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

Average Error: 0.01% → 0.02%

Time: 3.4s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

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 + (x * (y + 0.5));
}

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 + (x * (y + 0.5d0))
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 + (x * (y + 0.5));
}

Python

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

↓

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

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(x * Float64(y + 0.5)))
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = z + (x * (y + 0.5));
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[(x * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.01

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

2. Simplified 0.01

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

   [Start] 0.01	\[ \left(\frac{x}{2} + y \cdot x\right) + z \]
   associate-+l+ [=>] 0.01	\[ \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]
   *-commutative [=>] 0.01	\[ \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

3. Taylor expanded in x around -inf 0.02

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot y - 0.5\right)\right) + z} \]

4. Simplified 0.02

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

   [Start] 0.02	\[ -1 \cdot \left(x \cdot \left(-1 \cdot y - 0.5\right)\right) + z \]
   +-commutative [=>] 0.02	\[ \color{blue}{z + -1 \cdot \left(x \cdot \left(-1 \cdot y - 0.5\right)\right)} \]
   mul-1-neg [=>] 0.02	\[ z + \color{blue}{\left(-x \cdot \left(-1 \cdot y - 0.5\right)\right)} \]
   unsub-neg [=>] 0.02	\[ \color{blue}{z - x \cdot \left(-1 \cdot y - 0.5\right)} \]
   sub-neg [=>] 0.02	\[ z - x \cdot \color{blue}{\left(-1 \cdot y + \left(-0.5\right)\right)} \]
   metadata-eval [=>] 0.02	\[ z - x \cdot \left(-1 \cdot y + \color{blue}{-0.5}\right) \]
   +-commutative [=>] 0.02	\[ z - x \cdot \color{blue}{\left(-0.5 + -1 \cdot y\right)} \]
   mul-1-neg [=>] 0.02	\[ z - x \cdot \left(-0.5 + \color{blue}{\left(-y\right)}\right) \]
   neg-sub0 [=>] 0.02	\[ z - x \cdot \left(-0.5 + \color{blue}{\left(0 - y\right)}\right) \]
   associate-+r- [=>] 0.02	\[ z - x \cdot \color{blue}{\left(\left(-0.5 + 0\right) - y\right)} \]
   metadata-eval [=>] 0.02	\[ z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

5. Final simplification 0.02

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

Alternatives

Alternative 1
Error	45.36%
Cost	1380

\[\begin{array}{l} \mathbf{if}\;z \leq -5500000000000:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-45}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-78}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-211}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-264}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{-257}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-36}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 2.12 \cdot 10^{+43}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+73}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	18.86%
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -14500000000000 \lor \neg \left(z \leq -8 \cdot 10^{-43} \lor \neg \left(z \leq -2.4 \cdot 10^{-78}\right) \land z \leq 6.4 \cdot 10^{+88}\right):\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]

Alternative 3
Error	42.88%
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -13500000000000:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-112}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-55}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4
Error	1.29%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 1.4 \cdot 10^{-6}\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \]

Alternative 5
Error	25.51%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+103}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+120}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6
Error	53.63%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023115 
(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))