Average Error: 0.2 → 0.2
Time: 2.4s
Precision: binary64
\[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
↓
\[\mathsf{fma}\left(0.954929658551372, x, -0.12900613773279798 \cdot {x}^{3}\right) \]
(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))
↓
(FPCore (x)
 :precision binary64
 (fma 0.954929658551372 x (* -0.12900613773279798 (pow x 3.0))))
double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}
↓
double code(double x) {
	return fma(0.954929658551372, x, (-0.12900613773279798 * pow(x, 3.0)));
}

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end
↓
function code(x)
	return fma(0.954929658551372, x, Float64(-0.12900613773279798 * (x ^ 3.0)))
end

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(0.954929658551372 * x + N[(-0.12900613773279798 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded in x around 0 0.2

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

  4. Applied egg-rr0.2

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

  5. Taylor expanded in x around 0 0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))