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

Average Error: 0.1 → 0.0

Time: 5.8s

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

   \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]
   Proof
   (+.f64 1 (/.f64 4 (/.f64 y (+.f64 x (-.f64 (*.f64 y 1/4) z))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 4 (/.f64 y (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 x (*.f64 y 1/4)) z))))): 3 points increase in error, 0 points decrease in error
   (+.f64 1 (/.f64 4 (/.f64 y (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 (+.f64 x (*.f64 y 1/4)) z) 1))))): 0 points increase in error, 3 points decrease in error
   (+.f64 1 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 4 y) (/.f64 (-.f64 (+.f64 x (*.f64 y 1/4)) z) 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (*.f64 (/.f64 4 y) (Rewrite=> /-rgt-identity_binary64 (-.f64 (+.f64 x (*.f64 y 1/4)) z)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 4 (-.f64 (+.f64 x (*.f64 y 1/4)) z)) y))): 0 points increase in error, 0 points decrease in error

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
   (+.f64 (*.f64 4 (/.f64 (-.f64 x z) y)) 2): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 2 (*.f64 4 (/.f64 (-.f64 x z) y)))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	29.5
Cost	1504

\[\begin{array}{l} t_0 := -4 \cdot \frac{z}{y}\\ t_1 := 4 \cdot \frac{x}{y} + 1\\ t_2 := t_0 + 1\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-114}:\\ \;\;\;\;2\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-203}:\\ \;\;\;\;2\\ \mathbf{elif}\;x \leq -1.82 \cdot 10^{-261}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-62}:\\ \;\;\;\;2\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+86}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	30.3
Cost	1112

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{y} + 1\\ t_1 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;z \leq -2.75 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-225}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.0245:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+98}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	30.1
Cost	716

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+116}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-216}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+95}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 4
Error	12.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+106} \lor \neg \left(x \leq 8 \cdot 10^{+88}\right):\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \]

Alternative 5
Error	8.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -8600000000 \lor \neg \left(z \leq 6.8 \cdot 10^{+29}\right):\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6
Error	29.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+70} \lor \neg \left(z \leq 2.4 \cdot 10^{+100}\right):\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Error	36.1
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022343 
(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)))