Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

Alternatives: 11

Speedup: N/A×

• 26.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((((y + z) + z) + y) + t)) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((((y + z) + z) + y) + t)) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(Float64(Float64(y + z) + z) + y) + t)) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((((y + z) + z) + y) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(N[(y + z), $MachinePrecision] + z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(y + \left(z + z\right)\right) + \left(y + t\right)\right) + y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (* x (+ (+ y (+ z z)) (+ y t))) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return (x * ((y + (z + z)) + (y + t))) + (y * 5.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * ((y + (z + z)) + (y + t))) + (y * 5.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * ((y + (z + z)) + (y + t))) + (y * 5.0);
}

Python

def code(x, y, z, t):
	return (x * ((y + (z + z)) + (y + t))) + (y * 5.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(x * Float64(Float64(y + Float64(z + z)) + Float64(y + t))) + Float64(y * 5.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * ((y + (z + z)) + (y + t))) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(y + N[(z + z), $MachinePrecision]), $MachinePrecision] + N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(y + z\right) + z\right) + \left(y + t\right)\right)} + y \cdot 5 \]

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

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(y + z\right) + z\right) + \left(y + t\right)\right)} + y \cdot 5 \]

   3. associate-+l+ N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(y + \left(z + z\right)\right)} + \left(y + t\right)\right) + y \cdot 5 \]

   4. flip-+ N/A

      \[\leadsto x \cdot \left(\left(y + \color{blue}{\frac{z \cdot z - z \cdot z}{z - z}}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   5. +-inverses N/A

      \[\leadsto x \cdot \left(\left(y + \frac{\color{blue}{0}}{z - z}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   6. +-inverses N/A

      \[\leadsto x \cdot \left(\left(y + \frac{\color{blue}{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}}{z - z}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   7. +-inverses N/A

      \[\leadsto x \cdot \left(\left(y + \frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\color{blue}{0}}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   8. +-inverses N/A

      \[\leadsto x \cdot \left(\left(y + \frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\color{blue}{\left(y + z\right) - \left(y + z\right)}}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   9. flip-+ N/A

      \[\leadsto x \cdot \left(\left(y + \color{blue}{\left(\left(y + z\right) + \left(y + z\right)\right)}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   10. +-commutative N/A

       \[\leadsto x \cdot \left(\left(y + \left(\left(y + z\right) + \color{blue}{\left(z + y\right)}\right)\right) + \left(y + t\right)\right) + y \cdot 5 \]

   11. associate-+l+ N/A

       \[\leadsto x \cdot \left(\left(y + \color{blue}{\left(\left(\left(y + z\right) + z\right) + y\right)}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   12. +-lowering-+.f64 N/A

       \[\leadsto x \cdot \left(\color{blue}{\left(y + \left(\left(\left(y + z\right) + z\right) + y\right)\right)} + \left(y + t\right)\right) + y \cdot 5 \]

   13. associate-+l+ N/A

       \[\leadsto x \cdot \left(\left(y + \color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   14. +-commutative N/A

       \[\leadsto x \cdot \left(\left(y + \left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right)\right) + \left(y + t\right)\right) + y \cdot 5 \]

   15. flip-+ N/A

       \[\leadsto x \cdot \left(\left(y + \color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   16. +-inverses N/A

       \[\leadsto x \cdot \left(\left(y + \frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   17. +-inverses N/A

       \[\leadsto x \cdot \left(\left(y + \frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   18. +-inverses N/A

       \[\leadsto x \cdot \left(\left(y + \frac{z \cdot z - z \cdot z}{\color{blue}{0}}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   19. +-inverses N/A

       \[\leadsto x \cdot \left(\left(y + \frac{z \cdot z - z \cdot z}{\color{blue}{z - z}}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   20. flip-+ N/A

       \[\leadsto x \cdot \left(\left(y + \color{blue}{\left(z + z\right)}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   21. +-lowering-+.f64 N/A

       \[\leadsto x \cdot \left(\left(y + \color{blue}{\left(z + z\right)}\right) + \left(y + t\right)\right) + y \cdot 5 \]

   22. +-lowering-+.f64 99.9

       \[\leadsto x \cdot \left(\left(y + \left(z + z\right)\right) + \color{blue}{\left(y + t\right)}\right) + y \cdot 5 \]

4. Applied egg-rr 99.9%

   \[\leadsto x \cdot \color{blue}{\left(\left(y + \left(z + z\right)\right) + \left(y + t\right)\right)} + y \cdot 5 \]
5. Add Preprocessing

