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

Percentage Accurate: 96.1% → 98.3%

Time: 7.2s

Alternatives: 8

Speedup: 0.5×

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: 98.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1 \cdot 10^{-107}:\\ \;\;\;\;x\_m - \left(z \cdot \sqrt{x\_m}\right) \cdot \left(\sqrt{x\_m} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m - x\_m \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.1e-107)
    (- x_m (* (* z (sqrt x_m)) (* (sqrt x_m) y)))
    (- x_m (* x_m (* z 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 (x_m <= 2.1e-107) {
		tmp = x_m - ((z * sqrt(x_m)) * (sqrt(x_m) * y));
	} else {
		tmp = x_m - (x_m * (z * 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 (x_m <= 2.1d-107) then
        tmp = x_m - ((z * sqrt(x_m)) * (sqrt(x_m) * y))
    else
        tmp = x_m - (x_m * (z * y))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.1e-107) {
		tmp = x_m - ((z * Math.sqrt(x_m)) * (Math.sqrt(x_m) * y));
	} else {
		tmp = x_m - (x_m * (z * y));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 2.1e-107:
		tmp = x_m - ((z * math.sqrt(x_m)) * (math.sqrt(x_m) * y))
	else:
		tmp = x_m - (x_m * (z * 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 (x_m <= 2.1e-107)
		tmp = Float64(x_m - Float64(Float64(z * sqrt(x_m)) * Float64(sqrt(x_m) * y)));
	else
		tmp = Float64(x_m - Float64(x_m * Float64(z * y)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2.1e-107)
		tmp = x_m - ((z * sqrt(x_m)) * (sqrt(x_m) * y));
	else
		tmp = x_m - (x_m * (z * y));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2.1e-107], N[(x$95$m - N[(N[(z * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[x$95$m], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0999999999999999e-107

   1. Initial program 93.9%

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

      1. sub-neg 93.9%

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

      2. distribute-rgt-in 93.9%

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

      3. *-un-lft-identity 93.9%

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

      4. distribute-rgt-neg-in 93.9%

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

   4. Applied egg-rr 93.9%

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

      1. associate-*l* 93.4%

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

      2. add-sqr-sqrt 53.0%

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

      3. sqrt-unprod 60.3%

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

      4. sqr-neg 60.3%

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

      5. sqrt-unprod 20.0%

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

      6. add-sqr-sqrt 45.7%

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

      7. cancel-sign-sub-inv 45.7%

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

      8. associate-*l* 46.1%

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

      9. distribute-rgt-neg-out 46.1%

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

      10. distribute-lft-neg-in 46.1%

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

      11. associate-*r* 45.7%

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

      12. add-sqr-sqrt 20.0%

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

      13. sqrt-unprod 60.3%

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

      14. sqr-neg 60.3%

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

      15. sqrt-unprod 53.0%

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

      16. add-sqr-sqrt 93.4%

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

   6. Applied egg-rr 93.4%

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

   7. Taylor expanded in y around 0 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*r* 93.4%

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

      3. add-sqr-sqrt 16.7%

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

      4. associate-*l* 16.7%

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

      5. *-commutative 16.7%

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

      6. cancel-sign-sub-inv 16.7%

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

      7. add-sqr-sqrt 9.2%

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

      8. sqrt-unprod 10.7%

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

      9. sqr-neg 10.7%

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

      10. sqrt-unprod 5.8%

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

      11. add-sqr-sqrt 8.9%

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

      12. cancel-sign-sub-inv 8.9%

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

      13. associate-*l* 8.9%

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

      14. cancel-sign-sub-inv 8.9%

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

      15. *-commutative 8.9%

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

      16. add-sqr-sqrt 5.8%

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

      17. sqrt-unprod 11.2%

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

      18. sqr-neg 11.2%

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

      19. sqrt-unprod 9.2%

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

      20. add-sqr-sqrt 17.3%

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

   9. Applied egg-rr 17.3%

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

   if 2.0999999999999999e-107 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-rgt-in 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l* 87.4%

