Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, B

Average Error: 0.0 → 0

Time: 1.6s

Precision: binary64

Cost: 6720

Math

\[x \cdot \left(1 - x \cdot 0.5\right) \]

↓

\[\mathsf{fma}\left(x, x \cdot -0.5, x\right) \]

FPCore

(FPCore (x) :precision binary64 (* x (- 1.0 (* x 0.5))))

↓

(FPCore (x) :precision binary64 (fma x (* x -0.5) x))

C

double code(double x) {
	return x * (1.0 - (x * 0.5));
}

↓

double code(double x) {
	return fma(x, (x * -0.5), x);
}

Julia

function code(x)
	return Float64(x * Float64(1.0 - Float64(x * 0.5)))
end

↓

function code(x)
	return fma(x, Float64(x * -0.5), x)
end

Wolfram

code[x_] := N[(x * N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * N[(x * -0.5), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[x \cdot \left(1 - x \cdot 0.5\right) \]

2. Simplified 0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]
   Proof

   [Start] 0.0	\[ x \cdot \left(1 - x \cdot 0.5\right) \]
   sub-neg [=>] 0.0	\[ x \cdot \color{blue}{\left(1 + \left(-x \cdot 0.5\right)\right)} \]
   distribute-lft-in [=>] 0.0	\[ \color{blue}{x \cdot 1 + x \cdot \left(-x \cdot 0.5\right)} \]
   +-commutative [=>] 0.0	\[ \color{blue}{x \cdot \left(-x \cdot 0.5\right) + x \cdot 1} \]
   *-rgt-identity [=>] 0.0	\[ x \cdot \left(-x \cdot 0.5\right) + \color{blue}{x} \]
   fma-def [=>] 0	\[ \color{blue}{\mathsf{fma}\left(x, -x \cdot 0.5, x\right)} \]
   distribute-rgt-neg-in [=>] 0	\[ \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.5\right)}, x\right) \]
   metadata-eval [=>] 0	\[ \mathsf{fma}\left(x, x \cdot \color{blue}{-0.5}, x\right) \]

3. Final simplification 0

   \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, x\right) \]

Alternatives

Alternative 1
Error	1.8
Cost	585

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

Alternative 2
Error	0.0
Cost	448

\[x \cdot \left(1 + x \cdot -0.5\right) \]

Alternative 3
Error	21.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023125 
(FPCore (x)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (- 1.0 (* x 0.5))))