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

Average Accuracy: 100.0% → 100.0%

Time: 1.6s

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 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \left(\frac{x}{2} + y \cdot x\right) + z \]
   associate-+l+ [=>] 100.0	\[ \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]
   *-commutative [=>] 100.0	\[ \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]
   fma-def [=>] 100.0	\[ \frac{x}{2} + \color{blue}{\mathsf{fma}\left(x, y, z\right)} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	83.6%
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+72} \lor \neg \left(y \leq -1.75 \cdot 10^{+45} \lor \neg \left(y \leq -1.9 \cdot 10^{-11}\right) \land y \leq 5 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{x}{2} + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2} + z\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	576

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

Alternative 3
Accuracy	65.9%
Cost	320

\[\frac{x}{2} + z \]

Reproduce

herbie shell --seed 2023160 
(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))