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

Percentage Accurate: 96.0% → 97.9%

Time: 6.7s

Alternatives: 7

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: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (1.0d0 - (y * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-26}:\\ \;\;\;\;x\_m - \left(x\_m \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\_m - x\_m \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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 1e-26) (- x_m (* (* x_m y) z)) (- x_m (* x_m (* 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 <= 1e-26) {
		tmp = x_m - ((x_m * y) * z);
	} else {
		tmp = x_m - (x_m * (y * 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 (x_m <= 1d-26) then
        tmp = x_m - ((x_m * y) * z)
    else
        tmp = x_m - (x_m * (y * 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 (x_m <= 1e-26) {
		tmp = x_m - ((x_m * y) * z);
	} else {
		tmp = x_m - (x_m * (y * 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 x_m <= 1e-26:
		tmp = x_m - ((x_m * y) * z)
	else:
		tmp = x_m - (x_m * (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 <= 1e-26)
		tmp = Float64(x_m - Float64(Float64(x_m * y) * z));
	else
		tmp = Float64(x_m - Float64(x_m * Float64(y * 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 (x_m <= 1e-26)
		tmp = x_m - ((x_m * y) * z);
	else
		tmp = x_m - (x_m * (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]
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, 1e-26], N[(x$95$m - N[(N[(x$95$m * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e-26

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      2. distribute-rgt-in 95.0%

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

      3. *-un-lft-identity 95.0%

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

      4. distribute-rgt-neg-in 95.0%

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in y around 0 95.0%

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

      1. neg-mul-1 95.0%

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

      2. associate-*r* 94.0%

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

   7. Simplified 94.0%

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

   if 1e-26 < 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%

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

      1. distribute-rgt-neg-out 99.9%

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

      2. distribute-lft-neg-out 99.9%

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

      3. associate-*r* 87.4%

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

      4. *-commutative 87.4%

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

      5. distribute-lft-neg-in 87.4%

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

      6. add-sqr-sqrt 46.5%

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

      7. sqrt-unprod 51.0%

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

      8. sqr-neg 51.0%

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

      9. sqrt-unprod 18.4%

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

      10. add-sqr-sqrt 40.6%

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

      11. *-commutative 40.6%

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

      12. distribute-lft-neg-in 40.6%

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

      13. cancel-sign-sub-inv 40.6%

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

      14. associate-*l* 46.8%

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

      15. *-commutative 46.8%

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

      16. add-sqr-sqrt 27.9%

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

      17. sqrt-unprod 72.1%

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

      18. distribute-rgt-neg-out 72.1%

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

      19. distribute-rgt-neg-out 72.1%

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

      20. sqr-neg 72.1%

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

      21. sqrt-prod 53.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-26}:\\ \;\;\;\;x - \left(x \cdot y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+69} \lor \neg \left(y \leq 9.5 \cdot 10^{-77}\right):\\ \;\;\;\;x\_m \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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 (<= y -1.42e+69) (not (<= y 9.5e-77))) (* x_m (* y (- 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 ((y <= -1.42e+69) || !(y <= 9.5e-77)) {
		tmp = x_m * (y * -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 ((y <= (-1.42d+69)) .or. (.not. (y <= 9.5d-77))) 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);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -1.42e+69) || !(y <= 9.5e-77)) {
		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)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -1.42e+69) or not (y <= 9.5e-77):
		tmp = x_m * (y * -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 ((y <= -1.42e+69) || !(y <= 9.5e-77))
		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);
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 <= -1.42e+69) || ~((y <= 9.5e-77)))
		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]
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[y, -1.42e+69], N[Not[LessEqual[y, 9.5e-77]], $MachinePrecision]], N[(x$95$m * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.42e69 or 9.5000000000000005e-77 < y

