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

Average Error: 0.0 → 0.0

Time: 10.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma (- y z) x 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((y - z), x, 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(Float64(y - z), x, 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[(N[(y - z), $MachinePrecision] * x + z), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

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

Alternatives

Alternative 1
Error	26.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{+151}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;-z \cdot x\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-66}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-79}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	16.2
Cost	584

\[\begin{array}{l} t_0 := \left(1 - x\right) \cdot z\\ \mathbf{if}\;z \leq -7 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-119}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.4
Cost	584

\[\begin{array}{l} t_0 := \left(1 - x\right) \cdot z\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-79}:\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.9
Cost	584

\[\begin{array}{l} t_0 := \left(y - z\right) \cdot x\\ \mathbf{if}\;x \leq -1700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y \cdot x + z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	24.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-62}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-79}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

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

Alternative 7
Error	34.6
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))