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

Average Error: 0.0 → 0.0

Time: 3.2s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

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 * (-0.5 - y));
}

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 * ((-0.5d0) - y))
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 * (-0.5 - y));
}

Python

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

↓

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

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(-0.5 - y)))
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 * (-0.5 - y));
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[(-0.5 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ \left(\frac{x}{2} + y \cdot x\right) + z \]
   associate-+l+ [=>] 0.0	\[ \color{blue}{\frac{x}{2} + \left(y \cdot x + z\right)} \]
   *-commutative [=>] 0.0	\[ \frac{x}{2} + \left(\color{blue}{x \cdot y} + z\right) \]

3. Taylor expanded in x around -inf 0.0

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

4. Simplified 0.0

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

   [Start] 0.0	\[ -1 \cdot \left(x \cdot \left(-1 \cdot y - 0.5\right)\right) + z \]
   +-commutative [=>] 0.0	\[ \color{blue}{z + -1 \cdot \left(x \cdot \left(-1 \cdot y - 0.5\right)\right)} \]
   mul-1-neg [=>] 0.0	\[ z + \color{blue}{\left(-x \cdot \left(-1 \cdot y - 0.5\right)\right)} \]
   unsub-neg [=>] 0.0	\[ \color{blue}{z - x \cdot \left(-1 \cdot y - 0.5\right)} \]
   sub-neg [=>] 0.0	\[ z - x \cdot \color{blue}{\left(-1 \cdot y + \left(-0.5\right)\right)} \]
   metadata-eval [=>] 0.0	\[ z - x \cdot \left(-1 \cdot y + \color{blue}{-0.5}\right) \]
   +-commutative [=>] 0.0	\[ z - x \cdot \color{blue}{\left(-0.5 + -1 \cdot y\right)} \]
   mul-1-neg [=>] 0.0	\[ z - x \cdot \left(-0.5 + \color{blue}{\left(-y\right)}\right) \]
   neg-sub0 [=>] 0.0	\[ z - x \cdot \left(-0.5 + \color{blue}{\left(0 - y\right)}\right) \]
   associate-+r- [=>] 0.0	\[ z - x \cdot \color{blue}{\left(\left(-0.5 + 0\right) - y\right)} \]
   metadata-eval [=>] 0.0	\[ z - x \cdot \left(\color{blue}{-0.5} - y\right) \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	27.6
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -2.02 \cdot 10^{-25}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-172}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-135}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+38}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 2
Error	27.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-263}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-114}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+38}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3
Error	10.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-25} \lor \neg \left(z \leq 540000\right):\\ \;\;\;\;z + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \end{array} \]

Alternative 4
Error	0.8
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 0.0072\right):\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot 0.5\\ \end{array} \]

Alternative 5
Error	14.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+57}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 6
Error	34.3
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023060 
(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))