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

Average Error: 0.1 → 0

Time: 7.1s

Precision: binary64

Cost: 6592

Math

\[x - \frac{3}{8} \cdot y \]

↓

\[\mathsf{fma}\left(y, -0.375, x\right) \]

FPCore

(FPCore (x y) :precision binary64 (- x (* (/ 3.0 8.0) y)))

↓

(FPCore (x y) :precision binary64 (fma y -0.375 x))

C

double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

↓

double code(double x, double y) {
	return fma(y, -0.375, x);
}

Julia

function code(x, y)
	return Float64(x - Float64(Float64(3.0 / 8.0) * y))
end

↓

function code(x, y)
	return fma(y, -0.375, x)
end

Wolfram

code[x_, y_] := N[(x - N[(N[(3.0 / 8.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(y * -0.375 + x), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[x - \frac{3}{8} \cdot y \]

2. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.375, x\right)} \]
   Proof
   (fma.f64 y -3/8 x): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (Rewrite<= metadata-eval (neg.f64 3/8)) x): 0 points increase in error, 0 points decrease in error
   (fma.f64 y (neg.f64 (Rewrite<= metadata-eval (/.f64 3 8))) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (neg.f64 (/.f64 3 8))) x)): 16 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y (/.f64 3 8)))) x): 0 points increase in error, 0 points decrease in error
   (+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 3 8) y))) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (*.f64 (/.f64 3 8) y)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 x (*.f64 (/.f64 3 8) y))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0

   \[\leadsto \mathsf{fma}\left(y, -0.375, x\right) \]

Alternatives

Alternative 1
Error	17.0
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+90}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-39}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.375\\ \end{array} \]

Alternative 2
Error	0.1
Cost	320

\[x + y \cdot -0.375 \]

Alternative 3
Error	31.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3.0 8.0) y)))