Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J

Average Error: 3.5 → 0.2

Time: 6.7s

Precision: binary64

Cost: 1352

Math

\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+305}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 10^{+302}:\\ \;\;\;\;x \cdot \left(1 - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 y) z)))
   (if (<= t_0 -2e+305)
     (* z (* y x))
     (if (<= t_0 1e+302) (* x (- 1.0 t_0)) (* y (* z x))))))

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}

↓

double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -2e+305) {
		tmp = z * (y * x);
	} else if (t_0 <= 1e+302) {
		tmp = x * (1.0 - t_0);
	} else {
		tmp = y * (z * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (1.0d0 - ((1.0d0 - y) * 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - y) * z
    if (t_0 <= (-2d+305)) then
        tmp = z * (y * x)
    else if (t_0 <= 1d+302) then
        tmp = x * (1.0d0 - t_0)
    else
        tmp = y * (z * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -2e+305) {
		tmp = z * (y * x);
	} else if (t_0 <= 1e+302) {
		tmp = x * (1.0 - t_0);
	} else {
		tmp = y * (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	return x * (1.0 - ((1.0 - y) * z))

↓

def code(x, y, z):
	t_0 = (1.0 - y) * z
	tmp = 0
	if t_0 <= -2e+305:
		tmp = z * (y * x)
	elif t_0 <= 1e+302:
		tmp = x * (1.0 - t_0)
	else:
		tmp = y * (z * x)
	return tmp

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - y) * z)
	tmp = 0.0
	if (t_0 <= -2e+305)
		tmp = Float64(z * Float64(y * x));
	elseif (t_0 <= 1e+302)
		tmp = Float64(x * Float64(1.0 - t_0));
	else
		tmp = Float64(y * Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - ((1.0 - y) * z));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - y) * z;
	tmp = 0.0;
	if (t_0 <= -2e+305)
		tmp = z * (y * x);
	elseif (t_0 <= 1e+302)
		tmp = x * (1.0 - t_0);
	else
		tmp = y * (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+305], N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+302], N[(x * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.5
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 1 y) z) < -1.9999999999999999e305

   1. Initial program 61.0

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around inf 1.2

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} \]

   3. Simplified 1.2

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]
      Proof
      (*.f64 z (*.f64 y x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z y) x)): 0 points increase in error, 4 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 y z)) x): 1 points increase in error, 3 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 z x))): 4 points increase in error, 0 points decrease in error

   if -1.9999999999999999e305 < (*.f64 (-.f64 1 y) z) < 1.0000000000000001e302

   1. Initial program 0.1

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   if 1.0000000000000001e302 < (*.f64 (-.f64 1 y) z)

   1. Initial program 57.0

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around inf 3.1

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -2 \cdot 10^{+305}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 10^{+302}:\\ \;\;\;\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.9
Cost	716

\[\begin{array}{l} t_0 := -z \cdot x\\ \mathbf{if}\;z \leq -6500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6500:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	20.2
Cost	716

\[\begin{array}{l} t_0 := -z \cdot x\\ \mathbf{if}\;z \leq -6500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7500:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6500000:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 4
Error	11.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+150}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternative 5
Error	11.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+149}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternative 6
Error	19.4
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -6500000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;-z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	33.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))