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

Average Accuracy: 100.0% → 100.0%

Time: 4.8s

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	56.9%
Cost	984

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+84}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+21}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-99}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-151}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+37}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5\\ \end{array} \]

Alternative 2
Accuracy	73.9%
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+154}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 0.0095 \lor \neg \left(z \leq 1.7 \cdot 10^{+69}\right) \land z \leq 1.15 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Accuracy	83.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+16} \lor \neg \left(x \leq 1.12 \cdot 10^{+33}\right):\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 4
Accuracy	98.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -12500 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{2} + z\\ \end{array} \]

Alternative 5
Accuracy	100.0%
Cost	576

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

Alternative 6
Accuracy	58.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+36}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5\\ \end{array} \]

Alternative 7
Accuracy	100.0%
Cost	448

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

Alternative 8
Accuracy	46.2%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023147 
(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))