Linear.V3:cross from linear-1.19.1.3

Average Error: 0.02% → 0.01%

Time: 2.6s

Precision: binary64

Cost: 7296

Math

\[x \cdot y - z \cdot t \]

↓

\[\left(x \cdot y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right) - z \cdot t \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* x y) (* z t)))

↓

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x y) (fma (- z) t (* z t))) (* z t)))

C

double code(double x, double y, double z, double t) {
	return (x * y) - (z * t);
}

↓

double code(double x, double y, double z, double t) {
	return ((x * y) + fma(-z, t, (z * t))) - (z * t);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(x * y) - Float64(z * t))
end

↓

function code(x, y, z, t)
	return Float64(Float64(Float64(x * y) + fma(Float64(-z), t, Float64(z * t))) - Float64(z * t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(N[(x * y), $MachinePrecision] + N[((-z) * t + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.02

   \[x \cdot y - z \cdot t \]

2. Applied egg-rr 0.01

   \[\leadsto \color{blue}{z \cdot \left(-t\right) + \left(x \cdot y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right)} \]

3. Final simplification 0.01

   \[\leadsto \left(x \cdot y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right) - z \cdot t \]

Alternatives

Alternative 1
Error	33.99%
Cost	786

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-83} \lor \neg \left(t \leq 2.1 \cdot 10^{-38} \lor \neg \left(t \leq 6 \cdot 10^{+103}\right) \land t \leq 3.9 \cdot 10^{+158}\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Error	0.02%
Cost	448

\[x \cdot y - z \cdot t \]

Alternative 3
Error	47.84%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))