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

Percentage Accurate: 96.2% → 97.8%

Time: 7.0s

Alternatives: 8

Speedup: 0.6×

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: 96.2% 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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;x\_m - \left(x\_m \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m - x\_m \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

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 2.3e-42) (- x_m (* (* x_m z) y)) (- x_m (* x_m (* z y))))))

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 <= 2.3e-42) {
		tmp = x_m - ((x_m * z) * y);
	} else {
		tmp = x_m - (x_m * (z * y));
	}
	return x_s * tmp;
}

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
    real(8) :: tmp
    if (x_m <= 2.3d-42) then
        tmp = x_m - ((x_m * z) * y)
    else
        tmp = x_m - (x_m * (z * y))
    end if
    code = x_s * tmp
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) {
	double tmp;
	if (x_m <= 2.3e-42) {
		tmp = x_m - ((x_m * z) * y);
	} else {
		tmp = x_m - (x_m * (z * y));
	}
	return x_s * tmp;
}

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):
	tmp = 0
	if x_m <= 2.3e-42:
		tmp = x_m - ((x_m * z) * y)
	else:
		tmp = x_m - (x_m * (z * y))
	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 <= 2.3e-42)
		tmp = Float64(x_m - Float64(Float64(x_m * z) * y));
	else
		tmp = Float64(x_m - Float64(x_m * Float64(z * y)));
	end
	return Float64(x_s * tmp)
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_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2.3e-42)
		tmp = x_m - ((x_m * z) * y);
	else
		tmp = x_m - (x_m * (z * y));
	end
	tmp_2 = 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, 2.3e-42], N[(x$95$m - N[(N[(x$95$m * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.30000000000000004e-42

   1. Initial program 93.5%

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

   3. Taylor expanded in z around inf 89.0%

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

   4. Taylor expanded in z around 0 93.5%

      \[\leadsto \color{blue}{x + -1 \cdot \left(x \cdot \left(y \cdot z\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 96.7%

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

      2. *-commutative 96.7%

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

      3. neg-mul-1 96.7%

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

      4. sub-neg 96.7%

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

      5. associate-*r* 89.6%

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

      6. *-commutative 89.6%

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

   6. Simplified 89.6%

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

   if 2.30000000000000004e-42 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;x - \left(x \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \left(x\_m \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+61}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+226}:\\ \;\;\;\;z \cdot \left(x\_m \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(z \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

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 (<= z -4.9e-67)
    (* y (* x_m (- z)))
    (if (<= z 5.1e+61)
      x_m
      (if (<= z 1.05e+226) (* z (* x_m (- y))) (* x_m (* z (- y))))))))

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 (z <= -4.9e-67) {
		tmp = y * (x_m * -z);
	} else if (z <= 5.1e+61) {
		tmp = x_m;
	} else if (z <= 1.05e+226) {
		tmp = z * (x_m * -y);
	} else {
		tmp = x_m * (z * -y);
	}
	return x_s * tmp;
}

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
    real(8) :: tmp
    if (z <= (-4.9d-67)) then
        tmp = y * (x_m * -z)
    else if (z <= 5.1d+61) then
        tmp = x_m
    else if (z <= 1.05d+226) then
        tmp = z * (x_m * -y)
    else
        tmp = x_m * (z * -y)
    end if
    code = x_s * tmp
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) {
	double tmp;
	if (z <= -4.9e-67) {
		tmp = y * (x_m * -z);
	} else if (z <= 5.1e+61) {
		tmp = x_m;
	} else if (z <= 1.05e+226) {
		tmp = z * (x_m * -y);
	} else {
		tmp = x_m * (z * -y);
	}
	return x_s * tmp;
}

