Rosa's Benchmark

Average Error: 0.1 → 0.1

Time: 2.6s

Precision: 64

Math

\[0.95492965855137202 \cdot x - 0.129006137732797982 \cdot \left(\left(x \cdot x\right) \cdot x\right)\]

↓

\[\mathsf{fma}\left(x, 0.95492965855137202, -0.129006137732797982 \cdot {x}^{3}\right)\]

C

double f(double x) {
        double r16753 = 0.954929658551372;
        double r16754 = x;
        double r16755 = r16753 * r16754;
        double r16756 = 0.12900613773279798;
        double r16757 = r16754 * r16754;
        double r16758 = r16757 * r16754;
        double r16759 = r16756 * r16758;
        double r16760 = r16755 - r16759;
        return r16760;
}

↓

double f(double x) {
        double r16761 = x;
        double r16762 = 0.954929658551372;
        double r16763 = 0.12900613773279798;
        double r16764 = 3.0;
        double r16765 = pow(r16761, r16764);
        double r16766 = r16763 * r16765;
        double r16767 = -r16766;
        double r16768 = fma(r16761, r16762, r16767);
        return r16768;
}

Error

Bits error versus x

Derivation

1. Initial program 0.1

   \[0.95492965855137202 \cdot x - 0.129006137732797982 \cdot \left(\left(x \cdot x\right) \cdot x\right)\]

2. Simplified 0.1

   \[\leadsto \color{blue}{x \cdot \left(0.95492965855137202 - 0.129006137732797982 \cdot \left(x \cdot x\right)\right)}\]
3. Using strategy rm

4. Applied sub-neg 0.1

   \[\leadsto x \cdot \color{blue}{\left(0.95492965855137202 + \left(-0.129006137732797982 \cdot \left(x \cdot x\right)\right)\right)}\]

5. Applied distribute-lft-in 0.1

   \[\leadsto \color{blue}{x \cdot 0.95492965855137202 + x \cdot \left(-0.129006137732797982 \cdot \left(x \cdot x\right)\right)}\]

6. Simplified 0.1

   \[\leadsto x \cdot 0.95492965855137202 + \color{blue}{\left(-0.129006137732797982 \cdot {x}^{3}\right)}\]
7. Using strategy rm

8. Applied fma-def 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.95492965855137202, -0.129006137732797982 \cdot {x}^{3}\right)}\]

9. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(x, 0.95492965855137202, -0.129006137732797982 \cdot {x}^{3}\right)\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))