Average Error: 0.1 → 0.1
Time: 3.3s
Precision: 64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5\]
↓
\[\mathsf{fma}\left(x, \left(\mathsf{fma}\left(2, z, y\right) + y\right) + t, y \cdot 5\right)\]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5
↓
\mathsf{fma}\left(x, \left(\mathsf{fma}\left(2, z, y\right) + y\right) + t, y \cdot 5\right)
double code(double x, double y, double z, double t) {
	return ((x * ((((y + z) + z) + y) + t)) + (y * 5.0));
}
↓
double code(double x, double y, double z, double t) {
	return fma(x, ((fma(2.0, z, y) + y) + t), (y * 5.0));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied fma-def0.1

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

  4. Taylor expanded around 0 0.1

    \[\leadsto \mathsf{fma}\left(x, \left(\color{blue}{\left(2 \cdot z + y\right)} + y\right) + t, y \cdot 5\right)\]

  5. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, \left(\color{blue}{\mathsf{fma}\left(2, z, y\right)} + y\right) + t, y \cdot 5\right)\]

  6. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \left(\mathsf{fma}\left(2, z, y\right) + y\right) + t, y \cdot 5\right)\]

Reproduce

herbie shell --seed 2020103 +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)))