Average Error: 6.7 → 0.2
Time: 34.6s
Precision: 64
Internal Precision: 128
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[(\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot \left(-x.im\right)\right))_*\]

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.7
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 6.7

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

  3. Applied difference-of-squares6.7

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

  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. Using strategy rm

  6. Applied fma-neg0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019016 +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: 33.8s)Debug log

sample106.0ms

Algorithm
intervals
Results
470×(pre true 80)
470×(body real 80)

simplify388.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
388.0ms
(- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im))

prune21.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 6.0b

localize51.0ms

Local error

Found 2 expressions with local error:

6.5b
(* (* x.im x.im) (- x.re))
6.0b
(fma x.re (fma (- x.im) (+ x.im x.im) (* x.re x.re)) (* (* x.im x.im) (- x.re)))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
4×add-exp-log
4×associate-*r*
4×add-cbrt-cube
4×pow1
3×add-cube-cbrt
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×log1p-expm1-u
2×expm1-log1p-u
1×fma-udef
1×cbrt-unprod
1×distribute-rgt-neg-out
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×neg-mul-1
Counts
2 → 29
Calls
2 calls:
Slowest
7.0ms
(* (* x.im x.im) (- x.re))
0.0ms
(fma x.re (fma (- x.im) (+ x.im x.im) (* x.re x.re)) (* (* x.im x.im) (- x.re)))

series55.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
41.0ms
(fma x.re (fma (- x.im) (+ x.im x.im) (* x.re x.re)) (* (* x.im x.im) (- x.re)))
14.0ms
(* (* x.im x.im) (- x.re))

simplify766.0ms

Counts
12 → 35
Calls
12 calls:
Slowest
294.0ms
(* (* (* (* x.im x.im) (* x.im x.im)) (* x.im x.im)) (* (* (- x.re) (- x.re)) (- x.re)))
127.0ms
(- (pow x.re 3) (* 3 (* (pow x.im 2) x.re)))
117.0ms
(- (pow x.re 3) (* 3 (* (pow x.im 2) x.re)))
102.0ms
(- (pow x.re 3) (* 3 (* (pow x.im 2) x.re)))
37.0ms
(* -1 (* (pow x.im 2) x.re))

prune330.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 5.9b

localize23.0ms

Local error

Found 1 expressions with local error:

6.0b
(fma (* x.re -3) (* x.im x.im) (pow x.re 3))

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma (* x.re -3) (* x.im x.im) (pow x.re 3))

series43.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
43.0ms
(fma (* x.re -3) (* x.im x.im) (pow x.re 3))

simplify341.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
114.0ms
(- (pow x.re 3) (* 3 (* (pow x.im 2) x.re)))
114.0ms
(- (pow x.re 3) (* 3 (* (pow x.im 2) x.re)))
111.0ms
(- (pow x.re 3) (* 3 (* (pow x.im 2) x.re)))

prune149.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 5.9b

localize36.0ms

Local error

Found 4 expressions with local error:

6.0b
(* (- (* x.re x.re) (* x.im x.im)) x.re)
0.2b
(* (+ (* x.re x.im) (* x.im x.re)) x.im)
0.2b
(- (* (- (* 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))

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
12×associate-*l/
8×add-log-exp
7×associate-*l*
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×flip--
4×log1p-expm1-u
4×flip-+
4×frac-sub
4×add-exp-log
4×add-cbrt-cube
4×flip3--
4×flip3-+
4×pow1
4×expm1-log1p-u
2×*-commutative
1×difference-of-squares
1×prod-diff
1×diff-log
1×fma-neg
1×fma-def
1×sub-neg
1×sum-log
1×+-commutative
Counts
4 → 65
Calls
4 calls:
Slowest
17.0ms
(- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
9.0ms
(* (- (* x.re x.re) (* x.im x.im)) x.re)
9.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)
4.0ms
(+ (* x.re x.im) (* x.im x.re))

series122.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
44.0ms
(- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
40.0ms
(* (- (* x.re x.re) (* x.im x.im)) x.re)
20.0ms
(+ (* x.re x.im) (* x.im x.re))
17.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)

simplify6.3s

Counts
34 → 77
Calls
34 calls:
Slowest
1.4s
(* (+ (* (* 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)))
595.0ms
(* (- (* (* x.re x.re) (* x.re x.re)) (* (* x.im x.im) (* x.im x.im))) x.re)
481.0ms
(fma (- x.im) (+ (* x.re x.im) (* x.im x.re)) (* x.im (+ (* x.re x.im) (* x.im x.re))))
425.0ms
(- (* (* (- (pow (* x.re x.re) 3) (pow (* x.im x.im) 3)) x.re) (+ (* (* 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.re) (* x.re x.re)) (+ (* (* x.im x.im) (* x.im x.im)) (* (* x.re x.re) (* x.im x.im)))) (* (+ (pow (* x.re x.im) 3) (pow (* x.im x.re) 3)) x.im)))
404.0ms
(* (+ (pow (* x.re x.im) 3) (pow (* x.im x.re) 3)) x.im)

prune777.0ms

Pruning

8 alts after pruning (5 fresh and 3 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 4 expressions with local error:

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

rewrite37.0ms

Algorithm
rewrite-expression-head
Rules
44×associate-*l/
18×flip-+
18×flip3-+
16×frac-sub
12×frac-times
11×flip--
11×flip3--
9×associate-*l*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-log-exp
6×add-exp-log
6×associate-*r/
6×add-cbrt-cube
6×pow1
4×log1p-expm1-u
4×expm1-log1p-u
3×*-commutative
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×prod-diff
1×associate-*r*
1×diff-log
1×fma-neg
1×sub-neg
Counts
4 → 87
Calls
4 calls:
Slowest
13.0ms
(- (* (+ x.re x.im) (* (- x.re x.im) x.re)) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
11.0ms
(* (+ x.re x.im) (* (- x.re x.im) x.re))
5.0ms
(* (- x.re x.im) x.re)
5.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)

series149.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
47.0ms
(- (* (+ x.re x.im) (* (- x.re x.im) x.re)) (* (+ (* x.re x.im) (* x.im x.re)) x.im))
44.0ms
(* (+ x.re x.im) (* (- x.re x.im) x.re))
30.0ms
(* (- x.re x.im) x.re)
28.0ms
(* (+ (* x.re x.im) (* x.im x.re)) x.im)

simplify19.9s

Counts
73 → 99
Calls
73 calls:
Slowest
1.6s
(* (+ 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)))))
1.5s
(* (- 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)))))
1.3s
(* (* (- x.re x.im) (+ x.re x.im)) (- (* x.re x.im) (* x.im x.re)))
726.0ms
(- (* (* (+ x.re x.im) (* (- (pow x.re 3) (pow x.im 3)) x.re)) (+ (* (* 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.re) (+ (* x.im x.im) (* x.re x.im))) (* (+ (pow (* x.re x.im) 3) (pow (* x.im x.re) 3)) x.im)))
612.0ms
(* (+ (* x.re x.re) (- (* x.im x.im) (* x.re x.im))) (+ (* x.re x.re) (+ (* x.im x.im) (* x.re x.im))))

prune964.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes100.0ms

Accuracy

0% (0.2b remaining)

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

bsearch4.0ms

end0.0ms

sample3.0s

Algorithm
intervals
Results
14518×(pre true 80)
14518×(body real 80)