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

Average Error: 0.29% → 0.02%

Time: 6.3s

Precision: binary64

Cost: 832

Math

\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]

↓

\[4 \cdot \left(\frac{x}{y} + \left(0.75 - \frac{z}{y}\right)\right) + 1 \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

↓

(FPCore (x y z)
 :precision binary64
 (+ (* 4.0 (+ (/ x y) (- 0.75 (/ z y)))) 1.0))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

↓

double code(double x, double y, double z) {
	return (4.0 * ((x / y) + (0.75 - (z / y)))) + 1.0;
}

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 * 0.75d0)) - 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) + (0.75d0 - (z / y)))) + 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

↓

public static double code(double x, double y, double z) {
	return (4.0 * ((x / y) + (0.75 - (z / y)))) + 1.0;
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

↓

def code(x, y, z):
	return (4.0 * ((x / y) + (0.75 - (z / y)))) + 1.0

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

↓

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x / y) + Float64(0.75 - Float64(z / y)))) + 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

↓

function tmp = code(x, y, z)
	tmp = (4.0 * ((x / y) + (0.75 - (z / y)))) + 1.0;
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x / y), $MachinePrecision] + N[(0.75 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 0.29

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]

2. Simplified 0.47

   \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.75 - z\right)}}} \]
   Proof

   [Start] 0.29	\[ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   associate-/l* [=>] 0.47	\[ 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.75\right) - z}}} \]
   associate--l+ [=>] 0.47	\[ 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}} \]

3. Taylor expanded in x around inf 0.02

   \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \left(0.75 - \frac{z}{y}\right)\right)} \]

4. Simplified 0.02

   \[\leadsto \color{blue}{4 \cdot \left(\frac{x}{y} + \left(0.75 - \frac{z}{y}\right)\right) + 1} \]
   Proof

   [Start] 0.02	\[ 1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \left(0.75 - \frac{z}{y}\right)\right) \]
   +-commutative [=>] 0.02	\[ \color{blue}{\left(4 \cdot \frac{x}{y} + 4 \cdot \left(0.75 - \frac{z}{y}\right)\right) + 1} \]
   distribute-lft-out [=>] 0.02	\[ \color{blue}{4 \cdot \left(\frac{x}{y} + \left(0.75 - \frac{z}{y}\right)\right)} + 1 \]

5. Final simplification 0.02

   \[\leadsto 4 \cdot \left(\frac{x}{y} + \left(0.75 - \frac{z}{y}\right)\right) + 1 \]

Alternatives

Alternative 1
Error	46.52%
Cost	980

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := \frac{z}{\frac{y}{-4}}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+48}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 2
Error	14.39%
Cost	977

\[\begin{array}{l} t_0 := 4 + 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.3 \cdot 10^{+87}:\\ \;\;\;\;-4 \cdot \frac{z - x}{y}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-31} \lor \neg \left(x \leq 6.4 \cdot 10^{-10}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \]

Alternative 3
Error	18.37%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-18} \lor \neg \left(y \leq 2.7 \cdot 10^{+75}\right):\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{z - x}{y}\\ \end{array} \]

Alternative 4
Error	25.52%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+145}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+75}:\\ \;\;\;\;-4 \cdot \frac{z - x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 5
Error	47.04%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+52}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+63}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 6
Error	0.26%
Cost	576

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

Alternative 7
Error	57.9%
Cost	64

\[4 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))