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

      2. add-sqr-sqrt 44.2%

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

      3. sqrt-unprod 58.0%

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

      4. sqr-neg 58.0%

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

      5. sqrt-unprod 17.5%

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

      6. add-sqr-sqrt 44.5%

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

      7. cancel-sign-sub-inv 44.5%

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

      8. associate-*l* 49.3%

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

      9. distribute-rgt-neg-out 49.3%

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

      10. distribute-lft-neg-in 49.3%

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

      11. associate-*r* 44.5%

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

      12. add-sqr-sqrt 17.5%

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

      13. sqrt-unprod 58.0%

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

      14. sqr-neg 58.0%

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

      15. sqrt-unprod 44.2%

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

      16. add-sqr-sqrt 87.4%

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

   6. Applied egg-rr 87.4%

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

   7. Taylor expanded in y around 0 99.8%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-107}:\\ \;\;\;\;x - \left(z \cdot \sqrt{x}\right) \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{+111} \lor \neg \left(y \leq 9.5 \cdot 10^{-77}\right):\\ \;\;\;\;\left(x\_m \cdot z\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (or (<= y -3.85e+111) (not (<= y 9.5e-77))) (* (* x_m z) (- y)) 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 <= -3.85e+111) || !(y <= 9.5e-77)) {
		tmp = (x_m * z) * -y;
	} 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 <= (-3.85d+111)) .or. (.not. (y <= 9.5d-77))) then
        tmp = (x_m * z) * -y
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -3.85e+111) || !(y <= 9.5e-77)) {
		tmp = (x_m * z) * -y;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -3.85e+111) or not (y <= 9.5e-77):
		tmp = (x_m * z) * -y
	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 ((y <= -3.85e+111) || !(y <= 9.5e-77))
		tmp = Float64(Float64(x_m * z) * Float64(-y));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y <= -3.85e+111) || ~((y <= 9.5e-77)))
		tmp = (x_m * z) * -y;
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[y, -3.85e+111], N[Not[LessEqual[y, 9.5e-77]], $MachinePrecision]], N[(N[(x$95$m * z), $MachinePrecision] * (-y)), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.84999999999999977e111 or 9.5000000000000005e-77 < y

   1. Initial program 91.6%

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

   3. Taylor expanded in y around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-*r* 76.6%

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

      4. distribute-rgt-neg-in 76.6%

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

   5. Simplified 76.6%

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

   if -3.84999999999999977e111 < y < 9.5000000000000005e-77

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0 71.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{+111} \lor \neg \left(y \leq 9.5 \cdot 10^{-77}\right):\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{+111} \lor \neg \left(y \leq 3.5 \cdot 10^{-154}\right):\\ \;\;\;\;x\_m \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= y -3.85e+111) (not (<= y 3.5e-154))) (* x_m (* z (- y))) 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 <= -3.85e+111) || !(y <= 3.5e-154)) {
		tmp = x_m * (z * -y);
	} 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 <= (-3.85d+111)) .or. (.not. (y <= 3.5d-154))) then
        tmp = x_m * (z * -y)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((y <= -3.85e+111) || !(y <= 3.5e-154)) {
		tmp = x_m * (z * -y);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (y <= -3.85e+111) or not (y <= 3.5e-154):
		tmp = x_m * (z * -y)
	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 ((y <= -3.85e+111) || !(y <= 3.5e-154))
		tmp = Float64(x_m * Float64(z * Float64(-y)));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((y <= -3.85e+111) || ~((y <= 3.5e-154)))
		tmp = x_m * (z * -y);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[Or[LessEqual[y, -3.85e+111], N[Not[LessEqual[y, 3.5e-154]], $MachinePrecision]], N[(x$95$m * N[(z * (-y)), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.84999999999999977e111 or 3.5000000000000001e-154 < y

