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

Percentage Accurate: 99.7% → 99.7%

Time: 10.5s

Alternatives: 16

Speedup: 1.1×

• 20.9% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. neg-mul-1 N/A

      \[\leadsto x + \mathsf{fma}\left(y, 6, 6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z \]

   7. associate-*r* N/A

      \[\leadsto x + \mathsf{fma}\left(y, 6, \color{blue}{\left(6 \cdot -1\right) \cdot x}\right) \cdot z \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 2: 60.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-6 \cdot z\right)\\ \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* -6.0 z))))
   (if (<= z -0.17)
     t_0
     (if (<= z 4.9e-61) x (if (<= z 5.9e+60) (* y (* 6.0 z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (-6.0 * z);
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 4.9e-61) {
		tmp = x;
	} else if (z <= 5.9e+60) {
		tmp = y * (6.0 * z);
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = x * ((-6.0d0) * z)
    if (z <= (-0.17d0)) then
        tmp = t_0
    else if (z <= 4.9d-61) then
        tmp = x
    else if (z <= 5.9d+60) then
        tmp = y * (6.0d0 * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (-6.0 * z);
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 4.9e-61) {
		tmp = x;
	} else if (z <= 5.9e+60) {
		tmp = y * (6.0 * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (-6.0 * z)
	tmp = 0
	if z <= -0.17:
		tmp = t_0
	elif z <= 4.9e-61:
		tmp = x
	elif z <= 5.9e+60:
		tmp = y * (6.0 * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-6.0 * z))
	tmp = 0.0
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 4.9e-61)
		tmp = x;
	elseif (z <= 5.9e+60)
		tmp = Float64(y * Float64(6.0 * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (-6.0 * z);
	tmp = 0.0;
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 4.9e-61)
		tmp = x;
	elseif (z <= 5.9e+60)
		tmp = y * (6.0 * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(-6.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.17], t$95$0, If[LessEqual[z, 4.9e-61], x, If[LessEqual[z, 5.9e+60], N[(y * N[(6.0 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.170000000000000012 or 5.9000000000000002e60 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\left(-1 \cdot -6\right)} \cdot \left(y \cdot z\right) - \left(6 \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{-1 \cdot \left(-6 \cdot \left(y \cdot z\right)\right)} - \left(6 \cdot z\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \left(-6 \cdot \color{blue}{\left(z \cdot y\right)}\right) - \left(6 \cdot z\right) \cdot x \]

      8. associate-*r* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(-6 \cdot z\right) \cdot y\right)} - \left(6 \cdot z\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(y \cdot \left(-6 \cdot z\right)\right)} - \left(6 \cdot z\right) \cdot x \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right)} - \left(6 \cdot z\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{x \cdot \left(6 \cdot z\right)} \]

      12. metadata-eval N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right)} \cdot z\right) \]

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

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - x \cdot \color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(-6 \cdot z\right)\right)\right)} \]

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

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-6 \cdot z\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(-1 \cdot x\right)} \cdot \left(-6 \cdot z\right) \]

      17. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot \left(-1 \cdot y - -1 \cdot x\right)} \]

      18. distribute-lft-out-- N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \]

      19. neg-mul-1 N/A

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

      20. neg-sub0 N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(0 - \left(y - x\right)\right)} \]

      21. associate-+l- N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(\left(0 - y\right) + x\right)} \]

      22. neg-sub0 N/A

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

      23. mul-1-neg N/A

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

      24. *-lft-identity N/A

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

      25. *-inverses N/A

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

      26. associate-*l/ N/A

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

      27. associate-*r/ N/A

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

      28. associate-*r/ N/A

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

      29. *-rgt-identity N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 64.3%

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

   if -0.170000000000000012 < z < 4.90000000000000002e-61

   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

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

   5. Simplified 77.5%

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

   if 4.90000000000000002e-61 < z < 5.9000000000000002e60

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, 6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, \color{blue}{\left(6 \cdot -1\right) \cdot x}\right) \cdot z \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval 99.6