   1. Initial program 92.4%

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

   3. Taylor expanded in y around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-rgt-neg-in 66.7%

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

      3. distribute-rgt-neg-out 66.7%

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

   5. Simplified 66.7%

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

   if -1.42e69 < y < 9.5000000000000005e-77

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 69.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+69} \lor \neg \left(y \leq 9.5 \cdot 10^{-77}\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.68 \cdot 10^{+69}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-72}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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 -1.68e+69)
    (* x_m (* y (- z)))
    (if (<= y 2.05e-72) x_m (* y (* 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 (y <= -1.68e+69) {
		tmp = x_m * (y * -z);
	} else if (y <= 2.05e-72) {
		tmp = x_m;
	} else {
		tmp = y * (z * -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 (y <= (-1.68d+69)) then
        tmp = x_m * (y * -z)
    else if (y <= 2.05d-72) then
        tmp = x_m
    else
        tmp = y * (z * -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 (y <= -1.68e+69) {
		tmp = x_m * (y * -z);
	} else if (y <= 2.05e-72) {
		tmp = x_m;
	} else {
		tmp = y * (z * -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 y <= -1.68e+69:
		tmp = x_m * (y * -z)
	elif y <= 2.05e-72:
		tmp = x_m
	else:
		tmp = y * (z * -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 (y <= -1.68e+69)
		tmp = Float64(x_m * Float64(y * Float64(-z)));
	elseif (y <= 2.05e-72)
		tmp = x_m;
	else
		tmp = Float64(y * Float64(z * Float64(-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 (y <= -1.68e+69)
		tmp = x_m * (y * -z);
	elseif (y <= 2.05e-72)
		tmp = x_m;
	else
		tmp = y * (z * -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[LessEqual[y, -1.68e+69], N[(x$95$m * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e-72], x$95$m, N[(y * N[(z * (-x$95$m)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.68000000000000001e69

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. distribute-rgt-neg-in 81.4%

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

      3. distribute-rgt-neg-out 81.4%

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

   5. Simplified 81.4%

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

   if -1.68000000000000001e69 < y < 2.05000000000000002e-72

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 69.4%

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

   if 2.05000000000000002e-72 < y

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf 59.1%

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

      1. mul-1-neg 59.1%

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

      2. associate-*r* 65.2%

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

      3. distribute-rgt-neg-in 65.2%

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

      4. *-commutative 65.2%

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

      5. associate-*r* 67.0%

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

      6. distribute-rgt-neg-out 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.68 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+69}:\\ \;\;\;\;x\_m \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-71}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\right) \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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 -1.25e+69)
    (* x_m (* y (- z)))
    (if (<= y 2.5e-71) x_m (* (* x_m 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 (y <= -1.25e+69) {
		tmp = x_m * (y * -z);
	} else if (y <= 2.5e-71) {
		tmp = x_m;
	} else {
		tmp = (x_m * y) * -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 <= (-1.25d+69)) then
        tmp = x_m * (y * -z)
    else if (y <= 2.5d-71) then
        tmp = x_m
    else
        tmp = (x_m * y) * -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 <= -1.25e+69) {
		tmp = x_m * (y * -z);
	} else if (y <= 2.5e-71) {
		tmp = x_m;
	} else {
		tmp = (x_m * y) * -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 <= -1.25e+69:
		tmp = x_m * (y * -z)
	elif y <= 2.5e-71:
		tmp = x_m
	else:
		tmp = (x_m * 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 (y <= -1.25e+69)
		tmp = Float64(x_m * Float64(y * Float64(-z)));
	elseif (y <= 2.5e-71)
		tmp = x_m;
	else
		tmp = Float64(Float64(x_m * y) * Float64(-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 <= -1.25e+69)
		tmp = x_m * (y * -z);
	elseif (y <= 2.5e-71)
		tmp = x_m;
	else
		tmp = (x_m * 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]
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[y, -1.25e+69], N[(x$95$m * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e-71], x$95$m, N[(N[(x$95$m * y), $MachinePrecision] * (-z)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.25000000000000009e69