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):
	tmp = 0
	if z <= -4.9e-67:
		tmp = y * (x_m * -z)
	elif z <= 5.1e+61:
		tmp = x_m
	elif z <= 1.05e+226:
		tmp = z * (x_m * -y)
	else:
		tmp = x_m * (z * -y)
	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 (z <= -4.9e-67)
		tmp = Float64(y * Float64(x_m * Float64(-z)));
	elseif (z <= 5.1e+61)
		tmp = x_m;
	elseif (z <= 1.05e+226)
		tmp = Float64(z * Float64(x_m * Float64(-y)));
	else
		tmp = Float64(x_m * Float64(z * Float64(-y)));
	end
	return Float64(x_s * tmp)
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_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -4.9e-67)
		tmp = y * (x_m * -z);
	elseif (z <= 5.1e+61)
		tmp = x_m;
	elseif (z <= 1.05e+226)
		tmp = z * (x_m * -y);
	else
		tmp = x_m * (z * -y);
	end
	tmp_2 = 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[z, -4.9e-67], N[(y * N[(x$95$m * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.1e+61], x$95$m, If[LessEqual[z, 1.05e+226], N[(z * N[(x$95$m * (-y)), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(z * (-y)), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -4.89999999999999993e-67

   1. Initial program 91.4%

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

   3. Taylor expanded in y around inf 56.3%

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

      1. mul-1-neg 56.3%

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

      2. associate-*r* 62.5%

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

      3. distribute-rgt-neg-in 62.5%

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

      4. *-commutative 62.5%

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

      5. associate-*l* 62.5%

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

   5. Simplified 62.5%

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

   if -4.89999999999999993e-67 < z < 5.1000000000000001e61

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 74.0%

      \[\leadsto \color{blue}{x} \]

   if 5.1000000000000001e61 < z < 1.04999999999999997e226

   1. Initial program 88.3%

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

   3. Taylor expanded in z around inf 99.3%

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

   4. Taylor expanded in y around inf 88.3%

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

      1. neg-mul-1 88.3%

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

      2. *-commutative 88.3%

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

      3. distribute-rgt-neg-in 88.3%

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

   6. Simplified 88.3%

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

   if 1.04999999999999997e226 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 95.0%

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

      1. mul-1-neg 95.0%

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

      2. distribute-rgt-neg-in 95.0%

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

      3. distribute-rgt-neg-out 95.0%

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

   5. Simplified 95.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(-z\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-67}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+226}:\\ \;\;\;\;z \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-65} \lor \neg \left(z \leq 3.8 \cdot 10^{+59}\right):\\ \;\;\;\;y \cdot \left(x\_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (or (<= z -3e-65) (not (<= z 3.8e+59))) (* y (* x_m (- z))) 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) {
	double tmp;
	if ((z <= -3e-65) || !(z <= 3.8e+59)) {
		tmp = y * (x_m * -z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

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
    real(8) :: tmp
    if ((z <= (-3d-65)) .or. (.not. (z <= 3.8d+59))) then
        tmp = y * (x_m * -z)
    else
        tmp = x_m
    end if
    code = x_s * tmp
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) {
	double tmp;
	if ((z <= -3e-65) || !(z <= 3.8e+59)) {
		tmp = y * (x_m * -z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

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):
	tmp = 0
	if (z <= -3e-65) or not (z <= 3.8e+59):
		tmp = y * (x_m * -z)
	else:
		tmp = x_m
	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 ((z <= -3e-65) || !(z <= 3.8e+59))
		tmp = Float64(y * Float64(x_m * Float64(-z)));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
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_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -3e-65) || ~((z <= 3.8e+59)))
		tmp = y * (x_m * -z);
	else
		tmp = x_m;
	end
	tmp_2 = 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[Or[LessEqual[z, -3e-65], N[Not[LessEqual[z, 3.8e+59]], $MachinePrecision]], N[(y * N[(x$95$m * (-z)), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.99999999999999998e-65 or 3.8000000000000001e59 < z

   1. Initial program 91.8%

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

   3. Taylor expanded in y around inf 66.1%

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

      1. mul-1-neg 66.1%

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

      2. associate-*r* 72.1%

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

      3. distribute-rgt-neg-in 72.1%

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

      4. *-commutative 72.1%

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

      5. associate-*l* 68.8%

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

   5. Simplified 68.8%

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

   if -2.99999999999999998e-65 < z < 3.8000000000000001e59

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 74.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-65} \lor \neg \left(z \leq 3.8 \cdot 10^{+59}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-65} \lor \neg \left(z \leq 2.7 \cdot 10^{+60}\right):\\ \;\;\;\;x\_m \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (or (<= z -2.2e-65) (not (<= z 2.7e+60))) (* x_m (* z (- y))) 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) {
	double tmp;
	if ((z <= -2.2e-65) || !(z <= 2.7e+60)) {
		tmp = x_m * (z * -y);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

