Average Error: 0.1 → 0.1
Time: 2.0s
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 r647604 = 3.0;
        double r647605 = x;
        double r647606 = r647605 * r647604;
        double r647607 = r647606 * r647605;
        double r647608 = 4.0;
        double r647609 = r647605 * r647608;
        double r647610 = r647607 - r647609;
        double r647611 = 1.0;
        double r647612 = r647610 + r647611;
        double r647613 = r647604 * r647612;
        return r647613;
}
↓
double f(double x) {
        double r647614 = x;
        double r647615 = 9.0;
        double r647616 = r647615 * r647614;
        double r647617 = 12.0;
        double r647618 = r647616 - r647617;
        double r647619 = 3.0;
        double r647620 = fma(r647614, r647618, r647619);
        return r647620;
}

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. Taylor expanded around 0 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020064 +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)))