Average Error: 6.9 → 0.2
Time: 47.9s
Precision: 64
Internal Precision: 128
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im\]

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

Derivation

  1. Initial program 6.9

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

  3. Applied difference-of-squares6.9

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

  4. Applied associate-*l*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

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

Details

Time bar (total: 44.4s)Debug log

start68.0ms

Algorithm
intervals

setup372.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 6.2b

localize43.0ms

Local error

Found 4 expressions with local error:

6.2b
(* (- (* x.re x.re) (* x.im x.im)) x.re)
0.1b
(* (+ (* x.re x.im) (* x.im x.re)) x.im)
0.1b
(- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
0.0b
(+ (* x.re x.im) (* x.im x.re))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

8.0ms
(- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
4.0ms
(* (- (* x.re x.re) (* x.im x.im)) x.re)
4.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)

series120.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

42.0ms
(* (- (* x.re x.re) (* x.im x.im)) x.re)
40.0ms
(- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
20.0ms
(+ (* x.re x.im) (* x.im x.re))
18.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)

simplify6.7s

Counts
34 → 77
Calls

34 calls. Slowest were:

1.5s
(* (+ (* (* 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)))
638.0ms
(* (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))) x.re)
482.0ms
(fma (- x.im) (+ (* x.re x.im) (* x.im x.re)) (* x.im (+ (* x.re x.im) (* x.im x.re))))

prune847.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

0.2b
(* (+ x.re x.im) (* (- x.re x.im) x.re))
0.1b
(* (+ (* x.re x.im) (* x.im x.re)) x.im)
0.1b
(- (* (+ x.re x.im) (* (- x.re x.im) x.re)) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
0.0b
(+ (* x.re x.im) (* x.im x.re))

rewrite50.0ms

Algorithm
rewrite-expression-head
Counts
4 → 86
Calls

4 calls. Slowest were:

22.0ms
(- (* (+ x.re x.im) (* (- x.re x.im) x.re)) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
16.0ms
(* (+ x.re x.im) (* (- x.re x.im) x.re))
9.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)

series94.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

30.0ms
(- (* (+ x.re x.im) (* (- x.re x.im) x.re)) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
28.0ms
(* (+ x.re x.im) (* (- x.re x.im) x.re))
24.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)
11.0ms
(+ (* x.re x.im) (* x.im x.re))

simplify20.8s

Counts
70 → 98
Calls

70 calls. Slowest were:

1.7s
(* (+ 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)))))
1.7s
(* (- 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)))))
1.6s
(* (* (- x.re x.im) (+ x.re x.im)) (- (* x.re x.im) (* x.im x.re)))

prune1.1s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 4 expressions with local error:

0.2b
(fma (+ x.re x.im) (* (- x.re x.im) x.re) (- (* (+ (* x.re x.im) (* x.im x.re)) x.im)))
0.1b
(* (+ (* x.re x.im) (* x.im x.re)) x.im)
0.0b
(+ (* x.re x.im) (* x.im x.re))
0.0b
(* (- x.re x.im) x.re)

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

4.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)
2.0ms
(* (- x.re x.im) x.re)
2.0ms
(+ (* x.re x.im) (* x.im x.re))

series103.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

54.0ms
(fma (+ x.re x.im) (* (- x.re x.im) x.re) (- (* (+ (* x.re x.im) (* x.im x.re)) x.im)))
18.0ms
(* (- x.re x.im) x.re)
16.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)
14.0ms
(+ (* x.re x.im) (* x.im x.re))

simplify1.5s

Counts
21 → 66
Calls

21 calls. Slowest were:

442.0ms
(* (+ (pow (* x.re x.im) 3) (pow (* x.im x.re) 3)) x.im)
333.0ms
(* (- (* x.re x.re) (* x.im x.im)) x.re)
173.0ms
(* (- (* (* x.re x.im) (* x.re x.im)) (* (* x.im x.re) (* x.im x.re))) x.im)

prune667.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

26.4b
(cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im))
26.4b
(cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im))
26.4b
(cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im))
0.4b
(* (cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im)) (cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im)))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

21.0ms
(* (cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im)) (cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im)))
5.0ms
(cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im))
5.0ms
(cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im))

series608.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

215.0ms
(* (cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im)) (cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im)))
164.0ms
(cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im))
125.0ms
(cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im))
104.0ms
(cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im))

simplify9.9s

Counts
49 → 85
Calls

49 calls. Slowest were:

1.3s
(cbrt (+ (* (* x.re x.im) (* x.re x.im)) (- (* (* x.im x.re) (* x.im x.re)) (* (* x.re x.im) (* x.im x.re)))))
1.3s
(* (cbrt (+ (* (* x.re x.im) (* x.re x.im)) (- (* (* x.im x.re) (* x.im x.re)) (* (* x.re x.im) (* x.im x.re))))) (cbrt (+ (* (* x.re x.im) (* x.re x.im)) (- (* (* x.im x.re) (* x.im x.re)) (* (* x.re x.im) (* x.im x.re))))))
513.0ms
(* (cbrt (* (+ (pow (* x.re x.im) 3) (pow (* x.im x.re) 3)) x.im)) (cbrt (* (+ (* x.re x.im) (* x.im x.re)) x.im)))

prune1.2s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0b

regimes124.0ms

Accuracy

0% (0.2b remaining)

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

bsearch3.0ms