Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E

Percentage Accurate: 99.7% → 99.8%

Time: 9.4s

Alternatives: 11

Speedup: 1.0×

• 20.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (* (- y x) 6.0) z)))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * 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 + (((y - x) * 6.0d0) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - x\right) \cdot 6\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (* (- y x) 6.0) z)))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * 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 + (((y - x) * 6.0d0) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

Python

def code(x, y, z):
	return x + (((y - x) * 6.0) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (((y - x) * 6.0) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (* (- y x) z) 6.0 x))

C

double code(double x, double y, double z) {
	return fma(((y - x) * z), 6.0, x);
}

Julia

function code(x, y, z)
	return fma(Float64(Float64(y - x) * z), 6.0, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] * 6.0 + x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 99.8%

      \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

   2. +-commutative 99.8%

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

   3. *-commutative 99.8%

      \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

   4. associate-*r* 99.8%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

   5. fma-define 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right) \]
6. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, \left(x - y\right) \cdot -6, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z (* (- x y) -6.0) x))

C

double code(double x, double y, double z) {
	return fma(z, ((x - y) * -6.0), x);
}

Julia

function code(x, y, z)
	return fma(z, Float64(Float64(x - y) * -6.0), x)
end

Wolfram

code[x_, y_, z_] := N[(z * N[(N[(x - y), $MachinePrecision] * -6.0), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

   2. *-commutative 99.8%

      \[\leadsto \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

   3. fma-define 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \left(y - x\right) \cdot 6, x\right)} \]

   4. remove-double-neg 99.8%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{-\left(-\left(y - x\right) \cdot 6\right)}, x\right) \]

   5. distribute-lft-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(z, -\color{blue}{\left(-\left(y - x\right)\right) \cdot 6}, x\right) \]

   6. distribute-rgt-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(-\left(y - x\right)\right) \cdot \left(-6\right)}, x\right) \]

   7. sub-neg 99.8%

      \[\leadsto \mathsf{fma}\left(z, \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-6\right), x\right) \]

   8. distribute-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(\left(-y\right) + \left(-\left(-x\right)\right)\right)} \cdot \left(-6\right), x\right) \]

   9. remove-double-neg 99.8%

      \[\leadsto \mathsf{fma}\left(z, \left(\left(-y\right) + \color{blue}{x}\right) \cdot \left(-6\right), x\right) \]

   10. +-commutative 99.8%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(x + \left(-y\right)\right)} \cdot \left(-6\right), x\right) \]

   11. sub-neg 99.8%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(x - y\right)} \cdot \left(-6\right), x\right) \]

   12. metadata-eval 99.8%

       \[\leadsto \mathsf{fma}\left(z, \left(x - y\right) \cdot \color{blue}{-6}, x\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \left(x - y\right) \cdot -6, x\right)} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(z, \left(x - y\right) \cdot -6, x\right) \]
6. Add Preprocessing

