Difference of squares

Average Error: 0.0 → 0.0

Time: 1.5s

Precision: binary64

Math

\[a \cdot a - b \cdot b \]

↓

\[\mathsf{fma}\left(a, a, -b \cdot b\right) \]

FPCore

(FPCore (a b) :precision binary64 (- (* a a) (* b b)))

↓

(FPCore (a b) :precision binary64 (fma a a (- (* b b))))

C

double code(double a, double b) {
	return (a * a) - (b * b);
}

↓

double code(double a, double b) {
	return fma(a, a, -(b * b));
}

Julia

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

↓

function code(a, b)
	return fma(a, a, Float64(-Float64(b * b)))
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[(a * a + (-N[(b * b), $MachinePrecision])), $MachinePrecision]

Error

Bits error versus a

Bits error versus b

Target

Original	0.0
Target	0.0
Herbie	0.0

\[\left(a + b\right) \cdot \left(a - b\right) \]

Derivation

1. Initial program 0.0

   \[a \cdot a - b \cdot b \]

2. Applied fma-neg_binary64 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, -b \cdot b\right)} \]

3. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(a, a, -b \cdot b\right) \]

Reproduce

herbie shell --seed 2022129 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :herbie-target
  (* (+ a b) (- a b))

  (- (* a a) (* b b)))