Average Error: 0.1 → 0.1
Time: 8.1s
Precision: binary64
\[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
↓
\[\mathsf{fma}\left(5, y, \mathsf{fma}\left(2 \cdot \left(y + z\right), x, x \cdot t\right)\right) \]
(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 z)) x (* x t))))
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 + z)), x, (x * t)));
}

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end
↓
function code(x, y, z, t)
	return fma(5.0, y, fma(Float64(2.0 * Float64(y + z)), x, Float64(x * t)))
end

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(5.0 * y + N[(N[(2.0 * N[(y + z), $MachinePrecision]), $MachinePrecision] * x + N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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 \cdot \left(y + z\right), x, x \cdot t\right)\right)

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. Simplified0.1

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

  5. Applied fma-udef_binary640.1

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

  6. Applied distribute-rgt-in_binary640.1

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

  7. Applied fma-def_binary640.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2022129 
(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)))