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

Average Error: 0.2 → 0.0

Time: 11.7s

Precision: binary64

Cost: 768

Math

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

↓

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

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 (* 4.0 (+ (- (/ z y)) (/ x y)))))

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 + (4.0 * (-(z / y) + (x / 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 = 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 + (4.0d0 * (-(z / y) + (x / y)))
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 + (4.0 * (-(z / y) + (x / y)));
}

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 + (4.0 * (-(z / y) + (x / y)))

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(4.0 + Float64(4.0 * Float64(Float64(-Float64(z / y)) + Float64(x / y))))
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 + (4.0 * (-(z / y) + (x / y)));
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[(4.0 + N[(4.0 * N[((-N[(z / y), $MachinePrecision]) + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Taylor expanded in y around 0 0.0

   \[\leadsto \color{blue}{4 + 4 \cdot \frac{x - z}{y}} \]

3. Taylor expanded in x around 0 0.0

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

4. Simplified 0.0

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

   [Start] 0.0	\[ 4 + 4 \cdot \left(\frac{x}{y} + -1 \cdot \frac{z}{y}\right) \]
   rational.json-simplify-1 [=>] 0.0	\[ 4 + 4 \cdot \color{blue}{\left(-1 \cdot \frac{z}{y} + \frac{x}{y}\right)} \]
   rational.json-simplify-2 [=>] 0.0	\[ 4 + 4 \cdot \left(\color{blue}{\frac{z}{y} \cdot -1} + \frac{x}{y}\right) \]
   rational.json-simplify-9 [=>] 0.0	\[ 4 + 4 \cdot \left(\color{blue}{\left(-\frac{z}{y}\right)} + \frac{x}{y}\right) \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	30.3
Cost	1640

\[\begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ t_1 := \frac{x}{y} \cdot 4\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+34}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-69}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-74}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 2
Error	16.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+118}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+173}:\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 3
Error	11.7
Cost	712

\[\begin{array}{l} t_0 := \frac{x - z}{y} \cdot 4\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+42}:\\ \;\;\;\;4 + \frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	8.6
Cost	712

\[\begin{array}{l} t_0 := 4 + \frac{x}{y} \cdot 4\\ \mathbf{if}\;x \leq -7 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	30.6
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ \mathbf{if}\;x \leq -5 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.0
Cost	576

\[4 + 4 \cdot \frac{x - z}{y} \]

Alternative 7
Error	58.1
Cost	64

\[1 \]

Alternative 8
Error	36.5
Cost	64

\[4 \]

Reproduce

herbie shell --seed 2023077 
(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)))