Alternative 3: 61.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ t_1 := x \cdot \left(z \cdot \left(-6\right)\right)\\ \mathbf{if}\;z \leq -5.9 \cdot 10^{+203}:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -12.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+251}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* y z))) (t_1 (* x (* z (- 6.0)))))
   (if (<= z -5.9e+203)
     (* y (* z 6.0))
     (if (<= z -6.2e+153)
       t_1
       (if (<= z -3.9e+53)
         t_0
         (if (<= z -2.8e+17)
           t_1
           (if (<= z -12.5)
             t_0
             (if (<= z 9.5e-36) x (if (<= z 1.45e+251) t_0 t_1)))))))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (y * z);
	double t_1 = x * (z * -6.0);
	double tmp;
	if (z <= -5.9e+203) {
		tmp = y * (z * 6.0);
	} else if (z <= -6.2e+153) {
		tmp = t_1;
	} else if (z <= -3.9e+53) {
		tmp = t_0;
	} else if (z <= -2.8e+17) {
		tmp = t_1;
	} else if (z <= -12.5) {
		tmp = t_0;
	} else if (z <= 9.5e-36) {
		tmp = x;
	} else if (z <= 1.45e+251) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 6.0d0 * (y * z)
    t_1 = x * (z * -6.0d0)
    if (z <= (-5.9d+203)) then
        tmp = y * (z * 6.0d0)
    else if (z <= (-6.2d+153)) then
        tmp = t_1
    else if (z <= (-3.9d+53)) then
        tmp = t_0
    else if (z <= (-2.8d+17)) then
        tmp = t_1
    else if (z <= (-12.5d0)) then
        tmp = t_0
    else if (z <= 9.5d-36) then
        tmp = x
    else if (z <= 1.45d+251) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 6.0 * (y * z);
	double t_1 = x * (z * -6.0);
	double tmp;
	if (z <= -5.9e+203) {
		tmp = y * (z * 6.0);
	} else if (z <= -6.2e+153) {
		tmp = t_1;
	} else if (z <= -3.9e+53) {
		tmp = t_0;
	} else if (z <= -2.8e+17) {
		tmp = t_1;
	} else if (z <= -12.5) {
		tmp = t_0;
	} else if (z <= 9.5e-36) {
		tmp = x;
	} else if (z <= 1.45e+251) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 6.0 * (y * z)
	t_1 = x * (z * -6.0)
	tmp = 0
	if z <= -5.9e+203:
		tmp = y * (z * 6.0)
	elif z <= -6.2e+153:
		tmp = t_1
	elif z <= -3.9e+53:
		tmp = t_0
	elif z <= -2.8e+17:
		tmp = t_1
	elif z <= -12.5:
		tmp = t_0
	elif z <= 9.5e-36:
		tmp = x
	elif z <= 1.45e+251:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(y * z))
	t_1 = Float64(x * Float64(z * Float64(-6.0)))
	tmp = 0.0
	if (z <= -5.9e+203)
		tmp = Float64(y * Float64(z * 6.0));
	elseif (z <= -6.2e+153)
		tmp = t_1;
	elseif (z <= -3.9e+53)
		tmp = t_0;
	elseif (z <= -2.8e+17)
		tmp = t_1;
	elseif (z <= -12.5)
		tmp = t_0;
	elseif (z <= 9.5e-36)
		tmp = x;
	elseif (z <= 1.45e+251)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 6.0 * (y * z);
	t_1 = x * (z * -6.0);
	tmp = 0.0;
	if (z <= -5.9e+203)
		tmp = y * (z * 6.0);
	elseif (z <= -6.2e+153)
		tmp = t_1;
	elseif (z <= -3.9e+53)
		tmp = t_0;
	elseif (z <= -2.8e+17)
		tmp = t_1;
	elseif (z <= -12.5)
		tmp = t_0;
	elseif (z <= 9.5e-36)
		tmp = x;
	elseif (z <= 1.45e+251)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(z * (-6.0)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.9e+203], N[(y * N[(z * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -6.2e+153], t$95$1, If[LessEqual[z, -3.9e+53], t$95$0, If[LessEqual[z, -2.8e+17], t$95$1, If[LessEqual[z, -12.5], t$95$0, If[LessEqual[z, 9.5e-36], x, If[LessEqual[z, 1.45e+251], t$95$0, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.89999999999999972e203

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.9%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in y around inf 73.5%

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

      1. *-commutative 73.5%

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. associate-*r* 73.5%

         \[\leadsto \color{blue}{y \cdot \left(z \cdot 6\right)} \]

   7. Simplified 73.5%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot 6\right)} \]

   if -5.89999999999999972e203 < z < -6.2e153 or -3.89999999999999976e53 < z < -2.8e17 or 1.45e251 < z

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.7%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.7%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]

   6. Taylor expanded in y around 0 83.0%

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

      1. mul-1-neg 83.0%

         \[\leadsto 6 \cdot \color{blue}{\left(-x \cdot z\right)} \]

      2. distribute-lft-neg-out 83.0%

         \[\leadsto 6 \cdot \color{blue}{\left(\left(-x\right) \cdot z\right)} \]

