complex_multiply

std.elementwise.complex_multiply · Level L2

The elementwise product of two complex vectors, each stored as a [2, n] array of real parts and imaginary parts: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.

(ac − bd, ad + bc)

Signature

complex_multiply(z: f64[2, n], w: f64[2, n]) → f64[2, n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

zf64[2, n]wf64[2, n]SlicezrSlicewrSlicewiSliceziMultiplyrrMultiplyiiMultiplyriMultiplyirSubtractreAddimConcatzwzwf64[2, n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Equal to the reference z * w, as complex numbers in exact rational arithmetic, on all 40 test cases.
  • All 222 float64 results inside the running error bound; the closest uses 75% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
79%
bit-equal to the NumPy formula in float64
100%
largest error, in units in the last place
14

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls
—
Called by
circular_convolve
sha256:4fe3f634318fd76e207a798de0cd9a4bad0ac61492e5cc79fb978616b29d81b7

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol
Ω:std.elementwise.complex_multiply · Ω:complex_multiply
Pins
sha256:f7f9c24173c676751889be47bc14d88dcfe28a499e5cc800d8fa5e4b8970958athis graph alone
Evidence
sha256:d7043181fbf1e9c8d15e26ba8b140ea9c49547ba98303b1a84b18ee8456e5e57the hash of its verification record
Needs
no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.