Average Error: 0.1 → 0.0
Time: 4.1s
Precision: binary64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
↓
\[\mathsf{fma}\left(5, y, \mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(t, x, 2 \cdot \left(x \cdot z\right)\right)\right)\right) \]
x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5
↓
\mathsf{fma}\left(5, y, \mathsf{fma}\left(2, y \cdot x, \mathsf{fma}\left(t, x, 2 \cdot \left(x \cdot z\right)\right)\right)\right)

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))
↓
(FPCore (x y z t)
 :precision binary64
 (fma 5.0 y (fma 2.0 (* y x) (fma t x (* 2.0 (* x z))))))
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(5.0, y, fma(2.0, (y * x), fma(t, x, (2.0 * (x * z)))));
}

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

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

  3. Taylor expanded in x around 0 0.1

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

  4. Simplified0.0

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

  5. Taylor expanded in y around 0 0.0

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

  6. Applied fma-def_binary640.0

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

  7. Applied fma-def_binary640.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2022081 
(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 5.0)))