      3. *-commutative 83.0%

         \[\leadsto 6 \cdot \color{blue}{\left(z \cdot \left(-x\right)\right)} \]

   8. Simplified 83.0%

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

      1. *-commutative 83.0%

         \[\leadsto \color{blue}{\left(z \cdot \left(-x\right)\right) \cdot 6} \]

      2. distribute-rgt-neg-out 83.0%

         \[\leadsto \color{blue}{\left(-z \cdot x\right)} \cdot 6 \]

      3. distribute-lft-neg-out 83.0%

         \[\leadsto \color{blue}{-\left(z \cdot x\right) \cdot 6} \]

      4. *-commutative 83.0%

         \[\leadsto -\color{blue}{\left(x \cdot z\right)} \cdot 6 \]

      5. add-sqr-sqrt 30.2%

         \[\leadsto -\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot z\right) \cdot 6 \]

      6. sqrt-unprod 24.7%

         \[\leadsto -\left(\color{blue}{\sqrt{x \cdot x}} \cdot z\right) \cdot 6 \]

      7. sqr-neg 24.7%

         \[\leadsto -\left(\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \cdot z\right) \cdot 6 \]

      8. sqrt-unprod 0.6%

         \[\leadsto -\left(\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot z\right) \cdot 6 \]

      9. add-sqr-sqrt 4.0%

         \[\leadsto -\left(\color{blue}{\left(-x\right)} \cdot z\right) \cdot 6 \]

      10. associate-*l* 4.0%

          \[\leadsto -\color{blue}{\left(-x\right) \cdot \left(z \cdot 6\right)} \]

      11. add-sqr-sqrt 0.6%

          \[\leadsto -\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \left(z \cdot 6\right) \]

      12. sqrt-unprod 24.7%

          \[\leadsto -\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \left(z \cdot 6\right) \]

      13. sqr-neg 24.7%

          \[\leadsto -\sqrt{\color{blue}{x \cdot x}} \cdot \left(z \cdot 6\right) \]

      14. sqrt-unprod 30.2%

          \[\leadsto -\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(z \cdot 6\right) \]

      15. add-sqr-sqrt 83.1%

          \[\leadsto -\color{blue}{x} \cdot \left(z \cdot 6\right) \]

      16. *-commutative 83.1%

          \[\leadsto -x \cdot \color{blue}{\left(6 \cdot z\right)} \]

   10. Applied egg-rr 83.1%

       \[\leadsto \color{blue}{-x \cdot \left(6 \cdot z\right)} \]

   if -6.2e153 < z < -3.89999999999999976e53 or -2.8e17 < z < -12.5 or 9.5000000000000003e-36 < z < 1.45e251

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.6%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.6%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in y around inf 61.0%

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

      1. *-commutative 61.0%

         \[\leadsto 6 \cdot \color{blue}{\left(z \cdot y\right)} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot y\right)} \]

