Average Error: 0.0 → 0.0
Time: 1.6s
Precision: 64
\[x \cdot y + \left(1 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(z, 1, x \cdot \left(y - z\right)\right)\]
x \cdot y + \left(1 - x\right) \cdot z
↓
\mathsf{fma}\left(z, 1, x \cdot \left(y - z\right)\right)
double code(double x, double y, double z) {
	return ((x * y) + ((1.0 - x) * z));
}
↓
double code(double x, double y, double z) {
	return fma(z, 1.0, (x * (y - z)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[x \cdot y + \left(1 - x\right) \cdot z\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \left(1 - x\right) \cdot z\right)}\]
  3. Using strategy rm

  4. Applied flip--7.7

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 + x}} \cdot z\right)\]

  5. Applied associate-*l/9.8

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{\left(1 \cdot 1 - x \cdot x\right) \cdot z}{1 + x}}\right)\]
  6. Using strategy rm

  7. Applied *-commutative9.8

    \[\leadsto \mathsf{fma}\left(x, y, \frac{\color{blue}{z \cdot \left(1 \cdot 1 - x \cdot x\right)}}{1 + x}\right)\]

  8. Applied associate-/l*7.8

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{z}{\frac{1 + x}{1 \cdot 1 - x \cdot x}}}\right)\]

  9. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, y, \frac{z}{\color{blue}{\frac{1}{1 - x}}}\right)\]

  10. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(1 \cdot z + x \cdot y\right) - x \cdot z}\]

  11. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, 1, x \cdot \left(y - z\right)\right)}\]

  12. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(z, 1, x \cdot \left(y - z\right)\right)\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1 x) z)))