Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1

Average Error: 0.02% → 0.01%

Time: 3.2s

Precision: binary64

Cost: 448

Math

\[\left(\frac{x}{2} + y \cdot x\right) + z \]

↓

\[z + \left(0.5 + y\right) \cdot x \]

FPCore

(FPCore (x y z) :precision binary64 (+ (+ (/ x 2.0) (* y x)) z))

↓

(FPCore (x y z) :precision binary64 (+ z (* (+ 0.5 y) x)))

C

double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

↓

double code(double x, double y, double z) {
	return z + ((0.5 + y) * 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 / 2.0d0) + (y * 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 + ((0.5d0 + y) * x)
end function

Java

public static double code(double x, double y, double z) {
	return ((x / 2.0) + (y * x)) + z;
}

↓

public static double code(double x, double y, double z) {
	return z + ((0.5 + y) * x);
}

Python

def code(x, y, z):
	return ((x / 2.0) + (y * x)) + z

↓

def code(x, y, z):
	return z + ((0.5 + y) * x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / 2.0) + Float64(y * x)) + z)
end

↓

function code(x, y, z)
	return Float64(z + Float64(Float64(0.5 + y) * x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / 2.0) + (y * x)) + z;
end

↓

function tmp = code(x, y, z)
	tmp = z + ((0.5 + y) * x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / 2.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(z + N[(N[(0.5 + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.02

   \[\left(\frac{x}{2} + y \cdot x\right) + z \]

2. Simplified 0.02

   \[\leadsto \color{blue}{\frac{x}{2} + \left(x \cdot y + z\right)} \]
   Proof

   [Start] 0.02	\[ \left(\frac{x}{2} + y \cdot x\right) + z \]
   associate-+l+ [=>] 0.02	\[ \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]
   *-commutative [=>] 0.02	\[ \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

3. Taylor expanded in x around 0 0.01

   \[\leadsto \color{blue}{z + \left(0.5 + y\right) \cdot x} \]

4. Final simplification 0.01

   \[\leadsto z + \left(0.5 + y\right) \cdot x \]

Alternatives

Alternative 1
Error	45.4%
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+82}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{+42}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-9}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 50000000000000:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	17.27%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-7} \lor \neg \left(z \leq 6.7 \cdot 10^{+16}\right):\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \end{array} \]

Alternative 3
Error	1.29%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 0.5\right):\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot x\\ \end{array} \]

Alternative 4
Error	24.89%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+22}:\\ \;\;\;\;\left(0.5 + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 5
Error	44.11%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{-8}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 55000000000000:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6
Error	54.71%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Data.Histogram.Bin.BinF:$cfromIndex from histogram-fill-0.8.4.1"
  :precision binary64
  (+ (+ (/ x 2.0) (* y x)) z))