Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C

Average Error: 0.15% → 0.14%

Time: 6.1s

Precision: binary64

Cost: 832

Math

\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]

↓

\[\frac{4 \cdot x}{y} + \left(2 + \frac{z}{y} \cdot -4\right) \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

↓

(FPCore (x y z)
 :precision binary64
 (+ (/ (* 4.0 x) y) (+ 2.0 (* (/ z y) -4.0))))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

↓

double code(double x, double y, double z) {
	return ((4.0 * x) / y) + (2.0 + ((z / y) * -4.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.25d0)) - z)) / y)
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 = ((4.0d0 * x) / y) + (2.0d0 + ((z / y) * (-4.0d0)))
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

↓

public static double code(double x, double y, double z) {
	return ((4.0 * x) / y) + (2.0 + ((z / y) * -4.0));
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

↓

def code(x, y, z):
	return ((4.0 * x) / y) + (2.0 + ((z / y) * -4.0))

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

↓

function code(x, y, z)
	return Float64(Float64(Float64(4.0 * x) / y) + Float64(2.0 + Float64(Float64(z / y) * -4.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

↓

function tmp = code(x, y, z)
	tmp = ((4.0 * x) / y) + (2.0 + ((z / y) * -4.0));
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision] + N[(2.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.15

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]

2. Taylor expanded in z around inf 0.03

   \[\leadsto \color{blue}{1 + \left(-4 \cdot \frac{z}{y} + 4 \cdot \left(0.25 + \frac{x}{y}\right)\right)} \]

3. Simplified 0.14

   \[\leadsto \color{blue}{\frac{4 \cdot x}{y} + \left(\frac{z}{y} \cdot -4 + 2\right)} \]
   Proof

   [Start] 0.03	\[ 1 + \left(-4 \cdot \frac{z}{y} + 4 \cdot \left(0.25 + \frac{x}{y}\right)\right) \]
   associate-+r+ [=>] 0.03	\[ \color{blue}{\left(1 + -4 \cdot \frac{z}{y}\right) + 4 \cdot \left(0.25 + \frac{x}{y}\right)} \]
   distribute-lft-in [=>] 0.03	\[ \left(1 + -4 \cdot \frac{z}{y}\right) + \color{blue}{\left(4 \cdot 0.25 + 4 \cdot \frac{x}{y}\right)} \]
   metadata-eval [=>] 0.03	\[ \left(1 + -4 \cdot \frac{z}{y}\right) + \left(\color{blue}{1} + 4 \cdot \frac{x}{y}\right) \]
   associate-+r+ [=>] 0.02	\[ \color{blue}{\left(\left(1 + -4 \cdot \frac{z}{y}\right) + 1\right) + 4 \cdot \frac{x}{y}} \]
   +-commutative [=>] 0.02	\[ \color{blue}{4 \cdot \frac{x}{y} + \left(\left(1 + -4 \cdot \frac{z}{y}\right) + 1\right)} \]
   associate-*r/ [=>] 0.15	\[ \color{blue}{\frac{4 \cdot x}{y}} + \left(\left(1 + -4 \cdot \frac{z}{y}\right) + 1\right) \]
   +-commutative [=>] 0.15	\[ \frac{4 \cdot x}{y} + \left(\color{blue}{\left(-4 \cdot \frac{z}{y} + 1\right)} + 1\right) \]
   associate-+l+ [=>] 0.14	\[ \frac{4 \cdot x}{y} + \color{blue}{\left(-4 \cdot \frac{z}{y} + \left(1 + 1\right)\right)} \]
   *-commutative [=>] 0.14	\[ \frac{4 \cdot x}{y} + \left(\color{blue}{\frac{z}{y} \cdot -4} + \left(1 + 1\right)\right) \]
   metadata-eval [=>] 0.14	\[ \frac{4 \cdot x}{y} + \left(\frac{z}{y} \cdot -4 + \color{blue}{2}\right) \]

4. Final simplification 0.14

   \[\leadsto \frac{4 \cdot x}{y} + \left(2 + \frac{z}{y} \cdot -4\right) \]

Alternatives

Alternative 1
Error	47.8%
Cost	848

\[\begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-289}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 1.28 \cdot 10^{-284}:\\ \;\;\;\;\frac{4}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+59}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	47.75%
Cost	848

\[\begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-292}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-283}:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+62}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.27%
Cost	832

\[1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}} \]

Alternative 4
Error	17.96%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+35} \lor \neg \left(z \leq 3.9 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{-4}{y} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5
Error	13.84%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-18} \lor \neg \left(x \leq 3.2 \cdot 10^{-31}\right):\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + z \cdot \frac{-4}{y}\\ \end{array} \]

Alternative 6
Error	26.37%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+171}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{-4}{y} \cdot \left(z - x\right)\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Error	46.55%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+35} \lor \neg \left(z \leq 3.2 \cdot 10^{+59}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 8
Error	57.09%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))