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

Percentage Accurate: 96.1% → 97.4%

Time: 5.7s

Alternatives: 6

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: 96.1% 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: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.22 \cdot 10^{-116}:\\ \;\;\;\;x\_m - y \cdot \left(x\_m \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.22e-116) (- x_m (* y (* x_m z))) (* x_m (- 1.0 (* y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.22e-116) {
		tmp = x_m - (y * (x_m * z));
	} else {
		tmp = x_m * (1.0 - (y * z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
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 <= 2.22d-116) then
        tmp = x_m - (y * (x_m * z))
    else
        tmp = x_m * (1.0d0 - (y * z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.22e-116) {
		tmp = x_m - (y * (x_m * z));
	} else {
		tmp = x_m * (1.0 - (y * z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 2.22e-116:
		tmp = x_m - (y * (x_m * z))
	else:
		tmp = x_m * (1.0 - (y * z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.22e-116)
		tmp = Float64(x_m - Float64(y * Float64(x_m * z)));
	else
		tmp = Float64(x_m * Float64(1.0 - Float64(y * z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2.22e-116)
		tmp = x_m - (y * (x_m * z));
	else
		tmp = x_m * (1.0 - (y * 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]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2.22e-116], N[(x$95$m - N[(y * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.2200000000000001e-116

   1. Initial program 93.8%

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

   3. Taylor expanded in z around inf 93.7%

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

   4. Taylor expanded in z around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. associate-*r* 93.6%

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

      3. *-commutative 93.6%

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

      4. associate-*r* 95.4%

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

      5. distribute-rgt-neg-in 95.4%

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

      6. distribute-rgt-neg-in 95.4%

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

   6. Simplified 95.4%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 59.9%

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

      3. sqr-neg 59.9%

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

      4. sqrt-unprod 26.0%

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

      5. add-sqr-sqrt 48.1%

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

      6. cancel-sign-sub-inv 48.1%

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

      7. add-sqr-sqrt 22.8%

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

      8. sqrt-unprod 66.6%

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

      9. sqr-neg 66.6%

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

      10. sqrt-unprod 51.3%

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

      11. add-sqr-sqrt 95.4%

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

   8. Applied egg-rr 95.4%

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

   if 2.2200000000000001e-116 < 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 2: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \left(x\_m \cdot \left(-y\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq -50000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;y \cdot z \leq 2 \cdot 10^{+172}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (* x_m (- y)))))
   (*
    x_s
    (if (<= (* y z) -50000000000000.0)
      t_0
      (if (<= (* y z) 5e-7)
        x_m
        (if (<= (* y z) 2e+172) (* x_m (* y (- z))) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * (x_m * -y);
	double tmp;
	if ((y * z) <= -50000000000000.0) {
		tmp = t_0;
	} else if ((y * z) <= 5e-7) {
		tmp = x_m;
	} else if ((y * z) <= 2e+172) {
		tmp = x_m * (y * -z);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
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) :: t_0
    real(8) :: tmp
    t_0 = z * (x_m * -y)
    if ((y * z) <= (-50000000000000.0d0)) then
        tmp = t_0
    else if ((y * z) <= 5d-7) then
        tmp = x_m
    else if ((y * z) <= 2d+172) then
        tmp = x_m * (y * -z)
    else
        tmp = t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * (x_m * -y);
	double tmp;
	if ((y * z) <= -50000000000000.0) {
		tmp = t_0;
	} else if ((y * z) <= 5e-7) {
		tmp = x_m;
	} else if ((y * z) <= 2e+172) {
		tmp = x_m * (y * -z);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = z * (x_m * -y)
	tmp = 0
	if (y * z) <= -50000000000000.0:
		tmp = t_0
	elif (y * z) <= 5e-7:
		tmp = x_m
	elif (y * z) <= 2e+172:
		tmp = x_m * (y * -z)
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(z * Float64(x_m * Float64(-y)))
	tmp = 0.0
	if (Float64(y * z) <= -50000000000000.0)
		tmp = t_0;
	elseif (Float64(y * z) <= 5e-7)
		tmp = x_m;
	elseif (Float64(y * z) <= 2e+172)
		tmp = Float64(x_m * Float64(y * Float64(-z)));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = z * (x_m * -y);
	tmp = 0.0;
	if ((y * z) <= -50000000000000.0)
		tmp = t_0;
	elseif ((y * z) <= 5e-7)
		tmp = x_m;
	elseif ((y * z) <= 2e+172)
		tmp = x_m * (y * -z);
	else
		tmp = t_0;
	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]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(z * N[(x$95$m * (-y)), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(y * z), $MachinePrecision], -50000000000000.0], t$95$0, If[LessEqual[N[(y * z), $MachinePrecision], 5e-7], x$95$m, If[LessEqual[N[(y * z), $MachinePrecision], 2e+172], N[(x$95$m * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -5e13 or 2.0000000000000002e172 < (*.f64 y z)