   if -12.5 < z < 9.5000000000000003e-36

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 76.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+203}:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-6\right)\right)\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+53}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-6\right)\right)\\ \mathbf{elif}\;z \leq -12.5:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+251}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-6\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(y \cdot z\right)\\ t_1 := x \cdot \left(z \cdot \left(-6\right)\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+206}:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+154}:\\ \;\;\;\;6 \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -12.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+249}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* y z))) (t_1 (* x (* z (- 6.0)))))
   (if (<= z -1.1e+206)
     (* y (* z 6.0))
     (if (<= z -1.45e+154)
       (* 6.0 (* x (- z)))
       (if (<= z -1.4e+53)
         t_0
         (if (<= z -3.4e+20)
           t_1
           (if (<= z -12.5)
             t_0
             (if (<= z 1.05e-35) x (if (<= z 2.7e+249) t_0 t_1)))))))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (y * z);
	double t_1 = x * (z * -6.0);
	double tmp;
	if (z <= -1.1e+206) {
		tmp = y * (z * 6.0);
	} else if (z <= -1.45e+154) {
		tmp = 6.0 * (x * -z);
	} else if (z <= -1.4e+53) {
		tmp = t_0;
	} else if (z <= -3.4e+20) {
		tmp = t_1;
	} else if (z <= -12.5) {
		tmp = t_0;
	} else if (z <= 1.05e-35) {
		tmp = x;
	} else if (z <= 2.7e+249) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 6.0d0 * (y * z)
    t_1 = x * (z * -6.0d0)
    if (z <= (-1.1d+206)) then
        tmp = y * (z * 6.0d0)
    else if (z <= (-1.45d+154)) then
        tmp = 6.0d0 * (x * -z)
    else if (z <= (-1.4d+53)) then
        tmp = t_0
    else if (z <= (-3.4d+20)) then
        tmp = t_1
    else if (z <= (-12.5d0)) then
        tmp = t_0
    else if (z <= 1.05d-35) then
        tmp = x
    else if (z <= 2.7d+249) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 6.0 * (y * z);
	double t_1 = x * (z * -6.0);
	double tmp;
	if (z <= -1.1e+206) {
		tmp = y * (z * 6.0);
	} else if (z <= -1.45e+154) {
		tmp = 6.0 * (x * -z);
	} else if (z <= -1.4e+53) {
		tmp = t_0;
	} else if (z <= -3.4e+20) {
		tmp = t_1;
	} else if (z <= -12.5) {
		tmp = t_0;
	} else if (z <= 1.05e-35) {
		tmp = x;
	} else if (z <= 2.7e+249) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 6.0 * (y * z)
	t_1 = x * (z * -6.0)
	tmp = 0
	if z <= -1.1e+206:
		tmp = y * (z * 6.0)
	elif z <= -1.45e+154:
		tmp = 6.0 * (x * -z)
	elif z <= -1.4e+53:
		tmp = t_0
	elif z <= -3.4e+20:
		tmp = t_1
	elif z <= -12.5:
		tmp = t_0
	elif z <= 1.05e-35:
		tmp = x
	elif z <= 2.7e+249:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(y * z))
	t_1 = Float64(x * Float64(z * Float64(-6.0)))
	tmp = 0.0
	if (z <= -1.1e+206)
		tmp = Float64(y * Float64(z * 6.0));
	elseif (z <= -1.45e+154)
		tmp = Float64(6.0 * Float64(x * Float64(-z)));
	elseif (z <= -1.4e+53)
		tmp = t_0;
	elseif (z <= -3.4e+20)
		tmp = t_1;
	elseif (z <= -12.5)
		tmp = t_0;
	elseif (z <= 1.05e-35)
		tmp = x;
	elseif (z <= 2.7e+249)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 6.0 * (y * z);
	t_1 = x * (z * -6.0);
	tmp = 0.0;
	if (z <= -1.1e+206)
		tmp = y * (z * 6.0);
	elseif (z <= -1.45e+154)
		tmp = 6.0 * (x * -z);
	elseif (z <= -1.4e+53)
		tmp = t_0;
	elseif (z <= -3.4e+20)
		tmp = t_1;
	elseif (z <= -12.5)
		tmp = t_0;
	elseif (z <= 1.05e-35)
		tmp = x;
	elseif (z <= 2.7e+249)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(z * (-6.0)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+206], N[(y * N[(z * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.45e+154], N[(6.0 * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.4e+53], t$95$0, If[LessEqual[z, -3.4e+20], t$95$1, If[LessEqual[z, -12.5], t$95$0, If[LessEqual[z, 1.05e-35], x, If[LessEqual[z, 2.7e+249], t$95$0, t$95$1]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.10000000000000001e206

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.9%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in y around inf 73.5%

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

      1. *-commutative 73.5%

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. associate-*r* 73.5%

         \[\leadsto \color{blue}{y \cdot \left(z \cdot 6\right)} \]

   7. Simplified 73.5%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot 6\right)} \]

   if -1.10000000000000001e206 < z < -1.4499999999999999e154

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.6%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.6%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]

   6. Taylor expanded in y around 0 85.6%

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

      1. mul-1-neg 85.6%

         \[\leadsto 6 \cdot \color{blue}{\left(-x \cdot z\right)} \]

