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

Average Error: 0.25% → 0.01%

Time: 5.5s

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	0.25%
Target	0.01%
Herbie	0.01%

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

Derivation

1. Initial program 0.25

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

2. Simplified 0.27

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

   [Start] 0.25	\[ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   associate-/l* [=>] 0.27	\[ \color{blue}{\frac{4}{\frac{z}{\left(x - y\right) - z \cdot 0.5}}} \]

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	46.76%
Cost	1244

\[\begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ t_1 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+82}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 2
Error	47.22%
Cost	1115

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+141} \lor \neg \left(y \leq -6.6 \cdot 10^{+60} \lor \neg \left(y \leq -2.3 \cdot 10^{+22}\right) \land \left(y \leq 1.2 \cdot 10^{+55} \lor \neg \left(y \leq 8.5 \cdot 10^{+149}\right) \land y \leq 5.8 \cdot 10^{+187}\right)\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 3
Error	22.09%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+87} \lor \neg \left(x \leq 1.8 \cdot 10^{+160}\right):\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{y}{z} + 0.5\right)\\ \end{array} \]

Alternative 4
Error	13.77%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-6} \lor \neg \left(x \leq 1.42 \cdot 10^{-5}\right):\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{y}{z} + 0.5\right)\\ \end{array} \]

Alternative 5
Error	17.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+52} \lor \neg \left(z \leq 2.9 \cdot 10^{+15}\right):\\ \;\;\;\;-4 \cdot \left(\frac{y}{z} + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\frac{z}{x - y}}\\ \end{array} \]

Alternative 6
Error	57.6%
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023121 
(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))