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

Average Error: 0.01% → 0.01%

Time: 3.7s

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 0.01

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

2. Simplified 0.01

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

   [Start] 0.01	\[ \left(\frac{x}{2} + y \cdot x\right) + z \]
   associate-+l+ [=>] 0.01	\[ \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]
   *-commutative [=>] 0.01	\[ \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]
   fma-def [=>] 0.01	\[ \frac{x}{2} + \color{blue}{\mathsf{fma}\left(x, y, z\right)} \]

3. Taylor expanded in x around 0 0.01

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

4. Simplified 0.01

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

   [Start] 0.01	\[ z + \left(0.5 + y\right) \cdot x \]
   +-commutative [=>] 0.01	\[ \color{blue}{\left(0.5 + y\right) \cdot x + z} \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(0.5 + y, x, z\right)} \]

5. Final simplification 0.01

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

Alternatives

Alternative 1
Error	31.86%
Cost	1115

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+40} \lor \neg \left(x \leq -1.9 \cdot 10^{-41}\right) \land \left(x \leq -1.75 \cdot 10^{-111} \lor \neg \left(x \leq 1.55 \cdot 10^{-107}\right) \land \left(x \leq 8.1 \cdot 10^{-53} \lor \neg \left(x \leq 3.6 \cdot 10^{-13}\right)\right)\right):\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	43.9%
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+87}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{+75}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-96}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-75}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Error	18.64%
Cost	720

\[\begin{array}{l} t_0 := \left(0.5 + y\right) \cdot x\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 300000000:\\ \;\;\;\;z + 0.5 \cdot x\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+217}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	0.01%
Cost	448

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

Alternative 5
Error	52.94%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023088 
(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))