Average Error: 0.1 → 0.0
Time: 2.9s
Precision: 64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]
↓
\[\mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(5, y, t \cdot x\right)\right)\]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5
↓
\mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(5, y, t \cdot x\right)\right)
double f(double x, double y, double z, double t) {
        double r110542 = x;
        double r110543 = y;
        double r110544 = z;
        double r110545 = r110543 + r110544;
        double r110546 = r110545 + r110544;
        double r110547 = r110546 + r110543;
        double r110548 = t;
        double r110549 = r110547 + r110548;
        double r110550 = r110542 * r110549;
        double r110551 = 5.0;
        double r110552 = r110543 * r110551;
        double r110553 = r110550 + r110552;
        return r110553;
}
↓
double f(double x, double y, double z, double t) {
        double r110554 = y;
        double r110555 = z;
        double r110556 = r110554 + r110555;
        double r110557 = x;
        double r110558 = r110557 + r110557;
        double r110559 = 5.0;
        double r110560 = t;
        double r110561 = r110560 * r110557;
        double r110562 = fma(r110559, r110554, r110561);
        double r110563 = fma(r110556, r110558, r110562);
        return r110563;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(t, x, y \cdot 5\right)\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \mathsf{fma}\left(y + z, x + x, \color{blue}{5 \cdot y + t \cdot x}\right)\]

  4. Simplified0.0

    \[\leadsto \mathsf{fma}\left(y + z, x + x, \color{blue}{\mathsf{fma}\left(5, y, t \cdot x\right)}\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y + z, x + x, \mathsf{fma}\left(5, y, t \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5)))