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf 94.6%

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

   4. Taylor expanded in y around inf 94.6%

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

      1. mul-1-neg 94.6%

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

      2. *-commutative 94.6%

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

      3. distribute-rgt-neg-in 94.6%

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

   6. Simplified 94.6%

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

   if -5e13 < (*.f64 y z) < 4.99999999999999977e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.8%

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

   if 4.99999999999999977e-7 < (*.f64 y z) < 2.0000000000000002e172

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. distribute-rgt-neg-out 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -50000000000000:\\ \;\;\;\;z \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \cdot z \leq 2 \cdot 10^{+172}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := 1 - y \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100000000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;x\_m \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (*
    x_s
    (if (or (<= t_0 -100000000.0) (not (<= t_0 2.0)))
      (* x_m (* y (- z)))
      x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -100000000.0) || !(t_0 <= 2.0)) {
		tmp = x_m * (y * -z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y * z)
    if ((t_0 <= (-100000000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x_m * (y * -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);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = 1.0 - (y * z);
	double tmp;
	if ((t_0 <= -100000000.0) || !(t_0 <= 2.0)) {
		tmp = x_m * (y * -z);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if (t_0 <= -100000000.0) or not (t_0 <= 2.0):
		tmp = x_m * (y * -z)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(1.0 - Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -100000000.0) || !(t_0 <= 2.0))
		tmp = Float64(x_m * Float64(y * 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);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if ((t_0 <= -100000000.0) || ~((t_0 <= 2.0)))
		tmp = x_m * (y * -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]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[Or[LessEqual[t$95$0, -100000000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x$95$m * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 y z)) < -1e8 or 2 < (-.f64 #s(literal 1 binary64) (*.f64 y z))

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. distribute-rgt-neg-out 90.6%

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

   5. Simplified 90.6%

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

   if -1e8 < (-.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 98.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -100000000 \lor \neg \left(1 - y \cdot z \leq 2\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq 2 \cdot 10^{+172}:\\ \;\;\;\;x\_m \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x\_m \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= (* y z) 2e+172) (* x_m (- 1.0 (* y z))) (* z (* x_m (- y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * z) <= 2e+172) {
		tmp = x_m * (1.0 - (y * z));
	} else {
		tmp = z * (x_m * -y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
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 ((y * z) <= 2d+172) then
        tmp = x_m * (1.0d0 - (y * z))
    else
        tmp = z * (x_m * -y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * z) <= 2e+172) {
		tmp = x_m * (1.0 - (y * z));
	} else {
		tmp = z * (x_m * -y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y * z) <= 2e+172:
		tmp = x_m * (1.0 - (y * z))
	else:
		tmp = z * (x_m * -y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * z) <= 2e+172)
		tmp = Float64(x_m * Float64(1.0 - Float64(y * z)));
	else
		tmp = Float64(z * Float64(x_m * Float64(-y)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y * z) <= 2e+172)
		tmp = x_m * (1.0 - (y * z));
	else
		tmp = z * (x_m * -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]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * z), $MachinePrecision], 2e+172], N[(x$95$m * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(x$95$m * (-y)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < 2.0000000000000002e172

   1. Initial program 98.1%

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

   if 2.0000000000000002e172 < (*.f64 y z)

   1. Initial program 78.5%

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

   3. Taylor expanded in z around inf 96.7%

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

   4. Taylor expanded in y around inf 96.7%

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

      1. mul-1-neg 96.7%

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

      2. *-commutative 96.7%

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

      3. distribute-rgt-neg-in 96.7%

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

   6. Simplified 96.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq 2 \cdot 10^{+172}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\frac{x\_m \cdot z}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= (* y z) -2e+17) (/ (* x_m z) z) x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * z) <= -2e+17) {
		tmp = (x_m * z) / z;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
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 ((y * z) <= (-2d+17)) then
        tmp = (x_m * z) / 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);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y * z) <= -2e+17) {
		tmp = (x_m * z) / z;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y * z) <= -2e+17:
		tmp = (x_m * z) / z
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(y * z) <= -2e+17)
		tmp = Float64(Float64(x_m * z) / z);
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y * z) <= -2e+17)
		tmp = (x_m * z) / 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]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(y * z), $MachinePrecision], -2e+17], N[(N[(x$95$m * z), $MachinePrecision] / z), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -2e17

   1. Initial program 93.2%

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

   3. Taylor expanded in z around inf 95.0%

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

   4. Taylor expanded in y around 0 8.7%

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

      1. associate-*r/ 21.7%

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

      2. *-commutative 21.7%

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

   6. Applied egg-rr 21.7%

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

   if -2e17 < (*.f64 y z)

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0 62.8%

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

Alternative 6: 50.8% accurate, 7.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
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)
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);
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)
def code(x_s, x_m, y, z):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
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);
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]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in y around 0 49.9%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))