Average Error: 0.2 → 0.2
Time: 2.4s
Precision: binary64
\[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
↓
\[\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right) \]
0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)
↓
\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))
↓
(FPCore (x)
 :precision binary64
 (fma 0.954929658551372 x (* (pow x 3.0) -0.12900613773279798)))
double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}
↓
double code(double x) {
	return fma(0.954929658551372, x, (pow(x, 3.0) * -0.12900613773279798));
}

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

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

  3. Taylor expanded in x around 0 0.2

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

  4. Applied cube-mult_binary640.2

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

  5. Applied associate-*r*_binary640.2

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

  6. Applied fma-neg_binary640.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2021275 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))