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

Average Error: 0.23% → 0.01%

Time: 7.6s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (+ (* 4.0 (/ (- x y) 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 * ((x - y) / 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 * ((x - y) / 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 * ((x - y) / 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 * ((x - y) / 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(4.0 * Float64(Float64(x - y) / 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 * ((x - y) / 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[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]

Target

Original	0.23%
Target	0.01%
Herbie	0.01%

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

Derivation

1. Initial program 0.23

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

2. Simplified 0.37

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

   [Start] 0.23	\[ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   associate-*l/ [<=] 0.37	\[ \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]
   sub-neg [=>] 0.37	\[ \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]
   distribute-rgt-neg-in [=>] 0.37	\[ \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]
   metadata-eval [=>] 0.37	\[ \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

3. Taylor expanded in z around 0 0.01

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

4. Final simplification 0.01

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

Alternatives

Alternative 1
Error	31.81%
Cost	2011

\[\begin{array}{l} \mathbf{if}\;x - y \leq -2 \cdot 10^{-22} \lor \neg \left(x - y \leq 2 \cdot 10^{-123}\right) \land \left(x - y \leq 2 \cdot 10^{-106} \lor \neg \left(x - y \leq 8.2 \cdot 10^{+57} \lor \neg \left(x - y \leq 2 \cdot 10^{+74}\right) \land x - y \leq 2 \cdot 10^{+104}\right)\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 2
Error	49.26%
Cost	980

\[\begin{array}{l} t_0 := x \cdot \frac{4}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+176}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -37000000000:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-174}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 3
Error	49.16%
Cost	980

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+176}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -27000000000:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-174}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 4
Error	49.13%
Cost	980

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+176}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -20000000000:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 5
Error	19.9%
Cost	978

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+176} \lor \neg \left(z \leq -5.8 \cdot 10^{+125}\right) \land \left(z \leq -5 \cdot 10^{+23} \lor \neg \left(z \leq 7.4 \cdot 10^{+40}\right)\right):\\ \;\;\;\;-4 \cdot \frac{y}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \end{array} \]

Alternative 6
Error	13.96%
Cost	977

\[\begin{array}{l} t_0 := -4 \cdot \frac{y}{z} + -2\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.25 \cdot 10^{+125}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+31} \lor \neg \left(y \leq 1.6 \cdot 10^{+23}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 + 4 \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7
Error	46.35%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{+109} \lor \neg \left(y \leq 3.2 \cdot 10^{+64}\right):\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 8
Error	57.28%
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023090 
(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))