Average Error: 6.9 → 0.3
Time: 31.7s
Precision: 64
Internal Precision: 128
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\]

Error

Bits error versus x.re

Bits error versus x.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.9
Target0.3
Herbie0.3
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.9

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
  2. Using strategy rm

  3. Applied difference-of-squares6.9

    \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  4. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019004 
(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: 31.0s)Debug log

sample114.0ms

Algorithm
intervals

simplify155.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

155.0ms
(+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re))

prune18.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 8.1b

localize24.0ms

Local error

Found 4 expressions with local error:

8.1b
(* (- (* x.re x.re) (* x.im x.im)) x.im)
0.2b
(* (+ (* x.re x.im) (* x.im x.re)) x.re)
0.2b
(+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re))
0.0b
(- (* x.re x.re) (* x.im x.im))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

6.0ms
(+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re))
4.0ms
(* (- (* x.re x.re) (* x.im x.im)) x.im)
4.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.re)

series136.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

55.0ms
(* (- (* x.re x.re) (* x.im x.im)) x.im)
42.0ms
(+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re))
20.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.re)
19.0ms
(- (* x.re x.re) (* x.im x.im))

simplify6.1s

Counts
33 → 67
Calls

33 calls. Slowest were:

2.5s
(* (exp (* (- (* x.re x.re) (* x.im x.im)) x.im)) (exp (* (+ (* x.re x.im) (* x.im x.re)) x.re)))
520.0ms
(* (+ (pow (* x.re x.im) 3) (pow (* x.im x.re) 3)) x.re)
381.0ms
(* (+ (* (* x.re x.re) (* x.re x.re)) (+ (* (* x.im x.im) (* x.im x.im)) (* (* x.re x.re) (* x.im x.im)))) (- (* x.re x.im) (* x.im x.re)))

prune623.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.1b

localize24.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (+ x.re x.im) (* (- x.re x.im) x.im))
0.2b
(* (+ (* x.re x.im) (* x.im x.re)) x.re)
0.2b
(+ (* (+ x.re x.im) (* (- x.re x.im) x.im)) (* (+ (* x.re x.im) (* x.im x.re)) x.re))
0.0b
(* (- x.re x.im) x.im)

rewrite47.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

18.0ms
(+ (* (+ x.re x.im) (* (- x.re x.im) x.im)) (* (+ (* x.re x.im) (* x.im x.re)) x.re))
14.0ms
(* (+ x.re x.im) (* (- x.re x.im) x.im))
8.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.re)

series131.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

46.0ms
(+ (* (+ x.re x.im) (* (- x.re x.im) x.im)) (* (+ (* x.re x.im) (* x.im x.re)) x.re))
45.0ms
(* (+ x.re x.im) (* (- x.re x.im) x.im))
26.0ms
(* (- x.re x.im) x.im)
13.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.re)

simplify12.9s

Counts
70 → 88
Calls

70 calls. Slowest were:

608.0ms
(* (* (- x.re x.im) (+ (* x.re x.re) (+ (* x.im x.im) (* x.re x.im)))) (+ (* (* x.re x.im) (* x.re x.im)) (- (* (* x.im x.re) (* x.im x.re)) (* (* x.re x.im) (* x.im x.re)))))
587.0ms
(+ (* (* (- (* x.re x.re) (* x.im x.im)) (* (- (* x.re x.re) (* x.im x.im)) x.im)) (+ (* (* x.re x.im) (* x.re x.im)) (- (* (* x.im x.re) (* x.im x.re)) (* (* x.re x.im) (* x.im x.re))))) (* (* (- x.re x.im) (+ x.re x.im)) (* (+ (pow (* x.re x.im) 3) (pow (* x.im x.re) 3)) x.re)))
582.0ms
(* (- x.re x.im) (+ (* (* x.re x.im) (* x.re x.im)) (- (* (* x.im x.re) (* x.im x.re)) (* (* x.re x.im) (* x.im x.re)))))

prune847.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0b

localize25.0ms

Local error

Found 4 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 75
Calls

4 calls. Slowest were:

5.0ms
(* (cbrt (+ x.re x.im)) (cbrt (+ x.re x.im)))
1.0ms
(cbrt (+ x.re x.im))
1.0ms
(cbrt (+ x.re x.im))

series632.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

197.0ms
(* (cbrt (+ x.re x.im)) (cbrt (+ x.re x.im)))
165.0ms
(cbrt (+ x.re x.im))
139.0ms
(cbrt (+ x.re x.im))
130.0ms
(cbrt (+ x.re x.im))

simplify1.5s

Counts
59 → 87
Calls

59 calls. Slowest were:

206.0ms
(- (+ (* 1/3 (* (pow (/ 1 (pow x.im 2)) 1/3) x.re)) (pow x.im 1/3)) (* 1/9 (* (pow (/ 1 (pow x.im 5)) 1/3) (pow x.re 2))))
166.0ms
(- (+ (* 1/3 (* (pow (/ 1 (pow x.im 2)) 1/3) x.re)) (pow x.im 1/3)) (* 1/9 (* (pow (/ 1 (pow x.im 5)) 1/3) (pow x.re 2))))
147.0ms
(- (+ (* 1/3 (* (pow (/ 1 (pow x.im 2)) 1/3) x.re)) (pow x.im 1/3)) (* 1/9 (* (pow (/ 1 (pow x.im 5)) 1/3) (pow x.re 2))))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize35.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 59
Calls

4 calls. Slowest were:

6.0ms
(cbrt (* (cbrt (+ x.re x.im)) (cbrt (+ x.re x.im))))
2.0ms
(cbrt (cbrt (+ x.re x.im)))
1.0ms
(cbrt (+ x.re x.im))

series953.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

337.0ms
(cbrt (* (cbrt (+ x.re x.im)) (cbrt (+ x.re x.im))))
286.0ms
(cbrt (cbrt (+ x.re x.im)))
180.0ms
(cbrt (+ x.re x.im))
149.0ms
(cbrt (+ x.re x.im))

simplify1.4s

Counts
39 → 71
Calls

39 calls. Slowest were:

214.0ms
(- (+ (pow x.im 2/9) (* 2/9 (* (pow (/ 1 (pow x.im 7)) 1/9) x.re))) (* 7/81 (* (pow (/ 1 (pow x.im 16)) 1/9) (pow x.re 2))))
198.0ms
(- (+ (* 1/3 (* (pow (/ 1 (pow x.im 2)) 1/3) x.re)) (pow x.im 1/3)) (* 1/9 (* (pow (/ 1 (pow x.im 5)) 1/3) (pow x.re 2))))
155.0ms
(- (+ (* 1/3 (* (pow (/ 1 (pow x.im 2)) 1/3) x.re)) (pow x.im 1/3)) (* 1/9 (* (pow (/ 1 (pow x.im 5)) 1/3) (pow x.re 2))))

prune1.2s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes134.0ms

Accuracy

0% (0.2b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch3.0ms

end0.0ms

sample2.9s

Algorithm
intervals