Average Error: 0.2 → 0.1
Time: 19.0s
Precision: 64
\[3.0 \cdot \left(\left(\left(x \cdot 3.0\right) \cdot x - x \cdot 4.0\right) + 1.0\right)\]
↓
\[\mathsf{fma}\left(x, x \cdot 9.0 - 12.0, 3.0\right)\]
3.0 \cdot \left(\left(\left(x \cdot 3.0\right) \cdot x - x \cdot 4.0\right) + 1.0\right)
↓
\mathsf{fma}\left(x, x \cdot 9.0 - 12.0, 3.0\right)
double f(double x) {
        double r22159923 = 3.0;
        double r22159924 = x;
        double r22159925 = r22159924 * r22159923;
        double r22159926 = r22159925 * r22159924;
        double r22159927 = 4.0;
        double r22159928 = r22159924 * r22159927;
        double r22159929 = r22159926 - r22159928;
        double r22159930 = 1.0;
        double r22159931 = r22159929 + r22159930;
        double r22159932 = r22159923 * r22159931;
        return r22159932;
}
↓
double f(double x) {
        double r22159933 = x;
        double r22159934 = 9.0;
        double r22159935 = r22159933 * r22159934;
        double r22159936 = 12.0;
        double r22159937 = r22159935 - r22159936;
        double r22159938 = 3.0;
        double r22159939 = fma(r22159933, r22159937, r22159938);
        return r22159939;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.1
\[3.0 + \left(\left(9.0 \cdot x\right) \cdot x - 12.0 \cdot x\right)\]

Derivation

  1. Initial program 0.2

    \[3.0 \cdot \left(\left(\left(x \cdot 3.0\right) \cdot x - x \cdot 4.0\right) + 1.0\right)\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(9.0 \cdot {x}^{2} + 3.0\right) - 12.0 \cdot x}\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(9.0, x \cdot x, 3.0\right) - x \cdot 12.0}\]

  4. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(9.0 \cdot {x}^{2} + 3.0\right) - 12.0 \cdot x}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 9.0 \cdot x - 12.0, 3.0\right)}\]

  6. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, x \cdot 9.0 - 12.0, 3.0\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))