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

Average Error: 0.18% → 0.01%

Time: 7.8s

Precision: binary64

Cost: 832

Math

\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]

↓

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

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

↓

(FPCore (x y z)
 :precision binary64
 (+ (+ (* -4.0 (/ y z)) (* 4.0 (/ x z))) -2.0))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

↓

double code(double x, double y, double z) {
	return ((-4.0 * (y / z)) + (4.0 * (x / z))) + -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 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
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) * (y / z)) + (4.0d0 * (x / z))) + (-2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

↓

public static double code(double x, double y, double z) {
	return ((-4.0 * (y / z)) + (4.0 * (x / z))) + -2.0;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

↓

def code(x, y, z):
	return ((-4.0 * (y / z)) + (4.0 * (x / z))) + -2.0

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

↓

function code(x, y, z)
	return Float64(Float64(Float64(-4.0 * Float64(y / z)) + Float64(4.0 * Float64(x / z))) + -2.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

↓

function tmp = code(x, y, z)
	tmp = ((-4.0 * (y / z)) + (4.0 * (x / z))) + -2.0;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]

Target

Original	0.18%
Target	0.01%
Herbie	0.01%

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

Derivation

1. Initial program 0.18

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]

2. Simplified 0.01

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

   [Start] 0.18	\[ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   associate-*l/ [<=] 0.4	\[ \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]
   distribute-rgt-out-- [<=] 0.39	\[ \color{blue}{\left(x - y\right) \cdot \frac{4}{z} - \left(z \cdot 0.5\right) \cdot \frac{4}{z}} \]
   cancel-sign-sub-inv [=>] 0.39	\[ \color{blue}{\left(x - y\right) \cdot \frac{4}{z} + \left(-z \cdot 0.5\right) \cdot \frac{4}{z}} \]
   associate-*r/ [=>] 0.28	\[ \color{blue}{\frac{\left(x - y\right) \cdot 4}{z}} + \left(-z \cdot 0.5\right) \cdot \frac{4}{z} \]
   associate-*l/ [<=] 0.15	\[ \color{blue}{\frac{x - y}{z} \cdot 4} + \left(-z \cdot 0.5\right) \cdot \frac{4}{z} \]
   *-commutative [=>] 0.15	\[ \color{blue}{4 \cdot \frac{x - y}{z}} + \left(-z \cdot 0.5\right) \cdot \frac{4}{z} \]
   fma-def [=>] 0.15	\[ \color{blue}{\mathsf{fma}\left(4, \frac{x - y}{z}, \left(-z \cdot 0.5\right) \cdot \frac{4}{z}\right)} \]
   associate-*r/ [=>] 0.04	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \color{blue}{\frac{\left(-z \cdot 0.5\right) \cdot 4}{z}}\right) \]
   *-commutative [<=] 0.04	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{\color{blue}{4 \cdot \left(-z \cdot 0.5\right)}}{z}\right) \]
   *-commutative [=>] 0.04	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{4 \cdot \left(-\color{blue}{0.5 \cdot z}\right)}{z}\right) \]
   distribute-lft-neg-in [=>] 0.04	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{4 \cdot \color{blue}{\left(\left(-0.5\right) \cdot z\right)}}{z}\right) \]
   associate-*r* [=>] 0.04	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{\color{blue}{\left(4 \cdot \left(-0.5\right)\right) \cdot z}}{z}\right) \]
   associate-/l* [=>] 0.01	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \color{blue}{\frac{4 \cdot \left(-0.5\right)}{\frac{z}{z}}}\right) \]
   metadata-eval [=>] 0.01	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{4 \cdot \color{blue}{-0.5}}{\frac{z}{z}}\right) \]
   metadata-eval [=>] 0.01	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{\color{blue}{-2}}{\frac{z}{z}}\right) \]
   *-inverses [=>] 0.01	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \frac{-2}{\color{blue}{1}}\right) \]
   metadata-eval [=>] 0.01	\[ \mathsf{fma}\left(4, \frac{x - y}{z}, \color{blue}{-2}\right) \]

3. Taylor expanded in x around 0 0.01

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

4. Final simplification 0.01

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

Alternatives

Alternative 1
Error	49.62%
Cost	1115

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+111} \lor \neg \left(y \leq -5 \cdot 10^{+56} \lor \neg \left(y \leq -1.85 \cdot 10^{-34}\right) \land \left(y \leq 1.75 \cdot 10^{+17} \lor \neg \left(y \leq 5.6 \cdot 10^{+110}\right) \land y \leq 1.75 \cdot 10^{+193}\right)\right):\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 2
Error	49.06%
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{-4}{z}\\ t_1 := x \cdot \frac{4}{z}\\ \mathbf{if}\;z \leq -5.1 \cdot 10^{+70}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+96}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 3
Error	48.95%
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \frac{-4}{z}\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+70}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-117}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+96}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 4
Error	48.93%
Cost	1112

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-120}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+96}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+110}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 5
Error	14.71%
Cost	1108

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{z} + -2\\ t_1 := -4 \cdot \frac{y}{z} + -2\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+61}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	27.48%
Cost	977

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+80}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-77} \lor \neg \left(z \leq 3.2 \cdot 10^{-37}\right) \land z \leq 2.35 \cdot 10^{+114}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 7
Error	20.28%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+71} \lor \neg \left(z \leq 8 \cdot 10^{-82}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \end{array} \]

Alternative 8
Error	0.4%
Cost	704

\[\left(\left(x - y\right) + z \cdot -0.5\right) \cdot \frac{4}{z} \]

Alternative 9
Error	0.27%
Cost	704

\[\frac{4}{\frac{z}{\left(x - y\right) + z \cdot -0.5}} \]

Alternative 10
Error	0.18%
Cost	704

\[\frac{4 \cdot \left(\left(x - y\right) + z \cdot -0.5\right)}{z} \]

Alternative 11
Error	57.69%
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))