Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A

Average Error: 0.1 → 0.0

Time: 1.6s

Precision: binary64

Cost: 6720

Math

\[x - \left(y \cdot 4\right) \cdot z \]

↓

\[\mathsf{fma}\left(z, y \cdot -4, x\right) \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* (* y 4.0) z)))

↓

(FPCore (x y z) :precision binary64 (fma z (* y -4.0) x))

C

double code(double x, double y, double z) {
	return x - ((y * 4.0) * z);
}

↓

double code(double x, double y, double z) {
	return fma(z, (y * -4.0), x);
}

Julia

function code(x, y, z)
	return Float64(x - Float64(Float64(y * 4.0) * z))
end

↓

function code(x, y, z)
	return fma(z, Float64(y * -4.0), x)
end

Wolfram

code[x_, y_, z_] := N[(x - N[(N[(y * 4.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(z * N[(y * -4.0), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[x - \left(y \cdot 4\right) \cdot z \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, y \cdot -4, x\right)} \]
   Proof
   (fma.f64 z (*.f64 y -4) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 z (*.f64 y (Rewrite<= metadata-eval (neg.f64 4))) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 z (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y 4))) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z (neg.f64 (*.f64 y 4))) x)): 1 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 y 4)) z)) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (neg.f64 (*.f64 y 4)) z))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 x (*.f64 (*.f64 y 4) z))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(z, y \cdot -4, x\right) \]

Alternatives

Alternative 1
Error	17.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1950000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+41}:\\ \;\;\;\;z \cdot \left(y \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

\[x + z \cdot \left(y \cdot -4\right) \]

Alternative 3
Error	27.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022330 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (* y 4.0) z)))