Average Error: 0.0 → 0.0
Time: 48.5s
Precision: 64
Internal Precision: 128
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[(x.re \cdot y.re + \left(y.im \cdot \left(-x.im\right)\right))_*\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 0.0

    \[x.re \cdot y.re - x.im \cdot y.im\]
  2. Using strategy rm

  3. Applied fma-neg0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))

Details

Time bar (total: 48.1s)Debug log

sample27.0ms

Algorithm
intervals

simplify9.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
8.0ms
(- (* x.re y.re) (* x.im y.im))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 1 expressions with local error:

0.0b
(- (* x.re y.re) (* x.im y.im))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
3×add-log-exp
1×flip--
1×log1p-expm1-u
1×add-cube-cbrt
1×add-exp-log
1×prod-diff
1×diff-log
1×fma-neg
1×add-cbrt-cube
1×flip3--
1×*-un-lft-identity
1×sub-neg
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 16
Calls
1 calls:
Slowest
4.0ms
(- (* x.re y.re) (* x.im y.im))

series48.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
48.0ms
(- (* x.re y.re) (* x.im y.im))

simplify107.0ms

Counts
6 → 19
Calls
6 calls:
Slowest
34.0ms
(/ (exp (* x.re y.re)) (exp (* x.im y.im)))
24.0ms
(- (* y.re x.re) (* y.im x.im))
19.0ms
(fma x.re y.re (- (* y.im x.im)))
18.0ms
(- (* y.re x.re) (* y.im x.im))
9.0ms
(fma (- y.im) x.im (* y.im x.im))

prune149.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma x.re y.re (- (* x.im y.im)))

rewrite0.0ms

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

series31.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
31.0ms
(fma x.re y.re (- (* x.im y.im)))

simplify18.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
9.0ms
(- (* y.re x.re) (* y.im x.im))
9.0ms
(- (* y.re x.re) (* y.im x.im))
0.0ms
0

prune104.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

37.4b
(/ (- (pow (* x.re y.re) 3) (pow (* x.im y.im) 3)) (+ (* (* x.re y.re) (* x.re y.re)) (+ (* (* x.im y.im) (* x.im y.im)) (* (* x.re y.re) (* x.im y.im)))))
5.0b
(* (* x.re y.re) (* x.im y.im))
1.2b
(- (pow (* x.re y.re) 3) (pow (* x.im y.im) 3))
0.6b
(+ (* (* x.im y.im) (* x.im y.im)) (* (* x.re y.re) (* x.im y.im)))

rewrite77.0ms

Algorithm
rewrite-expression-head
Rules
49×prod-diff
40×*-un-lft-identity
40×add-sqr-sqrt
30×add-cube-cbrt
18×times-frac
15×cube-prod
15×unpow3
15×unpow-prod-down
15×cube-mult
8×add-log-exp
8×add-exp-log
8×add-cbrt-cube
7×fma-neg
6×associate-/l*
6×pow1
5×difference-of-squares
5×distribute-lft-out--
5×difference-cubes
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-/r*
2×associate-/l/
2×flip--
2×flip-+
2×associate-/r/
2×flip3--
2×flip3-+
1×div-inv
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×associate-*r*
1×div-exp
1×div-sub
1×diff-log
1×distribute-rgt-out
1×frac-2neg
1×fma-def
1×sub-neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
4 → 149
Calls
4 calls:
Slowest
54.0ms
(/ (- (pow (* x.re y.re) 3) (pow (* x.im y.im) 3)) (+ (* (* x.re y.re) (* x.re y.re)) (+ (* (* x.im y.im) (* x.im y.im)) (* (* x.re y.re) (* x.im y.im)))))
8.0ms
(- (pow (* x.re y.re) 3) (pow (* x.im y.im) 3))
7.0ms
(* (* x.re y.re) (* x.im y.im))
6.0ms
(+ (* (* x.im y.im) (* x.im y.im)) (* (* x.re y.re) (* x.im y.im)))

series301.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
111.0ms
(- (pow (* x.re y.re) 3) (pow (* x.im y.im) 3))
93.0ms
(/ (- (pow (* x.re y.re) 3) (pow (* x.im y.im) 3)) (+ (* (* x.re y.re) (* x.re y.re)) (+ (* (* x.im y.im) (* x.im y.im)) (* (* x.re y.re) (* x.im y.im)))))
56.0ms
(+ (* (* x.im y.im) (* x.im y.im)) (* (* x.re y.re) (* x.im y.im)))
40.0ms
(* (* x.re y.re) (* x.im y.im))

simplify43.0s

Counts
179 → 161
Calls
179 calls:
Slowest
817.0ms
(fma (* (* x.re y.re) (* x.re y.re)) (* x.re y.re) (- (* (pow y.im 3) (pow x.im 3))))
634.0ms
(* (* (* (* x.re y.re) (* x.re y.re)) (* x.re y.re)) (* (* (* x.im y.im) (* x.im y.im)) (* x.im y.im)))
512.0ms
(fma (* x.re y.re) (* (* x.re y.re) (* x.re y.re)) (- (* (pow y.im 3) (pow x.im 3))))
468.0ms
(fma (* (cbrt (pow (* x.re y.re) 3)) (cbrt (pow (* x.re y.re) 3))) (cbrt (pow (* x.re y.re) 3)) (- (* (* x.im y.im) (* (* x.im y.im) (* x.im y.im)))))
443.0ms
(fma (* (* x.re y.re) (* x.re y.re)) (* x.re y.re) (- (* (cbrt (pow (* x.im y.im) 3)) (* (cbrt (pow (* x.im y.im) 3)) (cbrt (pow (* x.im y.im) 3))))))

prune2.3s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes293.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample1.6s

Algorithm
intervals