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

Average Accuracy: 99.9% → 99.9%

Time: 11.4s

Precision: binary64

Cost: 6784

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

function code(x, y)
	return Float64(x * Float64(1.0 - Float64(x * y)))
end

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9	\[ x \cdot \left(1 - x \cdot y\right) \]
   sub-neg [=>] 99.9	\[ x \cdot \color{blue}{\left(1 + \left(-x \cdot y\right)\right)} \]
   distribute-lft-in [=>] 99.9	\[ \color{blue}{x \cdot 1 + x \cdot \left(-x \cdot y\right)} \]
   +-commutative [=>] 99.9	\[ \color{blue}{x \cdot \left(-x \cdot y\right) + x \cdot 1} \]
   distribute-lft-neg-in [=>] 99.9	\[ x \cdot \color{blue}{\left(\left(-x\right) \cdot y\right)} + x \cdot 1 \]
   *-commutative [=>] 99.9	\[ x \cdot \color{blue}{\left(y \cdot \left(-x\right)\right)} + x \cdot 1 \]
   associate-*r* [=>] 99.9	\[ \color{blue}{\left(x \cdot y\right) \cdot \left(-x\right)} + x \cdot 1 \]
   *-rgt-identity [=>] 99.9	\[ \left(x \cdot y\right) \cdot \left(-x\right) + \color{blue}{x} \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(x \cdot y, -x, x\right)} \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	76.3%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+181} \lor \neg \left(y \leq 1.3 \cdot 10^{+112}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	448

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

Alternative 3
Accuracy	99.9%
Cost	448

\[x - x \cdot \left(x \cdot y\right) \]

Alternative 4
Accuracy	65.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x y)
  :name "Numeric.SpecFunctions:log1p from math-functions-0.1.5.2, A"
  :precision binary64
  (* x (- 1.0 (* x y))))