Average Error: 0.1 → 0.1
Time: 2.4s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[\mathsf{fma}\left({x}^{2}, 9, 3 - 12 \cdot x\right)\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
\mathsf{fma}\left({x}^{2}, 9, 3 - 12 \cdot x\right)
double f(double x) {
        double r654869 = 3.0;
        double r654870 = x;
        double r654871 = r654870 * r654869;
        double r654872 = r654871 * r654870;
        double r654873 = 4.0;
        double r654874 = r654870 * r654873;
        double r654875 = r654872 - r654874;
        double r654876 = 1.0;
        double r654877 = r654875 + r654876;
        double r654878 = r654869 * r654877;
        return r654878;
}
↓
double f(double x) {
        double r654879 = x;
        double r654880 = 2.0;
        double r654881 = pow(r654879, r654880);
        double r654882 = 9.0;
        double r654883 = 3.0;
        double r654884 = 12.0;
        double r654885 = r654884 * r654879;
        double r654886 = r654883 - r654885;
        double r654887 = fma(r654881, r654882, r654886);
        return r654887;
}

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. Taylor expanded around 0 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020081 +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)))