Rosa's Benchmark

Average Error: 0.2 → 0.2

Time: 7.8s

Precision: binary64

Cost: 6848

Math

\[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

↓

\[x \cdot \mathsf{fma}\left(x, x \cdot -0.12900613773279798, 0.954929658551372\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

↓

(FPCore (x)
 :precision binary64
 (* x (fma x (* x -0.12900613773279798) 0.954929658551372)))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

↓

double code(double x) {
	return x * fma(x, (x * -0.12900613773279798), 0.954929658551372);
}

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

↓

function code(x)
	return Float64(x * fma(x, Float64(x * -0.12900613773279798), 0.954929658551372))
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * N[(x * N[(x * -0.12900613773279798), $MachinePrecision] + 0.954929658551372), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

2. Simplified 0.2

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot -0.12900613773279798, 0.954929658551372\right)} \]
   Proof

   [Start] 0.2	\[ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   associate-*r* [=>] 0.2	\[ 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]
   distribute-rgt-out-- [=>] 0.2	\[ \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]
   cancel-sign-sub-inv [=>] 0.2	\[ x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798\right) \cdot \left(x \cdot x\right)\right)} \]
   +-commutative [<=] 0.2	\[ x \cdot \color{blue}{\left(\left(-0.12900613773279798\right) \cdot \left(x \cdot x\right) + 0.954929658551372\right)} \]
   associate-*r* [=>] 0.2	\[ x \cdot \left(\color{blue}{\left(\left(-0.12900613773279798\right) \cdot x\right) \cdot x} + 0.954929658551372\right) \]
   *-commutative [=>] 0.2	\[ x \cdot \left(\color{blue}{x \cdot \left(\left(-0.12900613773279798\right) \cdot x\right)} + 0.954929658551372\right) \]
   fma-def [=>] 0.2	\[ x \cdot \color{blue}{\mathsf{fma}\left(x, \left(-0.12900613773279798\right) \cdot x, 0.954929658551372\right)} \]
   *-commutative [=>] 0.2	\[ x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12900613773279798\right)}, 0.954929658551372\right) \]
   metadata-eval [=>] 0.2	\[ x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

3. Final simplification 0.2

   \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot -0.12900613773279798, 0.954929658551372\right) \]

Alternatives

Alternative 1
Error	1.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2.75 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \]

Alternative 2
Error	1.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.75:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)\\ \mathbf{elif}\;x \leq 2.7:\\ \;\;\;\;x \cdot 0.954929658551372\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	704

\[x \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right) + x \cdot 0.954929658551372 \]

Alternative 4
Error	0.2
Cost	576

\[x \cdot \left(0.954929658551372 + -0.12900613773279798 \cdot \left(x \cdot x\right)\right) \]

Alternative 5
Error	16.3
Cost	192

\[x \cdot 0.954929658551372 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))