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

Percentage Accurate: 96.1% → 99.8%

Time: 8.0s

Alternatives: 5

Speedup: 0.3×

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: 96.1% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+294} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+230}\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (if (or (<= t_0 -2e+294) (not (<= t_0 5e+230)))
     (fma (* x z) (- y) x)
     (* x t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -2e+294) || !(t_0 <= 5e+230)) {
		tmp = fma((x * z), -y, x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -2e+294) || !(t_0 <= 5e+230))
		tmp = fma(Float64(x * z), Float64(-y), x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+294], N[Not[LessEqual[t$95$0, 5e+230]], $MachinePrecision]], N[(N[(x * z), $MachinePrecision] * (-y) + x), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -2.00000000000000013e294 or 5.0000000000000003e230 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -2.00000000000000013e294 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 5.0000000000000003e230

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -2 \cdot 10^{+294} \lor \neg \left(1 - y \cdot z \leq 5 \cdot 10^{+230}\right):\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, -y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+143} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+215}\right):\\ \;\;\;\;\left(-z\right) \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (if (or (<= t_0 -4e+143) (not (<= t_0 5e+215)))
     (* (- z) (* y x))
     (* x t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -4e+143) || !(t_0 <= 5e+215)) {
		tmp = -z * (y * x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this 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 <= (-4d+143)) .or. (.not. (t_0 <= 5d+215))) then
        tmp = -z * (y * x)
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -4e+143) || !(t_0 <= 5e+215)) {
		tmp = -z * (y * x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if (t_0 <= -4e+143) or not (t_0 <= 5e+215):
		tmp = -z * (y * x)
	else:
		tmp = x * t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -4e+143) || !(t_0 <= 5e+215))
		tmp = Float64(Float64(-z) * Float64(y * x));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if ((t_0 <= -4e+143) || ~((t_0 <= 5e+215)))
		tmp = -z * (y * x);
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e+143], N[Not[LessEqual[t$95$0, 5e+215]], $MachinePrecision]], N[((-z) * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -4.0000000000000001e143 or 5.0000000000000001e215 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 80.0%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(1 - {\left(y \cdot z\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \]

      3. sqrt-pow1 N/A

         \[\leadsto x \cdot \left(1 - \color{blue}{\sqrt{{\left(y \cdot z\right)}^{2}}}\right) \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 11.1

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

   4. Applied rewrites 11.1%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 0.4

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

   7. Applied rewrites 0.4%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot y} \]
   8. Step-by-step derivation

   9. Applied rewrites 0.4%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot z\right)} \]

       3. unpow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. mul-1-neg N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-neg.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 98.4

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

   12. Applied rewrites 98.4%

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

   if -4.0000000000000001e143 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 5.0000000000000001e215

   1. Initial program 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -4 \cdot 10^{+143} \lor \neg \left(1 - y \cdot z \leq 5 \cdot 10^{+215}\right):\\ \;\;\;\;\left(-z\right) \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\left(-z\right) \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (if (or (<= t_0 -2.0) (not (<= t_0 2.0))) (* (- z) (* y x)) (* x 1.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 2.0)) {
		tmp = -z * (y * x);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this 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 <= (-2.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = -z * (y * x)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 2.0)) {
		tmp = -z * (y * x);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if (t_0 <= -2.0) or not (t_0 <= 2.0):
		tmp = -z * (y * x)
	else:
		tmp = x * 1.0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -2.0) || !(t_0 <= 2.0))
		tmp = Float64(Float64(-z) * Float64(y * x));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if ((t_0 <= -2.0) || ~((t_0 <= 2.0)))
		tmp = -z * (y * x);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[((-z) * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -2 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 89.8%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(1 - {\left(y \cdot z\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \]

      3. sqrt-pow1 N/A

         \[\leadsto x \cdot \left(1 - \color{blue}{\sqrt{{\left(y \cdot z\right)}^{2}}}\right) \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. sqrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 13.3

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

   4. Applied rewrites 13.3%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 1.0

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

   7. Applied rewrites 1.0%

      \[\leadsto \color{blue}{\left(z \cdot x\right) \cdot y} \]
   8. Step-by-step derivation

   9. Applied rewrites 1.0%

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

   10. Taylor expanded in z around -inf

       \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \left(z \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot z\right)} \]

       3. unpow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. mul-1-neg N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lower-neg.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 91.6

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

   12. Applied rewrites 91.6%

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

   if -2 < (-.f64 #s(literal 1 binary64) (*.f64 y z)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.0%

      \[\leadsto x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -2 \lor \neg \left(1 - y \cdot z \leq 2\right):\\ \;\;\;\;\left(-z\right) \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(-y\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x 2e-36) (fma (* x (- y)) z x) (* x (- 1.0 (* y z)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e-36) {
		tmp = fma((x * -y), z, x);
	} else {
		tmp = x * (1.0 - (y * z));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= 2e-36)
		tmp = fma(Float64(x * Float64(-y)), z, x);
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * z)));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, 2e-36], N[(N[(x * (-y)), $MachinePrecision] * z + x), $MachinePrecision], N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-36

   1. Initial program 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. associate-*r* N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 91.5

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

   4. Applied rewrites 91.5%

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

   if 1.9999999999999999e-36 < x

   1. Initial program 99.9%

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

Alternative 5: 51.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ x \cdot 1 \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* x 1.0))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return x * 1.0;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return x * 1.0

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(x * 1.0)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = x * 1.0;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 48.9%

   \[\leadsto x \cdot \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))