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

Percentage Accurate: 96.0% → 98.8%

Time: 10.3s

Alternatives: 4

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 - y \cdot z\right) \end{array} \]

FPCore

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

C

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

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 - (y * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - y \cdot z\right) \end{array} \]

FPCore

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

C

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

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 - (y * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0 - y, x\_m \cdot z, x\_m\right)\\ t_1 := x\_m \cdot \left(1 - z \cdot y\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;0 - x\_m \cdot \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (fma (- 0.0 y) (* x_m z) x_m)) (t_1 (* x_m (- 1.0 (* z y)))))
   (*
    x_s
    (if (<= t_1 -5e+159)
      t_0
      (if (<= t_1 5e+305) (- 0.0 (* x_m (fma y z -1.0))) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = fma((0.0 - y), (x_m * z), x_m);
	double t_1 = x_m * (1.0 - (z * y));
	double tmp;
	if (t_1 <= -5e+159) {
		tmp = t_0;
	} else if (t_1 <= 5e+305) {
		tmp = 0.0 - (x_m * fma(y, z, -1.0));
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = fma(Float64(0.0 - y), Float64(x_m * z), x_m)
	t_1 = Float64(x_m * Float64(1.0 - Float64(z * y)))
	tmp = 0.0
	if (t_1 <= -5e+159)
		tmp = t_0;
	elseif (t_1 <= 5e+305)
		tmp = Float64(0.0 - Float64(x_m * fma(y, z, -1.0)));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(0.0 - y), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision] + x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -5e+159], t$95$0, If[LessEqual[t$95$1, 5e+305], N[(0.0 - N[(x$95$m * N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < -5.00000000000000003e159 or 5.00000000000000009e305 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z)))

   1. Initial program 87.8%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-*l* N/A

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

      6. *-lft-identity N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - y}, z \cdot x, x\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - y}, z \cdot x, x\right) \]

      10. *-lowering-*.f64 95.9

          \[\leadsto \mathsf{fma}\left(0 - y, \color{blue}{z \cdot x}, x\right) \]

   4. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y, z \cdot x, x\right)} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 95.9

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

   6. Applied egg-rr 95.9%

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

   if -5.00000000000000003e159 < (*.f64 x (-.f64 #s(literal 1 binary64) (*.f64 y z))) < 5.00000000000000009e305

   1. Initial program 99.9%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 99.9%

      \[\leadsto x \cdot \color{blue}{\left(-\mathsf{fma}\left(y, z, -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - z \cdot y\right) \leq -5 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, x \cdot z, x\right)\\ \mathbf{elif}\;x \cdot \left(1 - z \cdot y\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;0 - x \cdot \mathsf{fma}\left(y, z, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0 - y, x \cdot z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(0 - z, x\_m \cdot y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x\_m \cdot \mathsf{fma}\left(y, z, -1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-55)
    (fma (- 0.0 z) (* x_m y) x_m)
    (- 0.0 (* x_m (fma y z -1.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e-55) {
		tmp = fma((0.0 - z), (x_m * y), x_m);
	} else {
		tmp = 0.0 - (x_m * fma(y, z, -1.0));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5e-55)
		tmp = fma(Float64(0.0 - z), Float64(x_m * y), x_m);
	else
		tmp = Float64(0.0 - Float64(x_m * fma(y, z, -1.0)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-55], N[(N[(0.0 - z), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision] + x$95$m), $MachinePrecision], N[(0.0 - N[(x$95$m * N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-55

   1. Initial program 93.4%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - z}, y \cdot x, x\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - z}, y \cdot x, x\right) \]

      11. *-lowering-*.f64 93.6

          \[\leadsto \mathsf{fma}\left(0 - z, \color{blue}{y \cdot x}, x\right) \]

   4. Applied egg-rr 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - z, y \cdot x, x\right)} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 93.6

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

   6. Applied egg-rr 93.6%

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

   if 5.0000000000000002e-55 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - y \cdot z\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 99.9%

      \[\leadsto x \cdot \color{blue}{\left(-\mathsf{fma}\left(y, z, -1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(0 - z, x \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot \mathsf{fma}\left(y, z, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \left(0 - x\_m \cdot \mathsf{fma}\left(y, z, -1\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (- 0.0 (* x_m (fma y z -1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (0.0 - (x_m * fma(y, z, -1.0)));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(0.0 - Float64(x_m * fma(y, z, -1.0))))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(0.0 - N[(x$95$m * N[(y * z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

   \[x \cdot \left(1 - y \cdot z\right) \]
2. Add Preprocessing

4. Applied egg-rr 95.5%

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

5. Final simplification 95.5%

   \[\leadsto 0 - x \cdot \mathsf{fma}\left(y, z, -1\right) \]
6. Add Preprocessing

Alternative 4: 51.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 95.5%

   \[x \cdot \left(1 - y \cdot z\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 51.6%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))