Average Error: 0.0 → 0.0
Time: 4.9s
Precision: 64
Internal Precision: 128
\[re \cdot re - im \cdot im\]
↓
\[\left(im + re\right) \cdot \left(re - im\right)\]

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied difference-of-squares0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (- (* re re) (* im im)))

Details

Time bar (total: 4.7s)Debug log

sample144.0ms

Algorithm
intervals

simplify35.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
35.0ms
(- (* re re) (* im im))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 1 expressions with local error:

0.0b
(- (* re re) (* im im))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
3×add-log-exp
1×difference-of-squares
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 → 17
Calls
1 calls:
Slowest
6.0ms
(- (* re re) (* im im))

series50.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
50.0ms
(- (* re re) (* im im))

simplify85.0ms

Counts
8 → 20
Calls
8 calls:
Slowest
26.0ms
(/ (exp (* re re)) (exp (* im im)))
16.0ms
(fma re re (- (* im im)))
11.0ms
(- (pow re 2) (pow im 2))
11.0ms
(- (pow re 2) (pow im 2))
10.0ms
(- (pow re 2) (pow im 2))

prune205.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 1 expressions with local error:

0.0b
(* (+ re im) (- re im))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
4×frac-times
3×flip--
3×flip-+
3×add-cube-cbrt
3×add-exp-log
3×associate-*l*
3×associate-*r*
3×add-cbrt-cube
3×flip3--
3×flip3-+
3×*-un-lft-identity
3×pow1
3×add-sqr-sqrt
2×associate-*r/
2×associate-*l/
2×distribute-lft-in
2×sub-neg
2×distribute-rgt-in
1×add-log-exp
1×log1p-expm1-u
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×expm1-log1p-u
Counts
1 → 31
Calls
1 calls:
Slowest
6.0ms
(* (+ re im) (- re im))

series24.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
23.0ms
(* (+ re im) (- re im))

simplify2.4s

Counts
27 → 34
Calls
27 calls:
Slowest
527.0ms
(* (+ (* re re) (- (* im im) (* re im))) (+ (* re re) (+ (* im im) (* re im))))
436.0ms
(* (+ (pow re 3) (pow im 3)) (- (pow re 3) (pow im 3)))
199.0ms
(* (+ (pow re 3) (pow im 3)) (- re im))
185.0ms
(* (+ (pow re 3) (pow im 3)) (- (* re re) (* im im)))
179.0ms
(* (- re im) (+ (* re re) (+ (* im im) (* re im))))

prune275.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize6.0ms

Local error

Found 1 expressions with local error:

0.0b
(fma re re (- (* im 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 re re (- (* im im)))

series26.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
26.0ms
(fma re re (- (* im im)))

simplify18.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
6.0ms
(- (pow re 2) (pow im 2))
5.0ms
(- (pow re 2) (pow im 2))
5.0ms
(- (pow re 2) (pow im 2))

prune109.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes71.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0b and baseline of 0.0b

bsearch1.0ms

end0.0ms

sample1.2s

Algorithm
intervals