Rosa's Benchmark

Average Error: 0.1 → 0.1

Time: 1.9s

Precision: binary64

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);
}

Error

Bits error versus x

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

3. Taylor expanded in x around 0 0.1

   \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 - 0.12900613773279798 \cdot {x}^{2}\right)} \]

4. Simplified 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2021220 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))