Average Error: 0.1 → 0.1
Time: 15.3s
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 r460115 = 3.0;
        double r460116 = x;
        double r460117 = r460116 * r460115;
        double r460118 = r460117 * r460116;
        double r460119 = 4.0;
        double r460120 = r460116 * r460119;
        double r460121 = r460118 - r460120;
        double r460122 = 1.0;
        double r460123 = r460121 + r460122;
        double r460124 = r460115 * r460123;
        return r460124;
}
↓
double f(double x) {
        double r460125 = x;
        double r460126 = 9.0;
        double r460127 = 12.0;
        double r460128 = -r460127;
        double r460129 = fma(r460125, r460126, r460128);
        double r460130 = 3.0;
        double r460131 = fma(r460125, r460129, r460130);
        return r460131;
}

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. Using strategy rm

  6. Applied *-un-lft-identity0.1

    \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 9, -12\right), 3\right)}\]

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 +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)))