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 63.6

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

   9. Simplified 63.6%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;x \cdot \left(-6 \cdot z\right)\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-6 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-6 \cdot x\right) \cdot z\\ \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* -6.0 x) z)))
   (if (<= z -0.17)
     t_0
     (if (<= z 2.7e-60) x (if (<= z 5.9e+60) (* y (* 6.0 z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-6.0 * x) * z;
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 2.7e-60) {
		tmp = x;
	} else if (z <= 5.9e+60) {
		tmp = y * (6.0 * z);
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = ((-6.0d0) * x) * z
    if (z <= (-0.17d0)) then
        tmp = t_0
    else if (z <= 2.7d-60) then
        tmp = x
    else if (z <= 5.9d+60) then
        tmp = y * (6.0d0 * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-6.0 * x) * z;
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 2.7e-60) {
		tmp = x;
	} else if (z <= 5.9e+60) {
		tmp = y * (6.0 * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-6.0 * x) * z
	tmp = 0
	if z <= -0.17:
		tmp = t_0
	elif z <= 2.7e-60:
		tmp = x
	elif z <= 5.9e+60:
		tmp = y * (6.0 * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-6.0 * x) * z)
	tmp = 0.0
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 2.7e-60)
		tmp = x;
	elseif (z <= 5.9e+60)
		tmp = Float64(y * Float64(6.0 * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-6.0 * x) * z;
	tmp = 0.0;
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 2.7e-60)
		tmp = x;
	elseif (z <= 5.9e+60)
		tmp = y * (6.0 * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -0.17], t$95$0, If[LessEqual[z, 2.7e-60], x, If[LessEqual[z, 5.9e+60], N[(y * N[(6.0 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.170000000000000012 or 5.9000000000000002e60 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot -6\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(-6 \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 64.3

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

   5. Simplified 64.3%

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

   6. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 64.1

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

   8. Simplified 64.1%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 64.2

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

   10. Applied egg-rr 64.2%

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

   if -0.170000000000000012 < z < 2.7e-60

   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

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

   5. Simplified 77.5%

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

   if 2.7e-60 < z < 5.9000000000000002e60

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, 6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, \color{blue}{\left(6 \cdot -1\right) \cdot x}\right) \cdot z \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval 99.6

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 63.6

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

   9. Simplified 63.6%

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

Alternative 4: 60.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -6.0 (* x z))))
   (if (<= z -0.17)
     t_0
     (if (<= z 2.6e-60) x (if (<= z 5.9e+60) (* y (* 6.0 z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -6.0 * (x * z);
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 2.6e-60) {
		tmp = x;
	} else if (z <= 5.9e+60) {
		tmp = y * (6.0 * z);
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = (-6.0d0) * (x * z)
    if (z <= (-0.17d0)) then
        tmp = t_0
    else if (z <= 2.6d-60) then
        tmp = x
    else if (z <= 5.9d+60) then
        tmp = y * (6.0d0 * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -6.0 * (x * z);
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 2.6e-60) {
		tmp = x;
	} else if (z <= 5.9e+60) {
		tmp = y * (6.0 * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -6.0 * (x * z)
	tmp = 0
	if z <= -0.17:
		tmp = t_0
	elif z <= 2.6e-60:
		tmp = x
	elif z <= 5.9e+60:
		tmp = y * (6.0 * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-6.0 * Float64(x * z))
	tmp = 0.0
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 2.6e-60)
		tmp = x;
	elseif (z <= 5.9e+60)
		tmp = Float64(y * Float64(6.0 * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -6.0 * (x * z);
	tmp = 0.0;
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 2.6e-60)
		tmp = x;
	elseif (z <= 5.9e+60)
		tmp = y * (6.0 * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.17], t$95$0, If[LessEqual[z, 2.6e-60], x, If[LessEqual[z, 5.9e+60], N[(y * N[(6.0 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.170000000000000012 or 5.9000000000000002e60 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot -6\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(-6 \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 64.3

