Average Error: 0.0 → 0.0
Time: 1.5s
Precision: 64
\[x \cdot y + \left(1 - x\right) \cdot z\]
↓
\[\mathsf{fma}\left(x, y, 1 \cdot \left(z - x \cdot z\right)\right)\]
x \cdot y + \left(1 - x\right) \cdot z
↓
\mathsf{fma}\left(x, y, 1 \cdot \left(z - x \cdot z\right)\right)
double code(double x, double y, double z) {
	return ((double) (((double) (x * y)) + ((double) (((double) (1.0 - x)) * z))));
}
↓
double code(double x, double y, double z) {
	return ((double) fma(x, y, ((double) (1.0 * ((double) (z - ((double) (x * 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 flip3--11.8

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{{1}^{3} - {x}^{3}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}} \cdot z\right)\]

  5. Applied associate-*l/13.9

    \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\frac{\left({1}^{3} - {x}^{3}\right) \cdot z}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)}}\right)\]

  6. Taylor expanded around 0 0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020123 +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)))