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

Percentage Accurate: 99.9% → 100.0%

Time: 2.4s

Precision: binary64

Cost: 6848

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \frac{y}{2}, z \cdot -0.125\right) \end{array} \]

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 -0.125)))

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 * -0.125));
}

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 * -0.125))
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 * -0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{2}, z \cdot -0.125\right)} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \frac{x \cdot y}{2} - \frac{z}{8} \]
   associate-*r/ [<=] 100.0%	\[ \color{blue}{x \cdot \frac{y}{2}} - \frac{z}{8} \]
   *-commutative [<=] 100.0%	\[ \color{blue}{\frac{y}{2} \cdot x} - \frac{z}{8} \]
   *-commutative [=>] 100.0%	\[ \color{blue}{x \cdot \frac{y}{2}} - \frac{z}{8} \]
   fma-neg [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{2}, -\frac{z}{8}\right)} \]
   distribute-neg-frac [=>] 100.0%	\[ \mathsf{fma}\left(x, \frac{y}{2}, \color{blue}{\frac{-z}{8}}\right) \]
   neg-mul-1 [=>] 100.0%	\[ \mathsf{fma}\left(x, \frac{y}{2}, \frac{\color{blue}{-1 \cdot z}}{8}\right) \]
   associate-/l* [=>] 99.9%	\[ \mathsf{fma}\left(x, \frac{y}{2}, \color{blue}{\frac{-1}{\frac{8}{z}}}\right) \]
   associate-/r/ [=>] 100.0%	\[ \mathsf{fma}\left(x, \frac{y}{2}, \color{blue}{\frac{-1}{8} \cdot z}\right) \]
   *-commutative [=>] 100.0%	\[ \mathsf{fma}\left(x, \frac{y}{2}, \color{blue}{z \cdot \frac{-1}{8}}\right) \]
   metadata-eval [=>] 100.0%	\[ \mathsf{fma}\left(x, \frac{y}{2}, z \cdot \color{blue}{-0.125}\right) \]

3. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, \frac{y}{2}, z \cdot -0.125\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6848

\[\mathsf{fma}\left(x, \frac{y}{2}, z \cdot -0.125\right) \]

Alternative 2
Accuracy	67.9%
Cost	850

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+92} \lor \neg \left(x \leq -5.7 \cdot 10^{+71} \lor \neg \left(x \leq -5 \cdot 10^{+43}\right) \land x \leq 1.76 \cdot 10^{-140}\right):\\ \;\;\;\;y \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	576

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

Alternative 4
Accuracy	50.6%
Cost	192

\[z \cdot -0.125 \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))