Average Error: 0.1 → 0.1
Time: 2.3s
Precision: binary64
\[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
↓
\[\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right) \]
(FPCore (x y z)
 :precision binary64
 (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))
↓
(FPCore (x y z) :precision binary64 (fma y x (* 3.0 (* z z))))
double code(double x, double y, double z) {
	return (((x * y) + (z * z)) + (z * z)) + (z * z);
}
↓
double code(double x, double y, double z) {
	return fma(y, x, (3.0 * (z * z)));
}

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * y) + Float64(z * z)) + Float64(z * z)) + Float64(z * z))
end
↓
function code(x, y, z)
	return fma(y, x, Float64(3.0 * Float64(z * z)))
end

code[x_, y_, z_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(y * x + N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z
↓
\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(3 \cdot z\right) \cdot z + y \cdot x \]

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot 3, x \cdot y\right)} \]

  3. Taylor expanded in z around 0 0.1

    \[\leadsto \color{blue}{y \cdot x + 3 \cdot {z}^{2}} \]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right)} \]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right) \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))