   1. Initial program 92.9%

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

   3. Taylor expanded in y around inf 67.7%

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

      1. mul-1-neg 67.7%

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

      2. distribute-rgt-neg-out 67.7%

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

   5. Simplified 67.7%

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

   if -3.84999999999999977e111 < y < 3.5000000000000001e-154

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 72.5%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{+111} \lor \neg \left(y \leq 3.5 \cdot 10^{-154}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{+111}:\\ \;\;\;\;x\_m \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-154}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x\_m \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

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 -3.85e+111)
    (* x_m (* z (- y)))
    (if (<= y 3.5e-154) x_m (* 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 <= -3.85e+111) {
		tmp = x_m * (z * -y);
	} else if (y <= 3.5e-154) {
		tmp = x_m;
	} 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 <= (-3.85d+111)) then
        tmp = x_m * (z * -y)
    else if (y <= 3.5d-154) then
        tmp = x_m
    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 <= -3.85e+111) {
		tmp = x_m * (z * -y);
	} else if (y <= 3.5e-154) {
		tmp = x_m;
	} 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 <= -3.85e+111:
		tmp = x_m * (z * -y)
	elif y <= 3.5e-154:
		tmp = x_m
	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 (y <= -3.85e+111)
		tmp = Float64(x_m * Float64(z * Float64(-y)));
	elseif (y <= 3.5e-154)
		tmp = x_m;
	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 <= -3.85e+111)
		tmp = x_m * (z * -y);
	elseif (y <= 3.5e-154)
		tmp = x_m;
	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[y, -3.85e+111], N[(x$95$m * N[(z * (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-154], x$95$m, N[(z * N[(x$95$m * (-y)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.84999999999999977e111

   1. Initial program 89.6%

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

   3. Taylor expanded in y around inf 85.9%

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

      1. mul-1-neg 85.9%

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

      2. distribute-rgt-neg-out 85.9%

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

   5. Simplified 85.9%

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

   if -3.84999999999999977e111 < y < 3.5000000000000001e-154

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 72.5%

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

   if 3.5000000000000001e-154 < y

   1. Initial program 94.1%

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

      1. sub-neg 94.1%

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

      2. distribute-rgt-in 94.1%

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

      3. *-un-lft-identity 94.1%

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

      4. distribute-rgt-neg-in 94.1%

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

   4. Applied egg-rr 94.1%

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

   5. Taylor expanded in z around inf 85.5%

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

      1. +-commutative 85.5%

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

      2. mul-1-neg 85.5%

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

      3. unsub-neg 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in z around inf 61.0%

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

      1. neg-mul-1 61.0%

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

      2. *-commutative 61.0%

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

      3. distribute-rgt-neg-in 61.0%

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

   10. Simplified 61.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{+111}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot y \leq 5 \cdot 10^{+173}:\\ \;\;\;\;x\_m - x\_m \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot z\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

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 (<= (* z y) 5e+173) (- x_m (* x_m (* z y))) (* (* x_m z) (- 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 ((z * y) <= 5e+173) {
		tmp = x_m - (x_m * (z * y));
	} else {
		tmp = (x_m * z) * -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 ((z * y) <= 5d+173) then
        tmp = x_m - (x_m * (z * y))
    else
        tmp = (x_m * z) * -y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * y) <= 5e+173) {
		tmp = x_m - (x_m * (z * y));
	} else {
		tmp = (x_m * z) * -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 (z * y) <= 5e+173:
		tmp = x_m - (x_m * (z * y))
	else:
		tmp = (x_m * z) * -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(z * y) <= 5e+173)
		tmp = Float64(x_m - Float64(x_m * Float64(z * y)));
	else
		tmp = Float64(Float64(x_m * z) * 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 ((z * y) <= 5e+173)
		tmp = x_m - (x_m * (z * y));
	else
		tmp = (x_m * z) * -y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * y), $MachinePrecision], 5e+173], N[(x$95$m - N[(x$95$m * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * z), $MachinePrecision] * (-y)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < 5.00000000000000034e173

   1. Initial program 98.1%

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

      1. sub-neg 98.1%

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

      2. distribute-rgt-in 98.1%

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

      3. *-un-lft-identity 98.1%

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

      4. distribute-rgt-neg-in 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate-*l* 90.1%

