Average Error: 0.1 → 0.1
Time: 15.1s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
\mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)
double f(double x) {
        double r591810 = 3.0;
        double r591811 = x;
        double r591812 = r591811 * r591810;
        double r591813 = r591812 * r591811;
        double r591814 = 4.0;
        double r591815 = r591811 * r591814;
        double r591816 = r591813 - r591815;
        double r591817 = 1.0;
        double r591818 = r591816 + r591817;
        double r591819 = r591810 * r591818;
        return r591819;
}
↓
double f(double x) {
        double r591820 = x;
        double r591821 = 9.0;
        double r591822 = 12.0;
        double r591823 = -r591822;
        double r591824 = fma(r591820, r591821, r591823);
        double r591825 = 3.0;
        double r591826 = fma(r591820, r591824, r591825);
        return r591826;
}

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

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +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)))