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

Average Error: 0.1 → 0.2

Time: 6.2s

Precision: binary64

Cost: 6848

Math

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

↓

\[\mathsf{fma}\left(\frac{-4}{y}, z - x, 2\right) \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

↓

(FPCore (x y z) :precision binary64 (fma (/ -4.0 y) (- z x) 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 fma((-4.0 / y), (z - x), 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 fma(Float64(-4.0 / y), Float64(z - x), 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 / y), $MachinePrecision] * N[(z - x), $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

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-4}{y}, z - x, 2\right)} \]
   Proof

   [Start] 0.1	\[ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   associate-*l/ [<=] 0.2	\[ 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]
   +-commutative [=>] 0.2	\[ 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]
   associate--l+ [=>] 0.2	\[ 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]
   +-commutative [=>] 0.2	\[ 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x - z\right) + y \cdot 0.25\right)} \]
   distribute-lft-in [=>] 0.2	\[ 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right) + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]
   *-commutative [<=] 0.2	\[ 1 + \left(\color{blue}{\left(x - z\right) \cdot \frac{4}{y}} + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   associate-+r+ [=>] 0.2	\[ \color{blue}{\left(1 + \left(x - z\right) \cdot \frac{4}{y}\right) + \frac{4}{y} \cdot \left(y \cdot 0.25\right)} \]
   +-commutative [<=] 0.2	\[ \color{blue}{\left(\left(x - z\right) \cdot \frac{4}{y} + 1\right)} + \frac{4}{y} \cdot \left(y \cdot 0.25\right) \]
   associate-+r+ [<=] 0.2	\[ \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]
   *-commutative [=>] 0.2	\[ \color{blue}{\frac{4}{y} \cdot \left(x - z\right)} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   sub-neg [=>] 0.2	\[ \frac{4}{y} \cdot \color{blue}{\left(x + \left(-z\right)\right)} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   +-commutative [=>] 0.2	\[ \frac{4}{y} \cdot \color{blue}{\left(\left(-z\right) + x\right)} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   neg-sub0 [=>] 0.2	\[ \frac{4}{y} \cdot \left(\color{blue}{\left(0 - z\right)} + x\right) + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   associate-+l- [=>] 0.2	\[ \frac{4}{y} \cdot \color{blue}{\left(0 - \left(z - x\right)\right)} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   sub0-neg [=>] 0.2	\[ \frac{4}{y} \cdot \color{blue}{\left(-\left(z - x\right)\right)} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   neg-mul-1 [=>] 0.2	\[ \frac{4}{y} \cdot \color{blue}{\left(-1 \cdot \left(z - x\right)\right)} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   associate-*r* [=>] 0.2	\[ \color{blue}{\left(\frac{4}{y} \cdot -1\right) \cdot \left(z - x\right)} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   *-commutative [<=] 0.2	\[ \color{blue}{\left(-1 \cdot \frac{4}{y}\right)} \cdot \left(z - x\right) + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   fma-def [=>] 0.2	\[ \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{4}{y}, z - x, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]
   associate-*r/ [=>] 0.2	\[ \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot 4}{y}}, z - x, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{fma}\left(\frac{\color{blue}{-4}}{y}, z - x, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) \]
   *-commutative [=>] 0.2	\[ \mathsf{fma}\left(\frac{-4}{y}, z - x, 1 + \color{blue}{\left(y \cdot 0.25\right) \cdot \frac{4}{y}}\right) \]
   associate-*l* [=>] 0.2	\[ \mathsf{fma}\left(\frac{-4}{y}, z - x, 1 + \color{blue}{y \cdot \left(0.25 \cdot \frac{4}{y}\right)}\right) \]
   associate-*r/ [=>] 0.2	\[ \mathsf{fma}\left(\frac{-4}{y}, z - x, 1 + y \cdot \color{blue}{\frac{0.25 \cdot 4}{y}}\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{fma}\left(\frac{-4}{y}, z - x, 1 + y \cdot \frac{\color{blue}{1}}{y}\right) \]
   rgt-mult-inverse [=>] 0.2	\[ \mathsf{fma}\left(\frac{-4}{y}, z - x, 1 + \color{blue}{1}\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{fma}\left(\frac{-4}{y}, z - x, \color{blue}{2}\right) \]

3. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(\frac{-4}{y}, z - x, 2\right) \]

Alternatives

Alternative 1
Error	31.6
Cost	1376

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+158}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{+91}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 2
Error	31.6
Cost	1376

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+158}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{+91}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 3
Error	16.4
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+158}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{+132} \lor \neg \left(y \leq -2.7 \cdot 10^{+91}\right) \land y \leq 1.05 \cdot 10^{+154}:\\ \;\;\;\;-4 \cdot \frac{z - x}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 4
Error	30.3
Cost	849

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+158}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+132} \lor \neg \left(y \leq -5.4 \cdot 10^{+80}\right) \land y \leq 5.5 \cdot 10^{+98}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 5
Error	0.2
Cost	832

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

Alternative 6
Error	11.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-19} \lor \neg \left(x \leq 1.8 \cdot 10^{+26}\right):\\ \;\;\;\;-4 \cdot \frac{z - x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \]

Alternative 7
Error	8.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+22} \lor \neg \left(x \leq 1.5 \cdot 10^{-14}\right):\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + -4 \cdot \frac{z}{y}\\ \end{array} \]

Alternative 8
Error	0.1
Cost	576

\[2 + \frac{4 \cdot \left(x - z\right)}{y} \]

Alternative 9
Error	36.7
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022349 
(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)))