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

Average Error: 0.2 → 0.0

Time: 6.1s

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 0.2

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

2. Taylor expanded in x around inf 0.0

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

3. Simplified 0.0

   \[\leadsto \color{blue}{4 \cdot \left(\frac{x}{y} + \left(0.75 - \frac{z}{y}\right)\right) + 1} \]
   Proof
   (+.f64 (*.f64 4 (+.f64 (/.f64 x y) (-.f64 3/4 (/.f64 z y)))) 1): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 4 (/.f64 x y)) (*.f64 4 (-.f64 3/4 (/.f64 z y))))) 1): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 1 (+.f64 (*.f64 4 (/.f64 x y)) (*.f64 4 (-.f64 3/4 (/.f64 z y)))))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	30.5
Cost	1772

\[\begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ t_1 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+102}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-10}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-114}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-264}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 2
Error	16.5
Cost	976

\[\begin{array}{l} t_0 := 4 \cdot \frac{x - z}{y}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+129}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq -380000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.00013:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 3
Error	11.9
Cost	976

\[\begin{array}{l} t_0 := 4 + \frac{z}{y} \cdot -4\\ t_1 := 4 \cdot \frac{x - z}{y}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.3
Cost	832

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

Alternative 5
Error	0.2
Cost	832

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

Alternative 6
Error	10.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+58}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-37}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \]

Alternative 7
Error	10.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+58}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x - z}}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-37}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \]

Alternative 8
Error	30.5
Cost	584

\[\begin{array}{l} t_0 := \frac{z}{y} \cdot -4\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+163}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	36.9
Cost	64

\[4 \]

Reproduce

herbie shell --seed 2022338 
(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)))