      2. distribute-lft-neg-out 85.6%

         \[\leadsto 6 \cdot \color{blue}{\left(\left(-x\right) \cdot z\right)} \]

      3. *-commutative 85.6%

         \[\leadsto 6 \cdot \color{blue}{\left(z \cdot \left(-x\right)\right)} \]

   8. Simplified 85.6%

      \[\leadsto 6 \cdot \color{blue}{\left(z \cdot \left(-x\right)\right)} \]

   if -1.4499999999999999e154 < z < -1.4e53 or -3.4e20 < z < -12.5 or 1.05e-35 < z < 2.70000000000000018e249

   1. Initial program 99.5%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.6%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.6%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in y around inf 61.0%

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

      1. *-commutative 61.0%

         \[\leadsto 6 \cdot \color{blue}{\left(z \cdot y\right)} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot y\right)} \]

   if -1.4e53 < z < -3.4e20 or 2.70000000000000018e249 < z

   1. Initial program 99.8%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.7%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.7%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.6%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in z around inf 99.6%

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot \left(y - x\right)\right)} \]

   6. Taylor expanded in y around 0 82.3%

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

      1. mul-1-neg 82.3%

         \[\leadsto 6 \cdot \color{blue}{\left(-x \cdot z\right)} \]

      2. distribute-lft-neg-out 82.3%

         \[\leadsto 6 \cdot \color{blue}{\left(\left(-x\right) \cdot z\right)} \]

      3. *-commutative 82.3%

         \[\leadsto 6 \cdot \color{blue}{\left(z \cdot \left(-x\right)\right)} \]

   8. Simplified 82.3%

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

      1. *-commutative 82.3%

         \[\leadsto \color{blue}{\left(z \cdot \left(-x\right)\right) \cdot 6} \]

      2. distribute-rgt-neg-out 82.3%

         \[\leadsto \color{blue}{\left(-z \cdot x\right)} \cdot 6 \]

      3. distribute-lft-neg-out 82.3%

         \[\leadsto \color{blue}{-\left(z \cdot x\right) \cdot 6} \]

      4. *-commutative 82.3%

         \[\leadsto -\color{blue}{\left(x \cdot z\right)} \cdot 6 \]

      5. add-sqr-sqrt 30.6%

         \[\leadsto -\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot z\right) \cdot 6 \]

      6. sqrt-unprod 23.5%

         \[\leadsto -\left(\color{blue}{\sqrt{x \cdot x}} \cdot z\right) \cdot 6 \]

      7. sqr-neg 23.5%

         \[\leadsto -\left(\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \cdot z\right) \cdot 6 \]

      8. sqrt-unprod 0.6%

         \[\leadsto -\left(\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot z\right) \cdot 6 \]

      9. add-sqr-sqrt 4.8%

         \[\leadsto -\left(\color{blue}{\left(-x\right)} \cdot z\right) \cdot 6 \]

      10. associate-*l* 4.8%

          \[\leadsto -\color{blue}{\left(-x\right) \cdot \left(z \cdot 6\right)} \]

      11. add-sqr-sqrt 0.6%

          \[\leadsto -\color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \left(z \cdot 6\right) \]

      12. sqrt-unprod 23.6%

          \[\leadsto -\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \left(z \cdot 6\right) \]

      13. sqr-neg 23.6%

          \[\leadsto -\sqrt{\color{blue}{x \cdot x}} \cdot \left(z \cdot 6\right) \]

      14. sqrt-unprod 30.6%

          \[\leadsto -\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(z \cdot 6\right) \]

      15. add-sqr-sqrt 82.5%

          \[\leadsto -\color{blue}{x} \cdot \left(z \cdot 6\right) \]

      16. *-commutative 82.5%

          \[\leadsto -x \cdot \color{blue}{\left(6 \cdot z\right)} \]