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

      2. add-sqr-sqrt 50.5%

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

      3. sqrt-unprod 63.0%

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

      4. sqr-neg 63.0%

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

      5. sqrt-unprod 22.2%

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

      6. add-sqr-sqrt 52.4%

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

      7. cancel-sign-sub-inv 52.4%

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

      8. associate-*l* 54.5%

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

      9. distribute-rgt-neg-out 54.5%

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

      10. distribute-lft-neg-in 54.5%

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

      11. associate-*r* 52.4%

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

      12. add-sqr-sqrt 22.2%

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

      13. sqrt-unprod 63.0%

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

      14. sqr-neg 63.0%

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

      15. sqrt-unprod 50.5%

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

      16. add-sqr-sqrt 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in y around 0 98.1%

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

   if 5.00000000000000034e173 < (*.f64 y z)

   1. Initial program 81.1%

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

   3. Taylor expanded in y around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. *-commutative 81.1%

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

      3. associate-*r* 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq 5 \cdot 10^{+173}:\\ \;\;\;\;x - x \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot y \leq 5 \cdot 10^{+173}:\\ \;\;\;\;x\_m \cdot \left(1 - z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot z\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

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 (<= (* z y) 5e+173) (* x_m (- 1.0 (* z y))) (* (* x_m z) (- 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 ((z * y) <= 5e+173) {
		tmp = x_m * (1.0 - (z * y));
	} else {
		tmp = (x_m * z) * -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 ((z * y) <= 5d+173) then
        tmp = x_m * (1.0d0 - (z * y))
    else
        tmp = (x_m * z) * -y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z * y) <= 5e+173) {
		tmp = x_m * (1.0 - (z * y));
	} else {
		tmp = (x_m * z) * -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 (z * y) <= 5e+173:
		tmp = x_m * (1.0 - (z * y))
	else:
		tmp = (x_m * z) * -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(z * y) <= 5e+173)
		tmp = Float64(x_m * Float64(1.0 - Float64(z * y)));
	else
		tmp = Float64(Float64(x_m * z) * 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 ((z * y) <= 5e+173)
		tmp = x_m * (1.0 - (z * y));
	else
		tmp = (x_m * z) * -y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(z * y), $MachinePrecision], 5e+173], N[(x$95$m * N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * z), $MachinePrecision] * (-y)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < 5.00000000000000034e173

   1. Initial program 98.1%

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

   if 5.00000000000000034e173 < (*.f64 y z)

   1. Initial program 81.1%

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

   3. Taylor expanded in y around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. *-commutative 81.1%

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

      3. associate-*r* 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq 5 \cdot 10^{+173}:\\ \;\;\;\;x \cdot \left(1 - z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot z\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{x\_m \cdot z}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -1.2e+158) (/ (* 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 <= -1.2e+158) {
		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 <= (-1.2d+158)) 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 <= -1.2e+158) {
		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 <= -1.2e+158:
		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 (y <= -1.2e+158)
		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 <= -1.2e+158)
		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[y, -1.2e+158], N[(N[(x$95$m * z), $MachinePrecision] / z), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.20000000000000004e158

   1. Initial program 85.9%

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

      1. sub-neg 85.9%

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

      2. distribute-rgt-in 85.8%

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

      3. *-un-lft-identity 85.8%

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

      4. distribute-rgt-neg-in 85.8%

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

   4. Applied egg-rr 85.8%

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

   5. Taylor expanded in z around inf 85.5%

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

      1. +-commutative 85.5%

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

      2. mul-1-neg 85.5%

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

      3. unsub-neg 85.5%

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

   7. Simplified 85.5%

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

   8. Taylor expanded in z around 0 3.3%

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

      1. associate-*r/ 13.7%

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

      2. *-commutative 13.7%

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

   10. Applied egg-rr 13.7%

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

   if -1.20000000000000004e158 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 53.9%

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

Alternative 8: 52.0% 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 95.8%

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

3. Taylor expanded in y around 0 48.6%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))