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

Average Error: 0.1 → 0.0

Time: 11.1s

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	0.1
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.2

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

   [Start] 0.1	\[ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   rational.json-simplify-49 [=>] 0.2	\[ \color{blue}{\left(\left(x - y\right) - z \cdot 0.5\right) \cdot \frac{4}{z}} \]

3. Taylor expanded in z around 0 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	12.6
Cost	1504

\[\begin{array}{l} t_0 := -4 \cdot \frac{y}{z} - 2\\ t_1 := 4 \cdot \frac{x}{z} - 2\\ t_2 := 4 \cdot \frac{x - y}{z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+190}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	30.4
Cost	1244

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ t_1 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1.38 \cdot 10^{+29}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 3
Error	17.1
Cost	976

\[\begin{array}{l} t_0 := 4 \cdot \frac{x - y}{z}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+136}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+35}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 4
Error	30.7
Cost	848

\[\begin{array}{l} t_0 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+29}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-38}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 5
Error	12.4
Cost	712

\[\begin{array}{l} t_0 := -4 \cdot \frac{y}{z} - 2\\ \mathbf{if}\;z \leq -6200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-46}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	36.4
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023063 
(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))