Average Error: 0.0 → 0.0
Time: 2.6s
Precision: binary64
Cost: 6784
\[x \cdot y - z \cdot t \]
↓
\[\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right) \]
(FPCore (x y z t) :precision binary64 (- (* x y) (* z t)))
↓
(FPCore (x y z t) :precision binary64 (fma y x (* z (- t))))
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(y, x, (z * -t));
}

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

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

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

Error

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{-1 \cdot \left(t \cdot z\right) + y \cdot x} \]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, z \cdot \left(-t\right)\right)} \]
    Proof
    (fma.f64 y x (*.f64 z (neg.f64 t))): 0 points increase in error, 0 points decrease in error
    (fma.f64 y x (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 z t)))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 y x) (neg.f64 (*.f64 z t)))): 1 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 x y)) (neg.f64 (*.f64 z t))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 x y) (neg.f64 (Rewrite=> *-commutative_binary64 (*.f64 t z)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 x y) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 t z)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (*.f64 t z)) (*.f64 x y))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 -1 (*.f64 t z)) (Rewrite<= *-commutative_binary64 (*.f64 y x))): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error23.4
Cost784
\[\begin{array}{l} t_1 := z \cdot \left(-t\right)\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-222}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-124}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error0.0
Cost448
\[y \cdot x - z \cdot t \]
Alternative 3
Error30.8
Cost192
\[y \cdot x \]

Error

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))