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

Average Error: 0.0 → 0.0

Time: 5.4s

Precision: binary64

Cost: 6720

Math

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

↓

\[\mathsf{fma}\left(0.5 + y, x, z\right) \]

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma (+ 0.5 y) x 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 fma((0.5 + y), x, 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 fma(Float64(0.5 + y), x, 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[(0.5 + y), $MachinePrecision] * x + z), $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. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]

4. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + y, x, z\right)} \]
   Proof

   [Start] 0.0	\[ z + \left(0.5 + y\right) \cdot x \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(0.5 + y\right) \cdot x + z} \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(0.5 + y, x, z\right)} \]

5. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(0.5 + y, x, z\right) \]

Alternatives

Alternative 1
Error	29.2
Cost	1248

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+122}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{+110}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{+27}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-154}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-195}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-174}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+28}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	10.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+86} \lor \neg \left(x \leq 8.5 \cdot 10^{+69}\right):\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 3
Error	1.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -62 \lor \neg \left(y \leq 3.1 \cdot 10^{-12}\right):\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + \frac{x}{2}\\ \end{array} \]

Alternative 4
Error	15.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+15}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+42}:\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 5
Error	27.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-34}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

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

Alternative 7
Error	34.7
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023066 
(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))