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

Percentage Accurate: 95.8% → 97.8%

Time: 8.6s

Alternatives: 5

Speedup: 1.0×

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: 95.8% 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, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(-x\_m\right), z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
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 3.1e-76) (fma (* y (- x_m)) z x_m) (* x_m (- 1.0 (* y 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 (x_m <= 3.1e-76) {
		tmp = fma((y * -x_m), z, x_m);
	} else {
		tmp = x_m * (1.0 - (y * 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 (x_m <= 3.1e-76)
		tmp = fma(Float64(y * Float64(-x_m)), z, x_m);
	else
		tmp = Float64(x_m * Float64(1.0 - Float64(y * z)));
	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, 3.1e-76], N[(N[(y * (-x$95$m)), $MachinePrecision] * z + x$95$m), $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 < 3.0999999999999997e-76

   1. Initial program 96.0%

      \[x \cdot \left(1 - y \cdot z\right) \]
   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. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 96.0

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

   4. Applied rewrites 96.0%

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

   if 3.0999999999999997e-76 < 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 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(-x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5000000:\\ \;\;\;\;y \cdot \left(x\_m \cdot \left(-z\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 0.01:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(-x\_m\right)\right) \cdot z\\ \end{array} \end{array} \]

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 (<= (* y z) -5000000.0)
    (* y (* x_m (- z)))
    (if (<= (* y z) 0.01) (* x_m 1.0) (* (* y (- x_m)) 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 ((y * z) <= -5000000.0) {
		tmp = y * (x_m * -z);
	} else if ((y * z) <= 0.01) {
		tmp = x_m * 1.0;
	} else {
		tmp = (y * -x_m) * 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 ((y * z) <= (-5000000.0d0)) then
        tmp = y * (x_m * -z)
    else if ((y * z) <= 0.01d0) then
        tmp = x_m * 1.0d0
    else
        tmp = (y * -x_m) * 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 ((y * z) <= -5000000.0) {
		tmp = y * (x_m * -z);
	} else if ((y * z) <= 0.01) {
		tmp = x_m * 1.0;
	} else {
		tmp = (y * -x_m) * 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 (y * z) <= -5000000.0:
		tmp = y * (x_m * -z)
	elif (y * z) <= 0.01:
		tmp = x_m * 1.0
	else:
		tmp = (y * -x_m) * 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 (Float64(y * z) <= -5000000.0)
		tmp = Float64(y * Float64(x_m * Float64(-z)));
	elseif (Float64(y * z) <= 0.01)
		tmp = Float64(x_m * 1.0);
	else
		tmp = Float64(Float64(y * Float64(-x_m)) * 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 ((y * z) <= -5000000.0)
		tmp = y * (x_m * -z);
	elseif ((y * z) <= 0.01)
		tmp = x_m * 1.0;
	else
		tmp = (y * -x_m) * 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[N[(y * z), $MachinePrecision], -5000000.0], N[(y * N[(x$95$m * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 0.01], N[(x$95$m * 1.0), $MachinePrecision], N[(N[(y * (-x$95$m)), $MachinePrecision] * z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -5e6

   1. Initial program 93.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if -5e6 < (*.f64 y z) < 0.0100000000000000002

   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 97.5%

      \[\leadsto x \cdot \color{blue}{1} \]

   if 0.0100000000000000002 < (*.f64 y z)

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 95.2

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

   5. Applied rewrites 95.2%

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

   7. Applied rewrites 90.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5000000:\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 0.01:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(-x\right)\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := y \cdot \left(x\_m \cdot \left(-z\right)\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \cdot z \leq 0.01:\\ \;\;\;\;x\_m \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* y (* x_m (- z)))))
   (*
    x_s
    (if (<= (* y z) -5000000.0) t_0 (if (<= (* y z) 0.01) (* x_m 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 = y * (x_m * -z);
	double tmp;
	if ((y * z) <= -5000000.0) {
		tmp = t_0;
	} else if ((y * z) <= 0.01) {
		tmp = x_m * 1.0;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = y * (x_m * -z)
    if ((y * z) <= (-5000000.0d0)) then
        tmp = t_0
    else if ((y * z) <= 0.01d0) then
        tmp = x_m * 1.0d0
    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);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = y * (x_m * -z);
	double tmp;
	if ((y * z) <= -5000000.0) {
		tmp = t_0;
	} else if ((y * z) <= 0.01) {
		tmp = x_m * 1.0;
	} else {
		tmp = t_0;
	}
	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):
	t_0 = y * (x_m * -z)
	tmp = 0
	if (y * z) <= -5000000.0:
		tmp = t_0
	elif (y * z) <= 0.01:
		tmp = x_m * 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 = Float64(y * Float64(x_m * Float64(-z)))
	tmp = 0.0
	if (Float64(y * z) <= -5000000.0)
		tmp = t_0;
	elseif (Float64(y * z) <= 0.01)
		tmp = Float64(x_m * 1.0);
	else
		tmp = t_0;
	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)
	t_0 = y * (x_m * -z);
	tmp = 0.0;
	if ((y * z) <= -5000000.0)
		tmp = t_0;
	elseif ((y * z) <= 0.01)
		tmp = x_m * 1.0;
	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]
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[(y * N[(x$95$m * (-z)), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(y * z), $MachinePrecision], -5000000.0], t$95$0, If[LessEqual[N[(y * z), $MachinePrecision], 0.01], N[(x$95$m * 1.0), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5e6 or 0.0100000000000000002 < (*.f64 y z)

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if -5e6 < (*.f64 y z) < 0.0100000000000000002

   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 97.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5000000:\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 0.01:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 1.0× speedup

Math

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

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 (* x_m (- 1.0 (* y 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) {
	return x_s * (x_m * (1.0 - (y * z)));
}

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 * (1.0d0 - (y * z)))
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 * (1.0 - (y * z)));
}

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 * (1.0 - (y * z)))

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(x_m * Float64(1.0 - Float64(y * z))))
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 * (1.0 - (y * z)));
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[(x$95$m * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

Alternative 5: 50.5% accurate, 2.3× speedup

Math

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

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 (* x_m 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 * (x_m * 1.0);
}

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 * 1.0d0)
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 * 1.0);
}

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 * 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(x_m * 1.0))
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 * 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[(x$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 46.7%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))