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

   5. Simplified 64.3%

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

   6. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 64.1

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

   8. Simplified 64.1%

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

   if -0.170000000000000012 < z < 2.5999999999999998e-60

   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

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

   5. Simplified 77.5%

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

   if 2.5999999999999998e-60 < z < 5.9000000000000002e60

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, 6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, \color{blue}{\left(6 \cdot -1\right) \cdot x}\right) \cdot z \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval 99.6

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 63.6

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

   9. Simplified 63.6%

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

Alternative 5: 60.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+60}:\\ \;\;\;\;6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -6.0 (* x z))))
   (if (<= z -0.17)
     t_0
     (if (<= z 2.7e-60) x (if (<= z 5.9e+60) (* 6.0 (* y z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -6.0 * (x * z);
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 2.7e-60) {
		tmp = x;
	} else if (z <= 5.9e+60) {
		tmp = 6.0 * (y * z);
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = (-6.0d0) * (x * z)
    if (z <= (-0.17d0)) then
        tmp = t_0
    else if (z <= 2.7d-60) then
        tmp = x
    else if (z <= 5.9d+60) then
        tmp = 6.0d0 * (y * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -6.0 * (x * z);
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 2.7e-60) {
		tmp = x;
	} else if (z <= 5.9e+60) {
		tmp = 6.0 * (y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -6.0 * (x * z)
	tmp = 0
	if z <= -0.17:
		tmp = t_0
	elif z <= 2.7e-60:
		tmp = x
	elif z <= 5.9e+60:
		tmp = 6.0 * (y * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-6.0 * Float64(x * z))
	tmp = 0.0
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 2.7e-60)
		tmp = x;
	elseif (z <= 5.9e+60)
		tmp = Float64(6.0 * Float64(y * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -6.0 * (x * z);
	tmp = 0.0;
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 2.7e-60)
		tmp = x;
	elseif (z <= 5.9e+60)
		tmp = 6.0 * (y * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.17], t$95$0, If[LessEqual[z, 2.7e-60], x, If[LessEqual[z, 5.9e+60], N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.170000000000000012 or 5.9000000000000002e60 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot -6\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(-6 \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 64.3

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

   5. Simplified 64.3%

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

   6. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 64.1

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

   8. Simplified 64.1%

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

   if -0.170000000000000012 < z < 2.7e-60

   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

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

   5. Simplified 77.5%

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

   if 2.7e-60 < z < 5.9000000000000002e60

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 63.6

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

   5. Simplified 63.6%

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

Alternative 6: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-6 \cdot z\right) \cdot \left(x - y\right)\\ \mathbf{if}\;z \leq -16500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x + 6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* -6.0 z) (- x y))))
   (if (<= z -16500.0) t_0 (if (<= z 0.165) (+ x (* 6.0 (* y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-6.0 * z) * (x - y);
	double tmp;
	if (z <= -16500.0) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = x + (6.0 * (y * z));
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = ((-6.0d0) * z) * (x - y)
    if (z <= (-16500.0d0)) then
        tmp = t_0
    else if (z <= 0.165d0) then
        tmp = x + (6.0d0 * (y * z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-6.0 * z) * (x - y);
	double tmp;
	if (z <= -16500.0) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = x + (6.0 * (y * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-6.0 * z) * (x - y)
	tmp = 0
	if z <= -16500.0:
		tmp = t_0
	elif z <= 0.165:
		tmp = x + (6.0 * (y * z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-6.0 * z) * Float64(x - y))
	tmp = 0.0
	if (z <= -16500.0)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = Float64(x + Float64(6.0 * Float64(y * z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-6.0 * z) * (x - y);
	tmp = 0.0;
	if (z <= -16500.0)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = x + (6.0 * (y * z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -16500.0], t$95$0, If[LessEqual[z, 0.165], N[(x + N[(6.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -16500 or 0.165000000000000008 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\left(-1 \cdot -6\right)} \cdot \left(y \cdot z\right) - \left(6 \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{-1 \cdot \left(-6 \cdot \left(y \cdot z\right)\right)} - \left(6 \cdot z\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \left(-6 \cdot \color{blue}{\left(z \cdot y\right)}\right) - \left(6 \cdot z\right) \cdot x \]