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
    real(8) :: tmp
    if ((z <= (-2.2d-65)) .or. (.not. (z <= 2.7d+60))) then
        tmp = x_m * (z * -y)
    else
        tmp = x_m
    end if
    code = x_s * tmp
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) {
	double tmp;
	if ((z <= -2.2e-65) || !(z <= 2.7e+60)) {
		tmp = x_m * (z * -y);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

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):
	tmp = 0
	if (z <= -2.2e-65) or not (z <= 2.7e+60):
		tmp = x_m * (z * -y)
	else:
		tmp = x_m
	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 ((z <= -2.2e-65) || !(z <= 2.7e+60))
		tmp = Float64(x_m * Float64(z * Float64(-y)));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
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_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z <= -2.2e-65) || ~((z <= 2.7e+60)))
		tmp = x_m * (z * -y);
	else
		tmp = x_m;
	end
	tmp_2 = 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[Or[LessEqual[z, -2.2e-65], N[Not[LessEqual[z, 2.7e+60]], $MachinePrecision]], N[(x$95$m * N[(z * (-y)), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.20000000000000021e-65 or 2.6999999999999999e60 < z

   1. Initial program 91.8%

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

   3. Taylor expanded in y around inf 66.1%

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

      1. mul-1-neg 66.1%

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

      2. distribute-rgt-neg-in 66.1%

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

      3. distribute-rgt-neg-out 66.1%

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

   5. Simplified 66.1%

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

   if -2.20000000000000021e-65 < z < 2.6999999999999999e60

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 74.0%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-65} \lor \neg \left(z \leq 2.7 \cdot 10^{+60}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot y \leq -4 \cdot 10^{+250}:\\ \;\;\;\;y \cdot \left(x\_m \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z \cdot y\right)\\ \end{array} \end{array} \]

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 (<= (* z y) -4e+250) (* y (* x_m (- z))) (* x_m (- 1.0 (* z y))))))

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 ((z * y) <= -4e+250) {
		tmp = y * (x_m * -z);
	} else {
		tmp = x_m * (1.0 - (z * y));
	}
	return x_s * tmp;
}

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
    real(8) :: tmp
    if ((z * y) <= (-4d+250)) then
        tmp = y * (x_m * -z)
    else
        tmp = x_m * (1.0d0 - (z * y))
    end if
    code = x_s * tmp
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) {
	double tmp;
	if ((z * y) <= -4e+250) {
		tmp = y * (x_m * -z);
	} else {
		tmp = x_m * (1.0 - (z * y));
	}
	return x_s * tmp;
}

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):
	tmp = 0
	if (z * y) <= -4e+250:
		tmp = y * (x_m * -z)
	else:
		tmp = x_m * (1.0 - (z * y))
	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 (Float64(z * y) <= -4e+250)
		tmp = Float64(y * Float64(x_m * Float64(-z)));
	else
		tmp = Float64(x_m * Float64(1.0 - Float64(z * y)));
	end
	return Float64(x_s * tmp)
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_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((z * y) <= -4e+250)
		tmp = y * (x_m * -z);
	else
		tmp = x_m * (1.0 - (z * y));
	end
	tmp_2 = 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[N[(z * y), $MachinePrecision], -4e+250], N[(y * N[(x$95$m * (-z)), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -3.9999999999999997e250

   1. Initial program 75.9%

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

   3. Taylor expanded in y around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-*r* 96.9%

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

      3. distribute-rgt-neg-in 96.9%

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

      4. *-commutative 96.9%

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

      5. associate-*l* 99.9%

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

   5. Simplified 99.9%

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

   if -3.9999999999999997e250 < (*.f64 y z)