   10. Applied egg-rr 82.5%

       \[\leadsto \color{blue}{-x \cdot \left(6 \cdot z\right)} \]

   if -12.5 < z < 1.05e-35

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 76.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+206}:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+154}:\\ \;\;\;\;6 \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+53}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-6\right)\right)\\ \mathbf{elif}\;z \leq -12.5:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+249}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-6\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-27} \lor \neg \left(z \leq 1.05 \cdot 10^{-35}\right):\\ \;\;\;\;-6 \cdot \left(z \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6e-27) (not (<= z 1.05e-35))) (* -6.0 (* z (- x y))) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6e-27) || !(z <= 1.05e-35)) {
		tmp = -6.0 * (z * (x - y));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6d-27)) .or. (.not. (z <= 1.05d-35))) then
        tmp = (-6.0d0) * (z * (x - y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6e-27) || !(z <= 1.05e-35)) {
		tmp = -6.0 * (z * (x - y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6e-27) or not (z <= 1.05e-35):
		tmp = -6.0 * (z * (x - y))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6e-27) || !(z <= 1.05e-35))
		tmp = Float64(-6.0 * Float64(z * Float64(x - y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6e-27) || ~((z <= 1.05e-35)))
		tmp = -6.0 * (z * (x - y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6e-27], N[Not[LessEqual[z, 1.05e-35]], $MachinePrecision]], N[(-6.0 * N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -6.0000000000000002e-27 or 1.05e-35 < z

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

      2. *-commutative 99.6%

         \[\leadsto \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      3. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \left(y - x\right) \cdot 6, x\right)} \]

      4. remove-double-neg 99.7%

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{-\left(-\left(y - x\right) \cdot 6\right)}, x\right) \]

      5. distribute-lft-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(z, -\color{blue}{\left(-\left(y - x\right)\right) \cdot 6}, x\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(-\left(y - x\right)\right) \cdot \left(-6\right)}, x\right) \]

      7. sub-neg 99.7%

         \[\leadsto \mathsf{fma}\left(z, \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-6\right), x\right) \]

      8. distribute-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(\left(-y\right) + \left(-\left(-x\right)\right)\right)} \cdot \left(-6\right), x\right) \]

      9. remove-double-neg 99.7%

         \[\leadsto \mathsf{fma}\left(z, \left(\left(-y\right) + \color{blue}{x}\right) \cdot \left(-6\right), x\right) \]

      10. +-commutative 99.7%

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(x + \left(-y\right)\right)} \cdot \left(-6\right), x\right) \]

      11. sub-neg 99.7%

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(x - y\right)} \cdot \left(-6\right), x\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(z, \left(x - y\right) \cdot \color{blue}{-6}, x\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \left(x - y\right) \cdot -6, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 95.3%

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(x - y\right)\right)} \]

   if -6.0000000000000002e-27 < z < 1.05e-35

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 77.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-27} \lor \neg \left(z \leq 1.05 \cdot 10^{-35}\right):\\ \;\;\;\;-6 \cdot \left(z \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.165 \lor \neg \left(z \leq 0.17\right):\\ \;\;\;\;-6 \cdot \left(z \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + 6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -0.165) (not (<= z 0.17)))
   (* -6.0 (* z (- x y)))
   (+ x (* 6.0 (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.165) || !(z <= 0.17)) {
		tmp = -6.0 * (z * (x - y));
	} else {
		tmp = x + (6.0 * (y * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-0.165d0)) .or. (.not. (z <= 0.17d0))) then
        tmp = (-6.0d0) * (z * (x - y))
    else
        tmp = x + (6.0d0 * (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -0.165) || !(z <= 0.17)) {
		tmp = -6.0 * (z * (x - y));
	} else {
		tmp = x + (6.0 * (y * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -0.165) or not (z <= 0.17):
		tmp = -6.0 * (z * (x - y))
	else:
		tmp = x + (6.0 * (y * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -0.165) || !(z <= 0.17))
		tmp = Float64(-6.0 * Float64(z * Float64(x - y)));
	else
		tmp = Float64(x + Float64(6.0 * Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -0.165) || ~((z <= 0.17)))
		tmp = -6.0 * (z * (x - y));
	else
		tmp = x + (6.0 * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -0.165], N[Not[LessEqual[z, 0.17]], $MachinePrecision]], N[(-6.0 * N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.165000000000000008 or 0.170000000000000012 < z

