Average Error: 0.1 → 0.1
Time: 2.6s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[\mathsf{fma}\left(x, 9 \cdot x - 12, 3\right)\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
\mathsf{fma}\left(x, 9 \cdot x - 12, 3\right)
double f(double x) {
        double r786062 = 3.0;
        double r786063 = x;
        double r786064 = r786063 * r786062;
        double r786065 = r786064 * r786063;
        double r786066 = 4.0;
        double r786067 = r786063 * r786066;
        double r786068 = r786065 - r786067;
        double r786069 = 1.0;
        double r786070 = r786068 + r786069;
        double r786071 = r786062 * r786070;
        return r786071;
}
↓
double f(double x) {
        double r786072 = x;
        double r786073 = 9.0;
        double r786074 = r786073 * r786072;
        double r786075 = 12.0;
        double r786076 = r786074 - r786075;
        double r786077 = 3.0;
        double r786078 = fma(r786072, r786076, r786077);
        return r786078;
}

Error

Bits error versus x

Target

Original0.1
Target0.1
Herbie0.1
\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)\]

Derivation

  1. Initial program 0.1

    \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 3 - 4, 1\right) \cdot 3}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(9 \cdot {x}^{2} + 3\right) - 12 \cdot x}\]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 9, 3 - 12 \cdot x\right)}\]

  5. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(9 \cdot {x}^{2} + 3\right) - 12 \cdot x}\]

  6. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 9 \cdot x - 12, 3\right)}\]

  7. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 9 \cdot x - 12, 3\right)\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3 (- (* (* 9 x) x) (* 12 x)))

  (* 3 (+ (- (* (* x 3) x) (* x 4)) 1)))