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

Average Error: 3.4 → 0.7

Time: 7.4s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-134}:\\ \;\;\;\;x + z \cdot \left(x \cdot y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* (- 1.0 y) z)))))
   (if (<= x -4.8e+86)
     t_0
     (if (<= x 2.2e-134) (+ x (* z (- (* x y) x))) t_0))))

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 = x * (1.0 - ((1.0 - y) * z));
	double tmp;
	if (x <= -4.8e+86) {
		tmp = t_0;
	} else if (x <= 2.2e-134) {
		tmp = x + (z * ((x * y) - x));
	} else {
		tmp = t_0;
	}
	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 = x * (1.0d0 - ((1.0d0 - y) * z))
    if (x <= (-4.8d+86)) then
        tmp = t_0
    else if (x <= 2.2d-134) then
        tmp = x + (z * ((x * y) - x))
    else
        tmp = t_0
    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 = x * (1.0 - ((1.0 - y) * z));
	double tmp;
	if (x <= -4.8e+86) {
		tmp = t_0;
	} else if (x <= 2.2e-134) {
		tmp = x + (z * ((x * y) - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = x * (1.0 - ((1.0 - y) * z))
	tmp = 0
	if x <= -4.8e+86:
		tmp = t_0
	elif x <= 2.2e-134:
		tmp = x + (z * ((x * y) - x))
	else:
		tmp = t_0
	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(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
	tmp = 0.0
	if (x <= -4.8e+86)
		tmp = t_0;
	elseif (x <= 2.2e-134)
		tmp = Float64(x + Float64(z * Float64(Float64(x * y) - x)));
	else
		tmp = t_0;
	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 = x * (1.0 - ((1.0 - y) * z));
	tmp = 0.0;
	if (x <= -4.8e+86)
		tmp = t_0;
	elseif (x <= 2.2e-134)
		tmp = x + (z * ((x * y) - x));
	else
		tmp = t_0;
	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[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e+86], t$95$0, If[LessEqual[x, 2.2e-134], N[(x + N[(z * N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	3.4
Target	0.2
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if x < -4.8000000000000001e86 or 2.2e-134 < x

   1. Initial program 1.0

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

   if -4.8000000000000001e86 < x < 2.2e-134

   1. Initial program 5.4

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

   2. Simplified 0.4

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

      [Start] 5.4	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      rational_best_oopsla_all_46_json_45_simplify-87 [=>] 5.4	\[ \color{blue}{\left(-x\right) \cdot \left(\left(1 - y\right) \cdot z - 1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-45 [=>] 5.4	\[ \left(-x\right) \cdot \color{blue}{\left(\left(1 - y\right) \cdot z + -1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-37 [=>] 5.4	\[ \color{blue}{\left(\left(1 - y\right) \cdot z\right) \cdot \left(-x\right) + \left(-x\right) \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 5.4	\[ \left(\left(1 - y\right) \cdot z\right) \cdot \left(-x\right) + \color{blue}{\left(-\left(-x\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 5.4	\[ \color{blue}{\left(-\left(-x\right)\right) + \left(\left(1 - y\right) \cdot z\right) \cdot \left(-x\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 5.4	\[ \color{blue}{\left(0 - \left(-x\right)\right)} + \left(\left(1 - y\right) \cdot z\right) \cdot \left(-x\right) \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 5.4	\[ \left(0 - \color{blue}{\left(0 - x\right)}\right) + \left(\left(1 - y\right) \cdot z\right) \cdot \left(-x\right) \]
      rational_best_oopsla_all_46_json_45_simplify-36 [=>] 5.4	\[ \color{blue}{\left(x - \left(0 - 0\right)\right)} + \left(\left(1 - y\right) \cdot z\right) \cdot \left(-x\right) \]
      metadata-eval [=>] 5.4	\[ \left(x - \color{blue}{0}\right) + \left(\left(1 - y\right) \cdot z\right) \cdot \left(-x\right) \]
      rational_best_oopsla_all_46_json_45_simplify-81 [=>] 5.4	\[ \color{blue}{x} + \left(\left(1 - y\right) \cdot z\right) \cdot \left(-x\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 5.4	\[ x + \color{blue}{\left(-x\right) \cdot \left(\left(1 - y\right) \cdot z\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 5.4	\[ x + \left(-x\right) \cdot \color{blue}{\left(z \cdot \left(1 - y\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.4	\[ x + \color{blue}{z \cdot \left(\left(-x\right) \cdot \left(1 - y\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-87 [<=] 0.4	\[ x + z \cdot \color{blue}{\left(x \cdot \left(y - 1\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 0.4	\[ x + z \cdot \color{blue}{\left(y \cdot x - x \cdot 1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 0.4	\[ x + z \cdot \left(\color{blue}{x \cdot y} - x \cdot 1\right) \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.4	\[ x + z \cdot \left(x \cdot y - \color{blue}{x}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-134}:\\ \;\;\;\;x + z \cdot \left(x \cdot y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.8
Cost	1220

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

Alternative 2
Error	13.5
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot x\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{+126}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{+162}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Error	19.8
Cost	716

\[\begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.55 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.9
Cost	716

\[\begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	13.5
Cost	712

\[\begin{array}{l} t_0 := x \cdot \left(z \cdot \left(-1 + y\right)\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-46}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	10.2
Cost	712

\[\begin{array}{l} t_0 := z \cdot \left(\left(y - 1\right) \cdot x\right)\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-46}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	2.3
Cost	712

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

Alternative 8
Error	1.0
Cost	712

\[\begin{array}{l} t_0 := z \cdot \left(\left(y - 1\right) \cdot x\right)\\ \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(z \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	19.5
Cost	520

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

Alternative 10
Error	33.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))))