      8. associate-*r* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(-6 \cdot z\right) \cdot y\right)} - \left(6 \cdot z\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(y \cdot \left(-6 \cdot z\right)\right)} - \left(6 \cdot z\right) \cdot x \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right)} - \left(6 \cdot z\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{x \cdot \left(6 \cdot z\right)} \]

      12. metadata-eval N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right)} \cdot z\right) \]

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

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - x \cdot \color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(-6 \cdot z\right)\right)\right)} \]

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

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-6 \cdot z\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(-1 \cdot x\right)} \cdot \left(-6 \cdot z\right) \]

      17. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot \left(-1 \cdot y - -1 \cdot x\right)} \]

      18. distribute-lft-out-- N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \]

      19. neg-mul-1 N/A

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

      20. neg-sub0 N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(0 - \left(y - x\right)\right)} \]

      21. associate-+l- N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(\left(0 - y\right) + x\right)} \]

      22. neg-sub0 N/A

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

      23. mul-1-neg N/A

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

      24. *-lft-identity N/A

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

      25. *-inverses N/A

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

      26. associate-*l/ N/A

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

      27. associate-*r/ N/A

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

      28. associate-*r/ N/A

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

      29. *-rgt-identity N/A

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

   5. Simplified 99.8%

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

   if -16500 < z < 0.165000000000000008

   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

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -16500:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot \left(x - y\right)\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x + 6 \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-6 \cdot z\right) \cdot \left(x - y\right)\\ \mathbf{if}\;z \leq -16600000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* -6.0 z) (- x y))))
   (if (<= z -16600000000000.0)
     t_0
     (if (<= z 0.165) (fma (* y 6.0) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-6.0 * z) * (x - y);
	double tmp;
	if (z <= -16600000000000.0) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = fma((y * 6.0), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-6.0 * z) * Float64(x - y))
	tmp = 0.0
	if (z <= -16600000000000.0)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = fma(Float64(y * 6.0), z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -16600000000000.0], t$95$0, If[LessEqual[z, 0.165], N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.66e13 or 0.165000000000000008 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\left(-1 \cdot -6\right)} \cdot \left(y \cdot z\right) - \left(6 \cdot z\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{-1 \cdot \left(-6 \cdot \left(y \cdot z\right)\right)} - \left(6 \cdot z\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \left(-6 \cdot \color{blue}{\left(z \cdot y\right)}\right) - \left(6 \cdot z\right) \cdot x \]

      8. associate-*r* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\left(-6 \cdot z\right) \cdot y\right)} - \left(6 \cdot z\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(y \cdot \left(-6 \cdot z\right)\right)} - \left(6 \cdot z\right) \cdot x \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right)} - \left(6 \cdot z\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{x \cdot \left(6 \cdot z\right)} \]

      12. metadata-eval N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-6\right)\right)} \cdot z\right) \]

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

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - x \cdot \color{blue}{\left(\mathsf{neg}\left(-6 \cdot z\right)\right)} \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(-6 \cdot z\right)\right)\right)} \]

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

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-6 \cdot z\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot y\right) \cdot \left(-6 \cdot z\right) - \color{blue}{\left(-1 \cdot x\right)} \cdot \left(-6 \cdot z\right) \]

      17. distribute-rgt-out-- N/A

          \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot \left(-1 \cdot y - -1 \cdot x\right)} \]

      18. distribute-lft-out-- N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \]

      19. neg-mul-1 N/A

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

      20. neg-sub0 N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(0 - \left(y - x\right)\right)} \]

      21. associate-+l- N/A

          \[\leadsto \left(-6 \cdot z\right) \cdot \color{blue}{\left(\left(0 - y\right) + x\right)} \]

