Average Error: 0.2 → 0.1
Time: 2.1s
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 r665090 = 3.0;
        double r665091 = x;
        double r665092 = r665091 * r665090;
        double r665093 = r665092 * r665091;
        double r665094 = 4.0;
        double r665095 = r665091 * r665094;
        double r665096 = r665093 - r665095;
        double r665097 = 1.0;
        double r665098 = r665096 + r665097;
        double r665099 = r665090 * r665098;
        return r665099;
}
↓
double f(double x) {
        double r665100 = x;
        double r665101 = 9.0;
        double r665102 = r665101 * r665100;
        double r665103 = 12.0;
        double r665104 = r665102 - r665103;
        double r665105 = 3.0;
        double r665106 = fma(r665100, r665104, r665105);
        return r665106;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.1
\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)\]

Derivation

  1. Initial program 0.2

    \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]

  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 2020036 +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)))