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

Percentage Accurate: 98.0% → 100.0%

Time: 3.5s

Precision: binary64

Cost: 448

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \\ z + \left(y - z\right) \cdot x \end{array} \]

FPCore

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

↓

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

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 z + ((y - z) * x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + ((1.0d0 - x) * z)
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z + ((y - z) * x)
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) + ((1.0 - x) * z);
}

↓

public static double code(double x, double y, double z) {
	return z + ((y - z) * x);
}

Python

def code(x, y, z):
	return (x * y) + ((1.0 - x) * z)

↓

def code(x, y, z):
	return z + ((y - z) * x)

Julia

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

↓

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

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) + ((1.0 - x) * z);
end

↓

function tmp = code(x, y, z)
	tmp = z + ((y - z) * x);
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[(z + N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

2. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y - z, z\right)} \]
   Step-by-step derivation

   [Start] 98.4%	\[ x \cdot y + \left(1 - x\right) \cdot z \]
   sub-neg [=>] 98.4%	\[ x \cdot y + \color{blue}{\left(1 + \left(-x\right)\right)} \cdot z \]
   +-commutative [=>] 98.4%	\[ x \cdot y + \color{blue}{\left(\left(-x\right) + 1\right)} \cdot z \]
   distribute-lft1-in [<=] 98.4%	\[ x \cdot y + \color{blue}{\left(\left(-x\right) \cdot z + z\right)} \]
   associate-+r+ [=>] 98.4%	\[ \color{blue}{\left(x \cdot y + \left(-x\right) \cdot z\right) + z} \]
   +-commutative [<=] 98.4%	\[ \color{blue}{\left(\left(-x\right) \cdot z + x \cdot y\right)} + z \]
   *-commutative [=>] 98.4%	\[ \left(\color{blue}{z \cdot \left(-x\right)} + x \cdot y\right) + z \]
   neg-mul-1 [=>] 98.4%	\[ \left(z \cdot \color{blue}{\left(-1 \cdot x\right)} + x \cdot y\right) + z \]
   associate-*r* [=>] 98.4%	\[ \left(\color{blue}{\left(z \cdot -1\right) \cdot x} + x \cdot y\right) + z \]
   *-commutative [=>] 98.4%	\[ \left(\left(z \cdot -1\right) \cdot x + \color{blue}{y \cdot x}\right) + z \]
   distribute-rgt-out [=>] 100.0%	\[ \color{blue}{x \cdot \left(z \cdot -1 + y\right)} + z \]
   fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(x, z \cdot -1 + y, z\right)} \]
   +-commutative [=>] 100.0%	\[ \mathsf{fma}\left(x, \color{blue}{y + z \cdot -1}, z\right) \]
   *-commutative [=>] 100.0%	\[ \mathsf{fma}\left(x, y + \color{blue}{-1 \cdot z}, z\right) \]
   neg-mul-1 [<=] 100.0%	\[ \mathsf{fma}\left(x, y + \color{blue}{\left(-z\right)}, z\right) \]
   unsub-neg [=>] 100.0%	\[ \mathsf{fma}\left(x, \color{blue}{y - z}, z\right) \]

3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	448

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

Alternative 2
Accuracy	61.9%
Cost	784

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00115:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+36}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+168}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Accuracy	85.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0041 \lor \neg \left(x \leq 8.8 \cdot 10^{-23}\right):\\ \;\;\;\;\left(y - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 4
Accuracy	99.0%
Cost	585

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

Alternative 5
Accuracy	75.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+94}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 9500000:\\ \;\;\;\;z \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Accuracy	62.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00115:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-17}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Accuracy	36.4%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:$crender from diagrams-rasterific-1.3.1.3"
  :precision binary64
  (+ (* x y) (* (- 1.0 x) z)))