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

Average Error: 3.2 → 0.2

Time: 10.0s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ t_1 := z \cdot \left(y \cdot x - x\right)\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{+251}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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)) (t_1 (* z (- (* y x) x))))
   (if (<= t_0 -4e+159)
     t_1
     (if (<= t_0 1e+251) (* x (+ 1.0 (* z (+ y -1.0)))) t_1))))

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 t_1 = z * ((y * x) - x);
	double tmp;
	if (t_0 <= -4e+159) {
		tmp = t_1;
	} else if (t_0 <= 1e+251) {
		tmp = x * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 - y) * z
    t_1 = z * ((y * x) - x)
    if (t_0 <= (-4d+159)) then
        tmp = t_1
    else if (t_0 <= 1d+251) then
        tmp = x * (1.0d0 + (z * (y + (-1.0d0))))
    else
        tmp = t_1
    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 t_1 = z * ((y * x) - x);
	double tmp;
	if (t_0 <= -4e+159) {
		tmp = t_1;
	} else if (t_0 <= 1e+251) {
		tmp = x * (1.0 + (z * (y + -1.0)));
	} else {
		tmp = t_1;
	}
	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
	t_1 = z * ((y * x) - x)
	tmp = 0
	if t_0 <= -4e+159:
		tmp = t_1
	elif t_0 <= 1e+251:
		tmp = x * (1.0 + (z * (y + -1.0)))
	else:
		tmp = t_1
	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)
	t_1 = Float64(z * Float64(Float64(y * x) - x))
	tmp = 0.0
	if (t_0 <= -4e+159)
		tmp = t_1;
	elseif (t_0 <= 1e+251)
		tmp = Float64(x * Float64(1.0 + Float64(z * Float64(y + -1.0))));
	else
		tmp = t_1;
	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;
	t_1 = z * ((y * x) - x);
	tmp = 0.0;
	if (t_0 <= -4e+159)
		tmp = t_1;
	elseif (t_0 <= 1e+251)
		tmp = x * (1.0 + (z * (y + -1.0)));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(z * N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+159], t$95$1, If[LessEqual[t$95$0, 1e+251], N[(x * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	3.2
Target	0.3
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 2 regimes

2. if (*.f64 (-.f64 1 y) z) < -3.9999999999999997e159 or 1e251 < (*.f64 (-.f64 1 y) z)

   1. Initial program 17.8

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

   2. Taylor expanded in z around inf 0.8

      \[\leadsto \color{blue}{z \cdot \left(\left(y - 1\right) \cdot x\right)} \]

   3. Simplified 0.8

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x - x\right)} \]
      Proof
      (*.f64 z (-.f64 (*.f64 y x) x)): 0 points increase in error, 0 points decrease in error
      (*.f64 z (-.f64 (*.f64 y x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 z (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 y 1)))): 1 points increase in error, 1 points decrease in error
      (*.f64 z (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 y 1) x))): 0 points increase in error, 0 points decrease in error

   if -3.9999999999999997e159 < (*.f64 (-.f64 1 y) z) < 1e251

   1. Initial program 0.1

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -4 \cdot 10^{+159}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 10^{+251}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.6
Cost	848

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{+194}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+173}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.752150736049714 \cdot 10^{+32}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	19.3
Cost	716

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -0.0026:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.06108608436924 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.4
Cost	716

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -0.0026:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.06108608436924 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+42}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	3.7
Cost	712

\[\begin{array}{l} t_0 := x + x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -63.27571743571262:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.004301235560980504:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.1
Cost	712

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

Alternative 6
Error	19.9
Cost	520

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -0.0026:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	32.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022291 
(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))))