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

Average Error: 0.1 → 0.0

Time: 6.9s

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

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

   [Start] 0.1	\[ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   associate-/l* [=>] 0.2	\[ \color{blue}{\frac{4}{\frac{z}{\left(x - y\right) - z \cdot 0.5}}} \]

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	31.9
Cost	1376

\[\begin{array}{l} t_0 := \frac{y \cdot -4}{z}\\ t_1 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-164}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-243}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-90}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-45}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.2
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+103} \lor \neg \left(x \leq -2.3 \cdot 10^{-7}\right) \land \left(x \leq -1.06 \cdot 10^{-36} \lor \neg \left(x \leq 2 \cdot 10^{+100}\right)\right):\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{y}{z} + 0.5\right)\\ \end{array} \]

Alternative 3
Error	12.0
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+103} \lor \neg \left(x \leq -2.4 \cdot 10^{+28} \lor \neg \left(x \leq -1 \cdot 10^{-36}\right) \land x \leq 1.06 \cdot 10^{+29}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\frac{y}{z} + 0.5\right)\\ \end{array} \]

Alternative 4
Error	11.9
Cost	976

\[\begin{array}{l} t_0 := \left(x - y\right) \cdot \frac{4}{z}\\ t_1 := -4 \cdot \left(\frac{y}{z} + 0.5\right)\\ \mathbf{if}\;y \leq -120:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-45}:\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	11.9
Cost	976

\[\begin{array}{l} t_0 := \frac{4}{\frac{z}{x - y}}\\ t_1 := -4 \cdot \left(\frac{y}{z} + 0.5\right)\\ \mathbf{if}\;y \leq -55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.16 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-45}:\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	11.8
Cost	976

\[\begin{array}{l} t_0 := \frac{x - y}{\frac{z}{4}}\\ t_1 := -4 \cdot \left(\frac{y}{z} + 0.5\right)\\ \mathbf{if}\;y \leq -23:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-45}:\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	30.3
Cost	850

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+103} \lor \neg \left(x \leq -1.85 \cdot 10^{+27} \lor \neg \left(x \leq -1.8 \cdot 10^{-40}\right) \land x \leq 2 \cdot 10^{+54}\right):\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 8
Error	36.5
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023020 
(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))