Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[x \cdot y + \left(x - 1\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, y, 1 \cdot \left(x \cdot z - z\right)\right)\]
x \cdot y + \left(x - 1\right) \cdot z
↓
\mathsf{fma}\left(x, y, 1 \cdot \left(x \cdot z - z\right)\right)
double f(double x, double y, double z) {
        double r129100 = x;
        double r129101 = y;
        double r129102 = r129100 * r129101;
        double r129103 = 1.0;
        double r129104 = r129100 - r129103;
        double r129105 = z;
        double r129106 = r129104 * r129105;
        double r129107 = r129102 + r129106;
        return r129107;
}
↓
double f(double x, double y, double z) {
        double r129108 = x;
        double r129109 = y;
        double r129110 = 1.0;
        double r129111 = z;
        double r129112 = r129108 * r129111;
        double r129113 = r129112 - r129111;
        double r129114 = r129110 * r129113;
        double r129115 = fma(r129108, r129109, r129114);
        return r129115;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x \cdot y + \left(x - 1\right) \cdot z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \left(x - 1\right) \cdot z\right)}\]
  3. Using strategy rm

  4. Applied flip--8.1

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}} \cdot z\right)\]

  5. Applied associate-*l/10.0

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{\left(x \cdot x - 1 \cdot 1\right) \cdot z}{x + 1}}\right)\]

  6. Taylor expanded around 0 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, y, 1 \cdot \left(x \cdot z - z\right)\right)\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1) z)))