Average Error: 6.8 → 0.2
Time: 15.9s
Precision: 64
Internal Precision: 128
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[(\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(\left(x.re + x.re\right) \cdot \left(x.re \cdot x.im\right)\right))_*\]

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.8
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.8

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  2. Initial simplification0.2

    \[\leadsto (\left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + \left(\left(x.re + x.re\right) \cdot \left(x.re \cdot x.im\right)\right))_*\]

  3. Final simplification0.2

    \[\leadsto (\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(\left(x.re + x.re\right) \cdot \left(x.re \cdot x.im\right)\right))_*\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

Details

Time bar (total: 12.0s)Debug log

start105.0ms

Algorithm
intervals

setup215.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize18.0ms

Local error

Found 2 expressions with local error:

0.2b
(fma (+ x.re x.im) (* x.im (- x.re x.im)) (* (+ x.re x.re) (* x.re x.im)))
0.1b
(* (+ x.re x.re) (* x.re x.im))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 29
Calls

2 calls. Slowest were:

7.0ms
(* (+ x.re x.re) (* x.re x.im))
0.0ms
(fma (+ x.re x.im) (* x.im (- x.re x.im)) (* (+ x.re x.re) (* x.re x.im)))

series81.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

66.0ms
(fma (+ x.re x.im) (* x.im (- x.re x.im)) (* (+ x.re x.re) (* x.re x.im)))
14.0ms
(* (+ x.re x.re) (* x.re x.im))

simplify1.1s

Counts
13 → 35
Calls

13 calls. Slowest were:

637.0ms
(* (* (* (+ x.re x.re) (+ x.re x.re)) (+ x.re x.re)) (* (* (* x.re x.im) (* x.re x.im)) (* x.re x.im)))
107.0ms
(* (+ (pow x.re 3) (pow x.re 3)) (* x.re x.im))
87.0ms
(- (* 3 (* x.im (pow x.re 2))) (pow x.im 3))

prune345.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

24.3b
(cbrt (* (+ x.re x.re) (* x.re x.im)))
24.3b
(cbrt (* (+ x.re x.re) (* x.re x.im)))
24.3b
(cbrt (* (+ x.re x.re) (* x.re x.im)))
0.4b
(* (cbrt (* (+ x.re x.re) (* x.re x.im))) (cbrt (* (+ x.re x.re) (* x.re x.im))))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

16.0ms
(* (cbrt (* (+ x.re x.re) (* x.re x.im))) (cbrt (* (+ x.re x.re) (* x.re x.im))))
2.0ms
(cbrt (* (+ x.re x.re) (* x.re x.im)))
2.0ms
(cbrt (* (+ x.re x.re) (* x.re x.im)))

series601.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

165.0ms
(* (cbrt (* (+ x.re x.re) (* x.re x.im))) (cbrt (* (+ x.re x.re) (* x.re x.im))))
150.0ms
(cbrt (* (+ x.re x.re) (* x.re x.im)))
147.0ms
(cbrt (* (+ x.re x.re) (* x.re x.im)))
139.0ms
(cbrt (* (+ x.re x.re) (* x.re x.im)))

simplify4.7s

Counts
49 → 85
Calls

49 calls. Slowest were:

467.0ms
(* (* (+ x.re x.re) (* x.re x.im)) (* (+ x.re x.re) (* x.re x.im)))
403.0ms
(* (* (+ x.re x.re) (* x.re x.im)) (* (+ x.re x.re) (* x.re x.im)))
324.0ms
(* (cbrt 2) (exp (* -1/3 (+ (log (/ 1 x.im)) (* 2 (log (/ 1 x.re)))))))

prune1.1s

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (+ x.re x.re))
0.6b
(cbrt (+ x.re x.re))
0.6b
(cbrt (+ x.re x.re))
0.5b
(* (cbrt (+ x.re x.re)) (cbrt (+ x.re x.re)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

10.0ms
(* (cbrt (+ x.re x.re)) (cbrt (+ x.re x.re)))
1.0ms
(cbrt (+ x.re x.re))
1.0ms
(cbrt (+ x.re x.re))

series1.4s

Counts
4 → 12
Calls

4 calls. Slowest were:

459.0ms
(* (cbrt (+ x.re x.re)) (cbrt (+ x.re x.re)))
319.0ms
(cbrt (+ x.re x.re))
312.0ms
(cbrt (+ x.re x.re))
304.0ms
(cbrt (+ x.re x.re))

simplify412.0ms

Counts
59 → 95
Calls

59 calls. Slowest were:

25.0ms
(* (cbrt (+ (pow x.re 3) (pow x.re 3))) (cbrt (+ x.re x.re)))
19.0ms
(* (cbrt (- x.re x.re)) (cbrt (+ (* x.re x.re) (- (* x.re x.re) (* x.re x.re)))))
19.0ms
(* (+ x.re x.re) (+ x.re x.re))

prune1.1s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 2 expressions with local error:

8.1b
(* x.im (pow x.re 2))
0.2b
(* 3 (* x.im (pow x.re 2)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

2.0ms
(* 3 (* x.im (pow x.re 2)))
1.0ms
(* x.im (pow x.re 2))

series54.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

28.0ms
(* 3 (* x.im (pow x.re 2)))
26.0ms
(* x.im (pow x.re 2))

simplify108.0ms

Counts
10 → 31
Calls

10 calls. Slowest were:

31.0ms
(* 3 (* x.im (pow x.re 2)))
26.0ms
(* 3 (* x.im (pow x.re 2)))
15.0ms
(* 3 (* x.im (pow x.re 2)))

prune413.0ms

Pruning

5 alts after pruning (1 fresh and 4 done)

Merged error: 0.0b

regimes131.0ms

Accuracy

0% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 0.2b

bsearch3.0ms