   1. Initial program 98.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -4 \cdot 10^{+250}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4.3 \cdot 10^{-35}:\\ \;\;\;\;x\_m - \left(x\_m \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - z \cdot y\right)\\ \end{array} \end{array} \]

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 4.3e-35) (- x_m (* (* x_m z) y)) (* x_m (- 1.0 (* z y))))))

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 <= 4.3e-35) {
		tmp = x_m - ((x_m * z) * y);
	} else {
		tmp = x_m * (1.0 - (z * y));
	}
	return x_s * tmp;
}

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
    real(8) :: tmp
    if (x_m <= 4.3d-35) then
        tmp = x_m - ((x_m * z) * y)
    else
        tmp = x_m * (1.0d0 - (z * y))
    end if
    code = x_s * tmp
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) {
	double tmp;
	if (x_m <= 4.3e-35) {
		tmp = x_m - ((x_m * z) * y);
	} else {
		tmp = x_m * (1.0 - (z * y));
	}
	return x_s * tmp;
}

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):
	tmp = 0
	if x_m <= 4.3e-35:
		tmp = x_m - ((x_m * z) * y)
	else:
		tmp = x_m * (1.0 - (z * y))
	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 <= 4.3e-35)
		tmp = Float64(x_m - Float64(Float64(x_m * z) * y));
	else
		tmp = Float64(x_m * Float64(1.0 - Float64(z * y)));
	end
	return Float64(x_s * tmp)
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_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 4.3e-35)
		tmp = x_m - ((x_m * z) * y);
	else
		tmp = x_m * (1.0 - (z * y));
	end
	tmp_2 = 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, 4.3e-35], N[(x$95$m - N[(N[(x$95$m * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.3000000000000002e-35

   1. Initial program 93.5%

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

   3. Taylor expanded in z around inf 89.0%

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

   4. Taylor expanded in z around 0 93.5%

      \[\leadsto \color{blue}{x + -1 \cdot \left(x \cdot \left(y \cdot z\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 96.7%

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

      2. *-commutative 96.7%

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

      3. neg-mul-1 96.7%

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

      4. sub-neg 96.7%

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

      5. associate-*r* 89.6%

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

      6. *-commutative 89.6%

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

   6. Simplified 89.6%

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

   if 4.3000000000000002e-35 < 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. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-35}:\\ \;\;\;\;x - \left(x \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{+204}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot z}{z}\\ \end{array} \end{array} \]

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 (<= z 7.5e+204) x_m (/ (* x_m z) z))))

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 (z <= 7.5e+204) {
		tmp = x_m;
	} else {
		tmp = (x_m * z) / z;
	}
	return x_s * tmp;
}

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
    real(8) :: tmp
    if (z <= 7.5d+204) then
        tmp = x_m
    else
        tmp = (x_m * z) / z
    end if
    code = x_s * tmp
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) {
	double tmp;
	if (z <= 7.5e+204) {
		tmp = x_m;
	} else {
		tmp = (x_m * z) / z;
	}
	return x_s * tmp;
}

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):
	tmp = 0
	if z <= 7.5e+204:
		tmp = x_m
	else:
		tmp = (x_m * z) / z
	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 (z <= 7.5e+204)
		tmp = x_m;
	else
		tmp = Float64(Float64(x_m * z) / z);
	end
	return Float64(x_s * tmp)
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_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= 7.5e+204)
		tmp = x_m;
	else
		tmp = (x_m * z) / z;
	end
	tmp_2 = 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[z, 7.5e+204], x$95$m, N[(N[(x$95$m * z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.4999999999999998e204

   1. Initial program 95.0%

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

   3. Taylor expanded in y around 0 52.3%

      \[\leadsto \color{blue}{x} \]

   if 7.4999999999999998e204 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 95.5%

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

   4. Taylor expanded in y around 0 5.0%

      \[\leadsto z \cdot \color{blue}{\frac{x}{z}} \]
   5. Step-by-step derivation

      1. associate-*r/ 41.6%

         \[\leadsto \color{blue}{\frac{z \cdot x}{z}} \]

   6. Applied egg-rr 41.6%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{+204}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 7.0× 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 48.2%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))