   1. Initial program 95.5%

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

   3. Taylor expanded in y around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. distribute-rgt-neg-in 81.4%

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

      3. distribute-rgt-neg-out 81.4%

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

   5. Simplified 81.4%

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

   if -1.25000000000000009e69 < y < 2.49999999999999999e-71

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 69.4%

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

   if 2.49999999999999999e-71 < y

   1. Initial program 90.8%

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

      1. flip-- 65.6%

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

      2. associate-*r/ 65.5%

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

      3. metadata-eval 65.5%

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

      4. pow2 65.5%

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

      5. +-commutative 65.5%

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

      6. fma-def 65.5%

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

   4. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - {\left(y \cdot z\right)}^{2}\right)}{\mathsf{fma}\left(y, z, 1\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in y around inf 59.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{-1}{y \cdot z}}} \]
   8. Step-by-step derivation

      1. div-inv 59.1%

         \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{-1}{y \cdot z}}} \]

      2. frac-2neg 59.1%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{--1}{-y \cdot z}}} \]

      3. metadata-eval 59.1%

         \[\leadsto x \cdot \frac{1}{\frac{\color{blue}{1}}{-y \cdot z}} \]

      4. remove-double-div 59.1%

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

      5. distribute-rgt-neg-in 59.1%

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

      6. associate-*r* 65.2%

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

      7. distribute-lft-neg-in 65.2%

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

      8. distribute-rgt-neg-in 65.2%

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

   9. Applied egg-rr 65.2%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq 4 \cdot 10^{+121}:\\ \;\;\;\;x\_m \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\right) \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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) 4e+121) (* x_m (- 1.0 (* y z))) (* (* x_m 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 ((y * z) <= 4e+121) {
		tmp = x_m * (1.0 - (y * z));
	} else {
		tmp = (x_m * y) * -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) <= 4d+121) then
        tmp = x_m * (1.0d0 - (y * z))
    else
        tmp = (x_m * y) * -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) <= 4e+121) {
		tmp = x_m * (1.0 - (y * z));
	} else {
		tmp = (x_m * y) * -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) <= 4e+121:
		tmp = x_m * (1.0 - (y * z))
	else:
		tmp = (x_m * 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 (Float64(y * z) <= 4e+121)
		tmp = Float64(x_m * Float64(1.0 - Float64(y * z)));
	else
		tmp = Float64(Float64(x_m * y) * Float64(-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) <= 4e+121)
		tmp = x_m * (1.0 - (y * z));
	else
		tmp = (x_m * 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]
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], 4e+121], N[(x$95$m * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] * (-z)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < 4.00000000000000015e121

   1. Initial program 98.2%

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

   if 4.00000000000000015e121 < (*.f64 y z)

   1. Initial program 83.7%

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

      1. flip-- 35.8%

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

      2. associate-*r/ 30.5%

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

      3. metadata-eval 30.5%

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

      4. pow2 30.5%

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

      5. +-commutative 30.5%

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

      6. fma-def 30.5%

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

   4. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - {\left(y \cdot z\right)}^{2}\right)}{\mathsf{fma}\left(y, z, 1\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in y around inf 83.6%

      \[\leadsto \frac{x}{\color{blue}{\frac{-1}{y \cdot z}}} \]
   8. Step-by-step derivation

      1. div-inv 83.6%

         \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{-1}{y \cdot z}}} \]

      2. frac-2neg 83.6%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{--1}{-y \cdot z}}} \]

      3. metadata-eval 83.6%

         \[\leadsto x \cdot \frac{1}{\frac{\color{blue}{1}}{-y \cdot z}} \]

