Expanding a square

Average Error: 39.3 → 0

Time: 1.6s

Precision: binary64

Math

\[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

↓

\[\mathsf{fma}\left(x, x, x \cdot 2\right) \]

FPCore

(FPCore (x) :precision binary64 (- (* (+ x 1.0) (+ x 1.0)) 1.0))

↓

(FPCore (x) :precision binary64 (fma x x (* x 2.0)))

C

double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

↓

double code(double x) {
	return fma(x, x, (x * 2.0));
}

Julia

function code(x)
	return Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - 1.0)
end

↓

function code(x)
	return fma(x, x, Float64(x * 2.0))
end

Wolfram

code[x_] := N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[x_] := N[(x * x + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Derivation

1. Initial program 39.3

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

2. Simplified 0.0

   \[\leadsto \color{blue}{x \cdot \left(x + 2\right)} \]

3. Applied egg-rr 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot 2\right)} \]

4. Final simplification 0

   \[\leadsto \mathsf{fma}\left(x, x, x \cdot 2\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))