Alternative 2: 48.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+112}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-12}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+16}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+208}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.4e+112)
   (* x t)
   (if (<= x -3.9e+30)
     (* x (+ y y))
     (if (<= x -2.65e-12)
       (* x t)
       (if (<= x 4e+16)
         (* y 5.0)
         (if (<= x 7.2e+208) (* x t) (* x (+ z z))))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.4e+112) {
		tmp = x * t;
	} else if (x <= -3.9e+30) {
		tmp = x * (y + y);
	} else if (x <= -2.65e-12) {
		tmp = x * t;
	} else if (x <= 4e+16) {
		tmp = y * 5.0;
	} else if (x <= 7.2e+208) {
		tmp = x * t;
	} else {
		tmp = x * (z + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.4d+112)) then
        tmp = x * t
    else if (x <= (-3.9d+30)) then
        tmp = x * (y + y)
    else if (x <= (-2.65d-12)) then
        tmp = x * t
    else if (x <= 4d+16) then
        tmp = y * 5.0d0
    else if (x <= 7.2d+208) then
        tmp = x * t
    else
        tmp = x * (z + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.4e+112) {
		tmp = x * t;
	} else if (x <= -3.9e+30) {
		tmp = x * (y + y);
	} else if (x <= -2.65e-12) {
		tmp = x * t;
	} else if (x <= 4e+16) {
		tmp = y * 5.0;
	} else if (x <= 7.2e+208) {
		tmp = x * t;
	} else {
		tmp = x * (z + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.4e+112:
		tmp = x * t
	elif x <= -3.9e+30:
		tmp = x * (y + y)
	elif x <= -2.65e-12:
		tmp = x * t
	elif x <= 4e+16:
		tmp = y * 5.0
	elif x <= 7.2e+208:
		tmp = x * t
	else:
		tmp = x * (z + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.4e+112)
		tmp = Float64(x * t);
	elseif (x <= -3.9e+30)
		tmp = Float64(x * Float64(y + y));
	elseif (x <= -2.65e-12)
		tmp = Float64(x * t);
	elseif (x <= 4e+16)
		tmp = Float64(y * 5.0);
	elseif (x <= 7.2e+208)
		tmp = Float64(x * t);
	else
		tmp = Float64(x * Float64(z + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.4e+112)
		tmp = x * t;
	elseif (x <= -3.9e+30)
		tmp = x * (y + y);
	elseif (x <= -2.65e-12)
		tmp = x * t;
	elseif (x <= 4e+16)
		tmp = y * 5.0;
	elseif (x <= 7.2e+208)
		tmp = x * t;
	else
		tmp = x * (z + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.4e+112], N[(x * t), $MachinePrecision], If[LessEqual[x, -3.9e+30], N[(x * N[(y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.65e-12], N[(x * t), $MachinePrecision], If[LessEqual[x, 4e+16], N[(y * 5.0), $MachinePrecision], If[LessEqual[x, 7.2e+208], N[(x * t), $MachinePrecision], N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.4e112 or -3.90000000000000011e30 < x < -2.64999999999999982e-12 or 4e16 < x < 7.20000000000000005e208

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 46.9

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

   5. Simplified 46.9%

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

   if -2.4e112 < x < -3.90000000000000011e30

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. +-lowering-+.f64 100.0

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(y \cdot x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot y\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(2 \cdot y\right) \cdot x} \]

      4. count-2 N/A

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

      5. +-lowering-+.f64 69.0

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

   8. Simplified 69.0%

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

   if -2.64999999999999982e-12 < x < 4e16

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 58.8

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 58.8%

      \[\leadsto \color{blue}{5 \cdot y} \]

   if 7.20000000000000005e208 < x

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. +-lowering-+.f64 100.0

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(y \cdot x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot y\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(2 \cdot y\right) \cdot x} \]

      4. count-2 N/A

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

      5. +-lowering-+.f64 44.3

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

   8. Simplified 44.3%

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

      1. flip-+ N/A

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

      2. +-inverses N/A

         \[\leadsto \frac{\color{blue}{0}}{y - y} \cdot x \]

      3. +-inverses N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{z \cdot z - z \cdot z}{\color{blue}{0}} \cdot x \]

      5. +-inverses N/A

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

      6. flip-+ N/A

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

      7. count-2 N/A

         \[\leadsto \color{blue}{\left(2 \cdot z\right)} \cdot x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot 2\right)} \cdot x \]