      22. neg-sub0 N/A

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

      23. mul-1-neg N/A

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

      24. *-lft-identity N/A

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

      25. *-inverses N/A

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

      26. associate-*l/ N/A

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

      27. associate-*r/ N/A

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

      28. associate-*r/ N/A

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

      29. *-rgt-identity N/A

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

   5. Simplified 99.8%

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

   if -1.66e13 < z < 0.165000000000000008

   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

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 98.9

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -16600000000000:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot \left(x - y\right)\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-6 \cdot z\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{if}\;z \leq -16600000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 6.0 (* z (- y x)))))
   (if (<= z -16600000000000.0)
     t_0
     (if (<= z 0.165) (fma (* y 6.0) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 6.0 * (z * (y - x));
	double tmp;
	if (z <= -16600000000000.0) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = fma((y * 6.0), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(6.0 * Float64(z * Float64(y - x)))
	tmp = 0.0
	if (z <= -16600000000000.0)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = fma(Float64(y * 6.0), z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(6.0 * N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -16600000000000.0], t$95$0, If[LessEqual[z, 0.165], N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.66e13 or 0.165000000000000008 < z

   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. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, 6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, \color{blue}{\left(6 \cdot -1\right) \cdot x}\right) \cdot z \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. unsub-neg N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 99.6

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

   7. Simplified 99.6%

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

   if -1.66e13 < z < 0.165000000000000008

   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

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 98.9

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

   7. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot 6, z, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (* -6.0 z) x x)))
   (if (<= x -6.8e-91) t_0 (if (<= x 7.5e+51) (fma (* y 6.0) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((-6.0 * z), x, x);
	double tmp;
	if (x <= -6.8e-91) {
		tmp = t_0;
	} else if (x <= 7.5e+51) {
		tmp = fma((y * 6.0), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-6.0 * z), x, x)
	tmp = 0.0
	if (x <= -6.8e-91)
		tmp = t_0;
	elseif (x <= 7.5e+51)
		tmp = fma(Float64(y * 6.0), z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-6.0 * z), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[x, -6.8e-91], t$95$0, If[LessEqual[x, 7.5e+51], N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.80000000000000053e-91 or 7.4999999999999999e51 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot -6\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(-6 \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 89.0

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

   5. Simplified 89.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 89.2

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

   7. Applied egg-rr 89.2%

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

   if -6.80000000000000053e-91 < x < 7.4999999999999999e51

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 80.4

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

   5. Simplified 80.4%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 80.5

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

   7. Applied egg-rr 80.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-6 \cdot z, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, -6 \cdot x, x\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (* -6.0 x) x)))
   (if (<= x -1.65e-90) t_0 (if (<= x 8e+51) (fma (* y 6.0) z x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (-6.0 * x), x);
	double tmp;
	if (x <= -1.65e-90) {
		tmp = t_0;
	} else if (x <= 8e+51) {
		tmp = fma((y * 6.0), z, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, Float64(-6.0 * x), x)
	tmp = 0.0
	if (x <= -1.65e-90)
		tmp = t_0;
	elseif (x <= 8e+51)
		tmp = fma(Float64(y * 6.0), z, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-6.0 * x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -1.65e-90], t$95$0, If[LessEqual[x, 8e+51], N[(N[(y * 6.0), $MachinePrecision] * z + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e-90 or 8e51 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot -6\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(-6 \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 89.0

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

   5. Simplified 89.0%

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

   if -1.65e-90 < x < 8e51

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 80.4

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

   5. Simplified 80.4%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 80.5

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

   7. Applied egg-rr 80.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(z, -6 \cdot x, x\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 6, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, -6 \cdot x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, -6 \cdot x, x\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma z (* -6.0 x) x)))
   (if (<= x -4.5e-92) t_0 (if (<= x 8.2e+51) (fma (* 6.0 z) y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(z, (-6.0 * x), x);
	double tmp;
	if (x <= -4.5e-92) {
		tmp = t_0;
	} else if (x <= 8.2e+51) {
		tmp = fma((6.0 * z), y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(z, Float64(-6.0 * x), x)
	tmp = 0.0
	if (x <= -4.5e-92)
		tmp = t_0;
	elseif (x <= 8.2e+51)
		tmp = fma(Float64(6.0 * z), y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-6.0 * x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -4.5e-92], t$95$0, If[LessEqual[x, 8.2e+51], N[(N[(6.0 * z), $MachinePrecision] * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e-92 or 8.20000000000000021e51 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot -6\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(-6 \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 89.0