      4. remove-double-div 83.7%

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

      5. distribute-rgt-neg-in 83.7%

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

      6. associate-*r* 99.8%

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

      7. distribute-lft-neg-in 99.8%

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

      8. distribute-rgt-neg-in 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq 4 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \cdot z \leq 4 \cdot 10^{+121}:\\ \;\;\;\;x\_m - x\_m \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\right) \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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) 4e+121) (- x_m (* x_m (* y z))) (* (* x_m 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 ((y * z) <= 4e+121) {
		tmp = x_m - (x_m * (y * z));
	} else {
		tmp = (x_m * y) * -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) <= 4d+121) then
        tmp = x_m - (x_m * (y * z))
    else
        tmp = (x_m * y) * -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) <= 4e+121) {
		tmp = x_m - (x_m * (y * z));
	} else {
		tmp = (x_m * y) * -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) <= 4e+121:
		tmp = x_m - (x_m * (y * z))
	else:
		tmp = (x_m * 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 (Float64(y * z) <= 4e+121)
		tmp = Float64(x_m - Float64(x_m * Float64(y * z)));
	else
		tmp = Float64(Float64(x_m * y) * Float64(-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) <= 4e+121)
		tmp = x_m - (x_m * (y * z));
	else
		tmp = (x_m * 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]
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], 4e+121], N[(x$95$m - N[(x$95$m * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] * (-z)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < 4.00000000000000015e121

   1. Initial program 98.2%

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

      1. sub-neg 98.2%

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

      2. distribute-rgt-in 98.2%

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

      3. *-un-lft-identity 98.2%

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

      4. distribute-rgt-neg-in 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. distribute-rgt-neg-out 98.2%

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

      2. distribute-lft-neg-out 98.2%

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

      3. associate-*r* 90.5%

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

      4. *-commutative 90.5%

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

      5. distribute-lft-neg-in 90.5%

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

      6. add-sqr-sqrt 51.0%

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

      7. sqrt-unprod 63.2%

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

      8. sqr-neg 63.2%

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

      9. sqrt-unprod 30.7%

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

      10. add-sqr-sqrt 52.0%

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

      11. *-commutative 52.0%

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

      12. distribute-lft-neg-in 52.0%

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

      13. cancel-sign-sub-inv 52.0%

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

      14. associate-*l* 55.2%

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

      15. *-commutative 55.2%

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

      16. add-sqr-sqrt 33.5%

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

      17. sqrt-unprod 66.9%

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

      18. distribute-rgt-neg-out 66.9%

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

      19. distribute-rgt-neg-out 66.9%

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

      20. sqr-neg 66.9%

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

      21. sqrt-prod 46.5%

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

   6. Applied egg-rr 98.2%

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

   if 4.00000000000000015e121 < (*.f64 y z)

   1. Initial program 83.7%

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

      1. flip-- 35.8%

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

      2. associate-*r/ 30.5%

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

      3. metadata-eval 30.5%

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

      4. pow2 30.5%

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

      5. +-commutative 30.5%

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

      6. fma-def 30.5%

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

   4. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - {\left(y \cdot z\right)}^{2}\right)}{\mathsf{fma}\left(y, z, 1\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in y around inf 83.6%

      \[\leadsto \frac{x}{\color{blue}{\frac{-1}{y \cdot z}}} \]
   8. Step-by-step derivation

      1. div-inv 83.6%

         \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{-1}{y \cdot z}}} \]

      2. frac-2neg 83.6%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{--1}{-y \cdot z}}} \]

      3. metadata-eval 83.6%

         \[\leadsto x \cdot \frac{1}{\frac{\color{blue}{1}}{-y \cdot z}} \]

      4. remove-double-div 83.7%

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

      5. distribute-rgt-neg-in 83.7%

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

      6. associate-*r* 99.8%

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

      7. distribute-lft-neg-in 99.8%

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

      8. distribute-rgt-neg-in 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq 4 \cdot 10^{+121}:\\ \;\;\;\;x - x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.2% 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 1 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 96.2%

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

3. Taylor expanded in y around 0 49.1%

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

4. Final simplification 49.1%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))