Average Error: 0.1 → 0.0
Time: 4.9s
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 r149565 = x;
        double r149566 = y;
        double r149567 = z;
        double r149568 = r149566 + r149567;
        double r149569 = r149568 + r149567;
        double r149570 = r149569 + r149566;
        double r149571 = t;
        double r149572 = r149570 + r149571;
        double r149573 = r149565 * r149572;
        double r149574 = 5.0;
        double r149575 = r149566 * r149574;
        double r149576 = r149573 + r149575;
        return r149576;
}
↓
double f(double x, double y, double z, double t) {
        double r149577 = y;
        double r149578 = z;
        double r149579 = r149577 + r149578;
        double r149580 = x;
        double r149581 = r149580 + r149580;
        double r149582 = 5.0;
        double r149583 = t;
        double r149584 = r149583 * r149580;
        double r149585 = fma(r149582, r149577, r149584);
        double r149586 = fma(r149579, r149581, r149585);
        return r149586;
}

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. Simplified0.1

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

  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 2020034 +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)))