Average Error: 0.1 → 0.1
Time: 2.4s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[\mathsf{fma}\left(x, 9 \cdot x - 12, 3\right)\]
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 r534645 = 3.0;
        double r534646 = x;
        double r534647 = r534646 * r534645;
        double r534648 = r534647 * r534646;
        double r534649 = 4.0;
        double r534650 = r534646 * r534649;
        double r534651 = r534648 - r534650;
        double r534652 = 1.0;
        double r534653 = r534651 + r534652;
        double r534654 = r534645 * r534653;
        return r534654;
}
↓
double f(double x) {
        double r534655 = x;
        double r534656 = 9.0;
        double r534657 = r534656 * r534655;
        double r534658 = 12.0;
        double r534659 = r534657 - r534658;
        double r534660 = 3.0;
        double r534661 = fma(r534655, r534659, r534660);
        return r534661;
}

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

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

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

  5. Taylor expanded around 0 0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020034 +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)))