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

Average Error: 0.1 → 0.0

Time: 14.5s

Precision: binary64

Cost: 576

Math

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

↓

\[4 \cdot \frac{x - z}{y} + 2 \]

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 z) y)) 2.0))

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 - z) / y)) + 2.0;
}

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 - z) / y)) + 2.0d0
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 - z) / y)) + 2.0;
}

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 - z) / y)) + 2.0

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(4.0 * Float64(Float64(x - z) / y)) + 2.0)
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 - z) / y)) + 2.0;
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[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.2

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

   [Start] 0.1	\[ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   rational.json-simplify-49 [=>] 0.2	\[ 1 + \color{blue}{\left(\left(x + y \cdot 0.25\right) - z\right) \cdot \frac{4}{y}} \]
   rational.json-simplify-48 [=>] 0.2	\[ 1 + \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \cdot \frac{4}{y} \]

3. Taylor expanded in y around 0 0.0

   \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]

4. Simplified 0.0

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

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

5. Final simplification 0.0

   \[\leadsto 4 \cdot \frac{x - z}{y} + 2 \]

Alternatives

Alternative 1
Error	31.7
Cost	1376

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ t_1 := \frac{z}{y} \cdot -4\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{+24}:\\ \;\;\;\;2\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-187}:\\ \;\;\;\;2\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.04 \cdot 10^{-22}:\\ \;\;\;\;2\\ \mathbf{elif}\;x \leq 29000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+58}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.8
Cost	976

\[\begin{array}{l} t_0 := 4 \cdot \frac{x - z}{y}\\ t_1 := 2 + \frac{z}{y} \cdot -4\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -260000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	32.3
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ \mathbf{if}\;y \leq -1.08 \cdot 10^{+107}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -4200000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-109}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 4
Error	16.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+108}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+89}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 5
Error	8.4
Cost	712

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{y} + 2\\ \mathbf{if}\;x \leq -6 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+39}:\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	36.2
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023074 
(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)))