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

Average Error: 0.0 → 0

Time: 2.8s

Precision: binary64

Cost: 6848

Math

\[\frac{x \cdot y}{2} - \frac{z}{8} \]

↓

\[\mathsf{fma}\left(x, \frac{y}{2}, \frac{z}{-8}\right) \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

↓

(FPCore (x y z) :precision binary64 (fma x (/ y 2.0) (/ z -8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

↓

double code(double x, double y, double z) {
	return fma(x, (y / 2.0), (z / -8.0));
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

↓

function code(x, y, z)
	return fma(x, Float64(y / 2.0), Float64(z / -8.0))
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(x * N[(y / 2.0), $MachinePrecision] + N[(z / -8.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]

2. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{2}, \frac{z}{-8}\right)} \]
   Proof

   [Start] 0.0	\[ \frac{x \cdot y}{2} - \frac{z}{8} \]
   associate-*r/ [<=] 0.0	\[ \color{blue}{x \cdot \frac{y}{2}} - \frac{z}{8} \]
   fma-neg [=>] 0	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right)} \]
   distribute-neg-frac [=>] 0	\[ \mathsf{fma}\left(x, \frac{y}{2}, \color{blue}{\frac{-z}{8}}\right) \]
   neg-mul-1 [=>] 0	\[ \mathsf{fma}\left(x, \frac{y}{2}, \frac{\color{blue}{-1 \cdot z}}{8}\right) \]
   *-commutative [=>] 0	\[ \mathsf{fma}\left(x, \frac{y}{2}, \frac{\color{blue}{z \cdot -1}}{8}\right) \]
   associate-/l* [=>] 0	\[ \mathsf{fma}\left(x, \frac{y}{2}, \color{blue}{\frac{z}{\frac{8}{-1}}}\right) \]
   metadata-eval [=>] 0	\[ \mathsf{fma}\left(x, \frac{y}{2}, \frac{z}{\color{blue}{-8}}\right) \]

3. Final simplification 0

   \[\leadsto \mathsf{fma}\left(x, \frac{y}{2}, \frac{z}{-8}\right) \]

Alternatives

Alternative 1
Error	16.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-70}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]

Alternative 2
Error	0.0
Cost	576

\[\frac{x \cdot y}{2} - \frac{z}{8} \]

Alternative 3
Error	26.8
Cost	192

\[z \cdot -0.125 \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))