Average Error: 0.1 → 0.1
Time: 18.0s
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 r431571 = 3.0;
        double r431572 = x;
        double r431573 = r431572 * r431571;
        double r431574 = r431573 * r431572;
        double r431575 = 4.0;
        double r431576 = r431572 * r431575;
        double r431577 = r431574 - r431576;
        double r431578 = 1.0;
        double r431579 = r431577 + r431578;
        double r431580 = r431571 * r431579;
        return r431580;
}
↓
double f(double x) {
        double r431581 = x;
        double r431582 = 9.0;
        double r431583 = 12.0;
        double r431584 = -r431583;
        double r431585 = fma(r431581, r431582, r431584);
        double r431586 = 3.0;
        double r431587 = fma(r431581, r431585, r431586);
        return r431587;
}

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

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 +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)))