Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1

Average Error: 0.0 → 0.0

Time: 4.7s

Precision: binary64

Cost: 6848

Math

\[\left(\frac{x}{2} + y \cdot x\right) + z \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]

2. Simplified 0.0

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

   [Start] 0.0	\[ \left(\frac{x}{2} + y \cdot x\right) + z \]
   associate-+l+ [=>] 0.0	\[ \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]
   *-commutative [=>] 0.0	\[ \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]
   fma-def [=>] 0.0	\[ \frac{x}{2} + \color{blue}{\mathsf{fma}\left(x, y, z\right)} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	11.3
Cost	849

\[\begin{array}{l} t_0 := \frac{x}{2} + z\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq 10^{-13} \lor \neg \left(z \leq 0.35\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Error	11.3
Cost	849

\[\begin{array}{l} t_0 := \frac{x}{2} + z\\ \mathbf{if}\;z \leq -4 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{2} + x \cdot y\\ \mathbf{elif}\;z \leq 10^{-13} \lor \neg \left(z \leq 58\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Error	0.0
Cost	576

\[z + \left(\frac{x}{2} + x \cdot y\right) \]

Alternative 4
Error	29.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -0.5:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 0.5:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5
Error	28.5
Cost	320

\[x \cdot \left(y + 0.5\right) \]

Alternative 6
Error	44.9
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))