1 a)) (log (/ -1 g))))))
50.0ms
(* (cbrt 1/2) (exp (* 1/3 (- (log g) (log a)))))
5.0ms
(cbrt (* (cbrt (/ g (* 2 a))) (cbrt (/ g (* 2 a)))))
5.0ms
(cbrt (sqrt (/ g (* 2 a))))

prune189.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.3b

localize16.0ms

Local error

Found 3 expressions with local error:

0.6b
(cbrt (* 2 a))
0.6b
(cbrt g)
0.4b
(/ (cbrt g) (cbrt (* 2 a)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
12×add-cube-cbrt
12×times-frac
12×*-un-lft-identity
12×add-sqr-sqrt
5×add-exp-log
5×cbrt-prod
4×associate-/r*
3×add-log-exp
3×associate-/l*
3×log1p-expm1-u
3×add-cbrt-cube
3×pow1
3×expm1-log1p-u
2×pow1/3
1×div-inv
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
3 → 54
Calls
3 calls:
Slowest
5.0ms
(/ (cbrt g) (cbrt (* 2 a)))
1.0ms
(cbrt (* 2 a))
0.0ms
(cbrt g)

series712.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
319.0ms
(cbrt (* 2 a))
250.0ms
(cbrt g)
143.0ms
(/ (cbrt g) (cbrt (* 2 a)))

simplify585.0ms

Counts
42 → 63
Calls
42 calls:
Slowest
190.0ms
(/ (exp (* 1/3 (- (log (/ 1 a)) (log (/ 1 g))))) (cbrt 2))
167.0ms
(/ (* (exp (* 1/3 (- (log (/ -1 a)) (log (/ -1 g))))) (cbrt -1)) (cbrt -2))
45.0ms
(/ (exp (* 1/3 (- (log g) (log a)))) (cbrt 2))
35.0ms
(/ (* (cbrt (cbrt g)) (cbrt (cbrt g))) (* (cbrt (cbrt (* 2 a))) (cbrt (cbrt (* 2 a)))))
18.0ms
(/ (* (cbrt (cbrt g)) (cbrt (cbrt g))) 1)

prune696.0ms

Pruning

19 alts after pruning (18 fresh and 1 done)

Merged error: 0.1b

localize6.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (* 2 a))
0.6b
(cbrt g)
0.3b
(/ (cbrt (* 2 a)) (cbrt g))
0.3b
(/ 1 (/ (cbrt (* 2 a)) (cbrt g)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
24×times-frac
21×add-cube-cbrt
21×*-un-lft-identity
21×add-sqr-sqrt
19×associate-/r*
9×add-exp-log
8×cbrt-prod
5×pow1
4×add-log-exp
4×associate-/l*
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
3×div-inv
2×div-exp
2×pow1/3
2×frac-2neg
2×clear-num
2×rec-exp
1×inv-pow
1×pow-flip
1×associate-/r/
1×cbrt-undiv
Counts
4 → 87
Calls
4 calls:
Slowest
3.0ms
(/ (cbrt (* 2 a)) (cbrt g))
1.0ms
(/ 1 (/ (cbrt (* 2 a)) (cbrt g)))
0.0ms
(cbrt (* 2 a))
0.0ms
(cbrt g)

series889.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
343.0ms
(cbrt (* 2 a))
245.0ms
(cbrt g)
159.0ms
(/ 1 (/ (cbrt (* 2 a)) (cbrt g)))
141.0ms
(/ (cbrt (* 2 a)) (cbrt g))

simplify1.2s

Counts
63 → 99
Calls
63 calls:
Slowest
249.0ms
(* (exp (* 1/3 (- (log (/ 1 g)) (log (/ 1 a))))) (cbrt 2))
227.0ms
(/ (exp (* 1/3 (- (log (/ 1 a)) (log (/ 1 g))))) (cbrt 2))
204.0ms
(/ (* (cbrt -2) (exp (* 1/3 (- (log (/ -1 g)) (log (/ -1 a)))))) (cbrt -1))
158.0ms
(/ (* (exp (* 1/3 (- (log (/ -1 a)) (log (/ -1 g))))) (cbrt -1)) (cbrt -2))
55.0ms
(/ (exp (* 1/3 (- (log g) (log a)))) (cbrt 2))

prune991.0ms

Pruning

19 alts after pruning (18 fresh and 1 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (* 2 a))
0.6b
(cbrt g)
0.3b
(/ 1 (* (cbrt (* 2 a)) (/ 1 (cbrt g))))
0.3b
(/ 1 (cbrt g))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
10×add-exp-log
8×pow1
5×add-cube-cbrt
5×*-un-lft-identity
5×rec-exp
5×add-sqr-sqrt
4×add-log-exp
4×pow-flip
4×log1p-expm1-u
4×add-cbrt-cube
4×associate-/r*
4×expm1-log1p-u
3×pow1/3
2×inv-pow
2×div-inv
2×associate-/r/
2×prod-exp
2×frac-2neg
2×clear-num
1×un-div-inv
1×associate-*r/
1×pow-prod-down
1×cbrt-prod
Counts
4 → 61
Calls
4 calls:
Slowest
8.0ms
(/ 1 (* (cbrt (* 2 a)) (/ 1 (cbrt g))))
2.0ms
(/ 1 (cbrt g))
1.0ms
(cbrt (* 2 a))
0.0ms
(cbrt g)

series1.1s

Counts
4 → 12
Calls
4 calls:
Slowest
318.0ms
(/ 1 (cbrt g))
294.0ms
(cbrt (* 2 a))
292.0ms
(cbrt g)
149.0ms
(/ 1 (* (cbrt (* 2 a)) (/ 1 (cbrt g))))

simplify549.0ms

Counts
23 → 73
Calls
23 calls:
Slowest
213.0ms
(/ (exp (* 1/3 (- (log (/ 1 a)) (log (/ 1 g))))) (cbrt 2))
188.0ms
(/ (* (exp (* 1/3 (- (log (/ -1 a)) (log (/ -1 g))))) (cbrt -1)) (cbrt -2))
58.0ms
(/ (exp (* 1/3 (- (log g) (log a)))) (cbrt 2))
27.0ms
(* (/ 1 (cbrt -1)) (pow (/ -1 g) 1/3))
10.0ms
(* (pow (* a -1) 1/3) (cbrt -2))

prune695.0ms

Pruning

19 alts after pruning (17 fresh and 2 done)

Merged error: 0.0b

regimes715.0ms

Accuracy

0% (0.8b remaining)

Error of 0.9b against oracle of 0.1b and baseline of 0.9b

bsearch2.0ms

end0.0ms

sample1.6s

Algorithm
intervals
Results
683.0ms7878×body80nan
658.0ms8000×body80valid