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot 6\right) \cdot z + x} \]

      2. *-commutative 99.7%

         \[\leadsto \color{blue}{z \cdot \left(\left(y - x\right) \cdot 6\right)} + x \]

      3. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \left(y - x\right) \cdot 6, x\right)} \]

      4. remove-double-neg 99.7%

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{-\left(-\left(y - x\right) \cdot 6\right)}, x\right) \]

      5. distribute-lft-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(z, -\color{blue}{\left(-\left(y - x\right)\right) \cdot 6}, x\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(-\left(y - x\right)\right) \cdot \left(-6\right)}, x\right) \]

      7. sub-neg 99.7%

         \[\leadsto \mathsf{fma}\left(z, \left(-\color{blue}{\left(y + \left(-x\right)\right)}\right) \cdot \left(-6\right), x\right) \]

      8. distribute-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(\left(-y\right) + \left(-\left(-x\right)\right)\right)} \cdot \left(-6\right), x\right) \]

      9. remove-double-neg 99.7%

         \[\leadsto \mathsf{fma}\left(z, \left(\left(-y\right) + \color{blue}{x}\right) \cdot \left(-6\right), x\right) \]

      10. +-commutative 99.7%

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(x + \left(-y\right)\right)} \cdot \left(-6\right), x\right) \]

      11. sub-neg 99.7%

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(x - y\right)} \cdot \left(-6\right), x\right) \]

      12. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(z, \left(x - y\right) \cdot \color{blue}{-6}, x\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \left(x - y\right) \cdot -6, x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 97.9%

      \[\leadsto \color{blue}{-6 \cdot \left(z \cdot \left(x - y\right)\right)} \]

   if -0.165000000000000008 < z < 0.170000000000000012

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.2%

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

      1. *-commutative 99.2%

         \[\leadsto x + 6 \cdot \color{blue}{\left(z \cdot y\right)} \]

   5. Simplified 99.2%

      \[\leadsto x + \color{blue}{6 \cdot \left(z \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.165 \lor \neg \left(z \leq 0.17\right):\\ \;\;\;\;-6 \cdot \left(z \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + 6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -12.5 \lor \neg \left(z \leq 2.15 \cdot 10^{-39}\right):\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -12.5) (not (<= z 2.15e-39))) (* 6.0 (* y z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -12.5) || !(z <= 2.15e-39)) {
		tmp = 6.0 * (y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-12.5d0)) .or. (.not. (z <= 2.15d-39))) then
        tmp = 6.0d0 * (y * z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -12.5) || !(z <= 2.15e-39)) {
		tmp = 6.0 * (y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -12.5) or not (z <= 2.15e-39):
		tmp = 6.0 * (y * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -12.5) || !(z <= 2.15e-39))
		tmp = Float64(6.0 * Float64(y * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -12.5) || ~((z <= 2.15e-39)))
		tmp = 6.0 * (y * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -12.5], N[Not[LessEqual[z, 2.15e-39]], $MachinePrecision]], N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -12.5 or 2.15e-39 < z

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.7%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.7%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in y around inf 56.3%

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

      1. *-commutative 56.3%

         \[\leadsto 6 \cdot \color{blue}{\left(z \cdot y\right)} \]

   7. Simplified 56.3%

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot y\right)} \]

   if -12.5 < z < 2.15e-39

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 76.7%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -12.5 \lor \neg \left(z \leq 2.15 \cdot 10^{-39}\right):\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -12.5:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -12.5) (* y (* z 6.0)) (if (<= z 3.7e-37) x (* 6.0 (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -12.5) {
		tmp = y * (z * 6.0);
	} else if (z <= 3.7e-37) {
		tmp = x;
	} else {
		tmp = 6.0 * (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-12.5d0)) then
        tmp = y * (z * 6.0d0)
    else if (z <= 3.7d-37) then
        tmp = x
    else
        tmp = 6.0d0 * (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -12.5) {
		tmp = y * (z * 6.0);
	} else if (z <= 3.7e-37) {
		tmp = x;
	} else {
		tmp = 6.0 * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -12.5:
		tmp = y * (z * 6.0)
	elif z <= 3.7e-37:
		tmp = x
	else:
		tmp = 6.0 * (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -12.5)
		tmp = Float64(y * Float64(z * 6.0));
	elseif (z <= 3.7e-37)
		tmp = x;
	else
		tmp = Float64(6.0 * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -12.5)
		tmp = y * (z * 6.0);
	elseif (z <= 3.7e-37)
		tmp = x;
	else
		tmp = 6.0 * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -12.5], N[(y * N[(z * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.7e-37], x, N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -12.5

