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

Average Error: 0.13% → 0%

Time: 4.1s

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 0.13

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

2. Simplified 0

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

   [Start] 0.13	\[ x - \frac{3}{8} \cdot y \]
   cancel-sign-sub-inv [=>] 0.13	\[ \color{blue}{x + \left(-\frac{3}{8}\right) \cdot y} \]
   +-commutative [=>] 0.13	\[ \color{blue}{\left(-\frac{3}{8}\right) \cdot y + x} \]
   *-commutative [=>] 0.13	\[ \color{blue}{y \cdot \left(-\frac{3}{8}\right)} + x \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(y, -\frac{3}{8}, x\right)} \]
   metadata-eval [=>] 0	\[ \mathsf{fma}\left(y, -\color{blue}{0.375}, x\right) \]
   metadata-eval [=>] 0	\[ \mathsf{fma}\left(y, \color{blue}{-0.375}, x\right) \]

3. Final simplification 0

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

Alternatives

Alternative 1
Error	26.04%
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-149}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-21}:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.13%
Cost	320

\[x + y \cdot -0.375 \]

Alternative 3
Error	49.48%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3.0 8.0) y)))