Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3

Average Error: 0.0 → 0.0

Time: 3.9s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

(FPCore (x y z) :precision binary64 (+ (* x y) (* (- 1.0 x) z)))

↓

(FPCore (x y z) :precision binary64 (fma x (- y z) z))

C

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(x, (y - z), z);
}

Julia

function code(x, y, z)
	return Float64(Float64(x * y) + Float64(Float64(1.0 - x) * z))
end

↓

function code(x, y, z)
	return fma(x, Float64(y - z), z)
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ x \cdot y + \left(1 - x\right) \cdot z \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(1 - x\right) \cdot z + x \cdot y} \]
   *-commutative [=>] 0.0	\[ \color{blue}{z \cdot \left(1 - x\right)} + x \cdot y \]
   sub-neg [=>] 0.0	\[ z \cdot \color{blue}{\left(1 + \left(-x\right)\right)} + x \cdot y \]
   distribute-rgt-in [=>] 0.0	\[ \color{blue}{\left(1 \cdot z + \left(-x\right) \cdot z\right)} + x \cdot y \]
   associate-+l+ [=>] 0.0	\[ \color{blue}{1 \cdot z + \left(\left(-x\right) \cdot z + x \cdot y\right)} \]
   *-lft-identity [=>] 0.0	\[ \color{blue}{z} + \left(\left(-x\right) \cdot z + x \cdot y\right) \]
   +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(-x\right) \cdot z + x \cdot y\right) + z} \]
   *-commutative [=>] 0.0	\[ \left(\color{blue}{z \cdot \left(-x\right)} + x \cdot y\right) + z \]
   neg-mul-1 [=>] 0.0	\[ \left(z \cdot \color{blue}{\left(-1 \cdot x\right)} + x \cdot y\right) + z \]
   associate-*r* [=>] 0.0	\[ \left(\color{blue}{\left(z \cdot -1\right) \cdot x} + x \cdot y\right) + z \]
   *-commutative [=>] 0.0	\[ \left(\left(z \cdot -1\right) \cdot x + \color{blue}{y \cdot x}\right) + z \]
   distribute-rgt-out [=>] 0.0	\[ \color{blue}{x \cdot \left(z \cdot -1 + y\right)} + z \]
   fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(x, z \cdot -1 + y, z\right)} \]
   +-commutative [=>] 0.0	\[ \mathsf{fma}\left(x, \color{blue}{y + z \cdot -1}, z\right) \]
   *-commutative [=>] 0.0	\[ \mathsf{fma}\left(x, y + \color{blue}{-1 \cdot z}, z\right) \]
   metadata-eval [<=] 0.0	\[ \mathsf{fma}\left(x, y + \color{blue}{\left(-1\right)} \cdot z, z\right) \]
   cancel-sign-sub-inv [<=] 0.0	\[ \mathsf{fma}\left(x, \color{blue}{y - 1 \cdot z}, z\right) \]
   *-lft-identity [=>] 0.0	\[ \mathsf{fma}\left(x, y - \color{blue}{z}, z\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	12.8
Cost	848

\[\begin{array}{l} t_0 := x \cdot \left(y - z\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-88}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-96}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-76}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	26.4
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+48}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-16}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Error	13.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-16} \lor \neg \left(z \leq 0.00029\right):\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - z\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -6000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 5
Error	0.0
Cost	576

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

Alternative 6
Error	24.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-16}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7
Error	0.0
Cost	448

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

Alternative 8
Error	34.7
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023041 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))