   1. Initial program 99.7%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.8%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.8%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.7%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in y around inf 58.4%

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

      1. *-commutative 58.4%

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot 6} \]

      2. associate-*r* 58.4%

         \[\leadsto \color{blue}{y \cdot \left(z \cdot 6\right)} \]

   7. Simplified 58.4%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot 6\right)} \]

   if -12.5 < z < 3.7e-37

   1. Initial program 99.9%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 76.7%

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

   if 3.7e-37 < z

   1. Initial program 99.6%

      \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* 99.5%

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

      2. +-commutative 99.5%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right) + x} \]

      3. *-commutative 99.5%

         \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(z \cdot 6\right)} + x \]

      4. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot z\right) \cdot 6} + x \]

      5. fma-define 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - x\right) \cdot z, 6, x\right)} \]

   5. Taylor expanded in y around inf 54.2%

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

      1. *-commutative 54.2%

         \[\leadsto 6 \cdot \color{blue}{\left(z \cdot y\right)} \]

   7. Simplified 54.2%

      \[\leadsto \color{blue}{6 \cdot \left(z \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -12.5:\\ \;\;\;\;y \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + z \cdot \left(\left(y - x\right) \cdot 6\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* z (* (- y x) 6.0))))

C

double code(double x, double y, double z) {
	return x + (z * ((y - x) * 6.0));
}

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 + (z * ((y - x) * 6.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + (z * ((y - x) * 6.0));
}

Python

def code(x, y, z):
	return x + (z * ((y - x) * 6.0))

Julia

function code(x, y, z)
	return Float64(x + Float64(z * Float64(Float64(y - x) * 6.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (z * ((y - x) * 6.0));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(z * N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto x + z \cdot \left(\left(y - x\right) \cdot 6\right) \]
4. Add Preprocessing

Alternative 10: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y - x\right) \cdot \left(z \cdot 6\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) (* z 6.0))))

C

double code(double x, double y, double z) {
	return x + ((y - x) * (z * 6.0));
}

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 + ((y - x) * (z * 6.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y - x) * (z * 6.0));
}

Python

def code(x, y, z):
	return x + ((y - x) * (z * 6.0))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) * Float64(z * 6.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y - x) * (z * 6.0));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - x), $MachinePrecision] * N[(z * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x + \left(y - x\right) \cdot \left(6 \cdot z\right)} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto x + \left(y - x\right) \cdot \left(z \cdot 6\right) \]
6. Add Preprocessing

Alternative 11: 36.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

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

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
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in z around 0 38.0%

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

4. Final simplification 38.0%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \left(6 \cdot z\right) \cdot \left(x - y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* (* 6.0 z) (- x y))))

C

double code(double x, double y, double z) {
	return x - ((6.0 * z) * (x - y));
}

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 - ((6.0d0 * z) * (x - y))
end function

Java

public static double code(double x, double y, double z) {
	return x - ((6.0 * z) * (x - y));
}

Python

def code(x, y, z):
	return x - ((6.0 * z) * (x - y))

Julia

function code(x, y, z)
	return Float64(x - Float64(Float64(6.0 * z) * Float64(x - y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - ((6.0 * z) * (x - y));
end

Wolfram

code[x_, y_, z_] := N[(x - N[(N[(6.0 * z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024079 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :alt
  (- x (* (* 6.0 z) (- x y)))

  (+ x (* (* (- y x) 6.0) z)))