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

Average Error: 0.0 → 0.0

Time: 3.0s

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 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{2}, \frac{z}{-8}\right)} \]
   Proof
   (fma.f64 x (/.f64 y 2) (/.f64 z -8)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (/.f64 y 2) (/.f64 z (Rewrite<= metadata-eval (/.f64 8 -1)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (/.f64 y 2) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z -1) 8))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (/.f64 y 2) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 z)) 8)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (/.f64 y 2) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 z 8)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (/.f64 y 2) (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 z 8)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x (/.f64 y 2)) (/.f64 z 8))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y 2) x)) (/.f64 z 8)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y x) 2)) (/.f64 z 8)): 8 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) 2) (/.f64 z 8)): 0 points increase in error, 8 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	17.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-12}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]

Alternative 2
Error	0.1
Cost	576

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

Alternative 3
Error	0.0
Cost	576

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

Alternative 4
Error	26.9
Cost	192

\[z \cdot -0.125 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))