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

Average Error: 0.2 → 0.0

Time: 1.1min

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

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 1.0 + (4.0 * ((x / y) - ((z / y) + -0.75)));
}

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 = 1.0d0 + (4.0d0 * ((x / y) - ((z / y) + (-0.75d0))))
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 1.0 + (4.0 * ((x / y) - ((z / y) + -0.75)));
}

Python

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

↓

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

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(1.0 + Float64(4.0 * Float64(Float64(x / y) - Float64(Float64(z / y) + -0.75))))
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 = 1.0 + (4.0 * ((x / y) - ((z / y) + -0.75)));
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[(1.0 + N[(4.0 * N[(N[(x / y), $MachinePrecision] - N[(N[(z / y), $MachinePrecision] + -0.75), $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 x around inf 0.0

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

3. Simplified 0.0

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

Alternatives

Alternative 1
Error	30.5
Cost	1112

\[\begin{array}{l} t_0 := \frac{4}{y} \cdot x\\ t_1 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-247}:\\ \;\;\;\;4\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-152}:\\ \;\;\;\;4\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+169}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	30.6
Cost	1112

\[\begin{array}{l} t_0 := \frac{4 \cdot x}{y}\\ t_1 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-246}:\\ \;\;\;\;4\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-273}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-160}:\\ \;\;\;\;4\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+169}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.7
Cost	976

\[\begin{array}{l} t_0 := 4 + \frac{z}{y} \cdot -4\\ t_1 := \frac{4}{y} \cdot \left(x - z\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+137}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+120}:\\ \;\;\;\;\frac{4}{y} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 5
Error	8.5
Cost	712

\[\begin{array}{l} t_0 := 4 + \frac{\frac{x}{y}}{0.25}\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+39}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	30.2
Cost	584

\[\begin{array}{l} t_0 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+171}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	0.0
Cost	576

\[4 + \frac{\frac{x - z}{y}}{0.25} \]

Alternative 8
Error	36.5
Cost	64

\[4 \]

Reproduce

herbie shell --seed 2023033 
(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)))