Rosa's Benchmark

Average Accuracy: 99.6% → 99.7%

Time: 8.5s

Precision: binary64

Cost: 13184

Math

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

↓

\[\mathsf{fma}\left(-0.12900613773279798, {x}^{3}, x \cdot 0.954929658551372\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma -0.12900613773279798 (pow x 3.0) (* x 0.954929658551372)))

C

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

↓

double code(double x) {
	return fma(-0.12900613773279798, pow(x, 3.0), (x * 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 fma(-0.12900613773279798, (x ^ 3.0), Float64(x * 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[(-0.12900613773279798 * N[Power[x, 3.0], $MachinePrecision] + N[(x * 0.954929658551372), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Simplified 99.7%

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

   [Start] 99.6	\[ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   associate-*r* [=>] 99.7	\[ 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]
   distribute-rgt-out-- [=>] 99.7	\[ \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]
   cancel-sign-sub-inv [=>] 99.7	\[ x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798\right) \cdot \left(x \cdot x\right)\right)} \]
   +-commutative [<=] 99.7	\[ x \cdot \color{blue}{\left(\left(-0.12900613773279798\right) \cdot \left(x \cdot x\right) + 0.954929658551372\right)} \]
   associate-*r* [=>] 99.7	\[ x \cdot \left(\color{blue}{\left(\left(-0.12900613773279798\right) \cdot x\right) \cdot x} + 0.954929658551372\right) \]
   *-commutative [=>] 99.7	\[ x \cdot \left(\color{blue}{x \cdot \left(\left(-0.12900613773279798\right) \cdot x\right)} + 0.954929658551372\right) \]
   fma-def [=>] 99.7	\[ x \cdot \color{blue}{\mathsf{fma}\left(x, \left(-0.12900613773279798\right) \cdot x, 0.954929658551372\right)} \]
   *-commutative [=>] 99.7	\[ x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12900613773279798\right)}, 0.954929658551372\right) \]
   metadata-eval [=>] 99.7	\[ x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

3. Taylor expanded in x around 0 99.7%

   \[\leadsto \color{blue}{-0.12900613773279798 \cdot {x}^{3} + 0.954929658551372 \cdot x} \]

4. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.12900613773279798, {x}^{3}, 0.954929658551372 \cdot x\right)} \]
   Proof

   [Start] 99.7	\[ -0.12900613773279798 \cdot {x}^{3} + 0.954929658551372 \cdot x \]
   fma-def [=>] 99.7	\[ \color{blue}{\mathsf{fma}\left(-0.12900613773279798, {x}^{3}, 0.954929658551372 \cdot x\right)} \]

5. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(-0.12900613773279798, {x}^{3}, x \cdot 0.954929658551372\right) \]

Alternatives

Alternative 1
Accuracy	98.2%
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
Accuracy	98.2%
Cost	712

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

Alternative 3
Accuracy	99.6%
Cost	704

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

Alternative 4
Accuracy	99.7%
Cost	576

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

Alternative 5
Accuracy	73.8%
Cost	192

\[x \cdot 0.954929658551372 \]

Reproduce

herbie shell --seed 2023140 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))