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

   5. Simplified 89.0%

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

   if -4.5e-92 < x < 8.20000000000000021e51

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 80.4

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

   5. Simplified 80.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 80.4

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

   7. Applied egg-rr 80.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(z, -6 \cdot x, x\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(6 \cdot z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, -6 \cdot x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+135}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(z, -6 \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.85e+135)
   (* z (* y 6.0))
   (if (<= y 5.8e+169) (fma z (* -6.0 x) x) (* y (* 6.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.85e+135) {
		tmp = z * (y * 6.0);
	} else if (y <= 5.8e+169) {
		tmp = fma(z, (-6.0 * x), x);
	} else {
		tmp = y * (6.0 * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.85e+135)
		tmp = Float64(z * Float64(y * 6.0));
	elseif (y <= 5.8e+169)
		tmp = fma(z, Float64(-6.0 * x), x);
	else
		tmp = Float64(y * Float64(6.0 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.85e+135], N[(z * N[(y * 6.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+169], N[(z * N[(-6.0 * x), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(6.0 * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8500000000000001e135

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 83.8

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

   5. Simplified 83.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 83.8

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

   7. Applied egg-rr 83.8%

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

   if -2.8500000000000001e135 < y < 5.8000000000000001e169

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot -6\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(-6 \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 79.5

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

   5. Simplified 79.5%

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

   if 5.8000000000000001e169 < y

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. neg-mul-1 N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, 6 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot z \]

      7. associate-*r* N/A

         \[\leadsto x + \mathsf{fma}\left(y, 6, \color{blue}{\left(6 \cdot -1\right) \cdot x}\right) \cdot z \]

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. metadata-eval 99.9

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

   4. Applied egg-rr 99.9%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 63.7

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

   9. Simplified 63.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+135}:\\ \;\;\;\;z \cdot \left(y \cdot 6\right)\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+169}:\\ \;\;\;\;\mathsf{fma}\left(z, -6 \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(6 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -0.17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -6.0 (* x z)))) (if (<= z -0.17) t_0 (if (<= z 0.165) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -6.0 * (x * z);
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = (-6.0d0) * (x * z)
    if (z <= (-0.17d0)) then
        tmp = t_0
    else if (z <= 0.165d0) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -6.0 * (x * z);
	double tmp;
	if (z <= -0.17) {
		tmp = t_0;
	} else if (z <= 0.165) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -6.0 * (x * z)
	tmp = 0
	if z <= -0.17:
		tmp = t_0
	elif z <= 0.165:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-6.0 * Float64(x * z))
	tmp = 0.0
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -6.0 * (x * z);
	tmp = 0.0;
	if (z <= -0.17)
		tmp = t_0;
	elseif (z <= 0.165)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-6.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.17], t$95$0, If[LessEqual[z, 0.165], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.170000000000000012 or 0.165000000000000008 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-6 \cdot z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-6 \cdot z\right) \cdot x + 1 \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot -6\right)} \cdot x + 1 \cdot x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(-6 \cdot x\right)} + 1 \cdot x \]

      5. *-lft-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 63.1

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

   5. Simplified 63.1%

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

   6. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 63.0

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

   8. Simplified 63.0%

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

   if -0.170000000000000012 < z < 0.165000000000000008

   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

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

   5. Simplified 72.8%

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

Alternative 14: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 15: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 99.8

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

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%

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

Alternative 16: 36.8% accurate, 17.0× 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

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

5. Simplified 35.8%

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

Developer Target 1: 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 2024199 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (* 6 z) (- x y))))

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