Linear.V3:cross from linear-1.19.1.3

Average Error: 0.0 → 0.0

Time: 2.6s

Precision: binary64

Cost: 7296

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (- (+ (fma (- z) t (* z t)) (* x y)) (* 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 (fma(-z, t, (z * t)) + (x * y)) - (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(fma(Float64(-z), t, Float64(z * t)) + Float64(x * y)) - 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[((-z) * t + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	21.4
Cost	1051

\[\begin{array}{l} \mathbf{if}\;t \leq -5.9 \cdot 10^{-77} \lor \neg \left(t \leq 2.05 \cdot 10^{-48} \lor \neg \left(t \leq 6.4 \cdot 10^{+37}\right) \land \left(t \leq 7.2 \cdot 10^{+54} \lor \neg \left(t \leq 5.5 \cdot 10^{+79}\right) \land t \leq 1.8 \cdot 10^{+97}\right)\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	31.3
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))