Difference of squares

Average Error: 0.0 → 0.0

Time: 1.3s

Precision: binary64

Cost: 6784

Math

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

↓

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

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(b * Float64(-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]

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 egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	14.5
Cost	1556

\[\begin{array}{l} t_0 := b \cdot \left(-b\right)\\ \mathbf{if}\;a \cdot a \leq 1.3 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \cdot a \leq 2.4 \cdot 10^{-226}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;a \cdot a \leq 2.6 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \cdot a \leq 2.2 \cdot 10^{-120}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;a \cdot a \leq 6.8 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	27.8
Cost	192

\[a \cdot a \]

Reproduce

herbie shell --seed 2022325 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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