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

         \[\leadsto \color{blue}{\left(z \cdot 2\right) \cdot x} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(2 \cdot z\right)} \cdot x \]

      11. count-2 N/A

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

      12. +-lowering-+.f64 53.7

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

   10. Applied egg-rr 53.7%

       \[\leadsto \color{blue}{\left(z + z\right) \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+112}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-12}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+16}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+208}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y + y\right)\\ \mathbf{if}\;x \leq -7 \cdot 10^{+112}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-12}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ y y))))
   (if (<= x -7e+112)
     (* x t)
     (if (<= x -1.45e+30)
       t_1
       (if (<= x -6.4e-12) (* x t) (if (<= x 2.5) (* y 5.0) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y + y);
	double tmp;
	if (x <= -7e+112) {
		tmp = x * t;
	} else if (x <= -1.45e+30) {
		tmp = t_1;
	} else if (x <= -6.4e-12) {
		tmp = x * t;
	} else if (x <= 2.5) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y + y)
    if (x <= (-7d+112)) then
        tmp = x * t
    else if (x <= (-1.45d+30)) then
        tmp = t_1
    else if (x <= (-6.4d-12)) then
        tmp = x * t
    else if (x <= 2.5d0) then
        tmp = y * 5.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (y + y);
	double tmp;
	if (x <= -7e+112) {
		tmp = x * t;
	} else if (x <= -1.45e+30) {
		tmp = t_1;
	} else if (x <= -6.4e-12) {
		tmp = x * t;
	} else if (x <= 2.5) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y + y)
	tmp = 0
	if x <= -7e+112:
		tmp = x * t
	elif x <= -1.45e+30:
		tmp = t_1
	elif x <= -6.4e-12:
		tmp = x * t
	elif x <= 2.5:
		tmp = y * 5.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y + y))
	tmp = 0.0
	if (x <= -7e+112)
		tmp = Float64(x * t);
	elseif (x <= -1.45e+30)
		tmp = t_1;
	elseif (x <= -6.4e-12)
		tmp = Float64(x * t);
	elseif (x <= 2.5)
		tmp = Float64(y * 5.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y + y);
	tmp = 0.0;
	if (x <= -7e+112)
		tmp = x * t;
	elseif (x <= -1.45e+30)
		tmp = t_1;
	elseif (x <= -6.4e-12)
		tmp = x * t;
	elseif (x <= 2.5)
		tmp = y * 5.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+112], N[(x * t), $MachinePrecision], If[LessEqual[x, -1.45e+30], t$95$1, If[LessEqual[x, -6.4e-12], N[(x * t), $MachinePrecision], If[LessEqual[x, 2.5], N[(y * 5.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.99999999999999994e112 or -1.4499999999999999e30 < x < -6.4000000000000002e-12

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 48.0

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

   5. Simplified 48.0%

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

   if -6.99999999999999994e112 < x < -1.4499999999999999e30 or 2.5 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. +-lowering-+.f64 99.9

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(y \cdot x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(2 \cdot y\right) \cdot x} \]

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

         \[\leadsto \color{blue}{\left(2 \cdot y\right) \cdot x} \]

      4. count-2 N/A

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

      5. +-lowering-+.f64 47.4

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

   8. Simplified 47.4%

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

   if -6.4000000000000002e-12 < x < 2.5

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 59.5

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 59.5%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+112}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-12}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(x, t, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -7.5e-45)
     t_1
     (if (<= y 5.5e-23)
       (* x (fma 2.0 z t))
       (if (<= y 3.05e+147) (fma x t (* y 5.0)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * fma(x, 2.0, 5.0);
	double tmp;
	if (y <= -7.5e-45) {
		tmp = t_1;
	} else if (y <= 5.5e-23) {
		tmp = x * fma(2.0, z, t);
	} else if (y <= 3.05e+147) {
		tmp = fma(x, t, (y * 5.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * fma(x, 2.0, 5.0))
	tmp = 0.0
	if (y <= -7.5e-45)
		tmp = t_1;
	elseif (y <= 5.5e-23)
		tmp = Float64(x * fma(2.0, z, t));
	elseif (y <= 3.05e+147)
		tmp = fma(x, t, Float64(y * 5.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0 + 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e-45], t$95$1, If[LessEqual[y, 5.5e-23], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.05e+147], N[(x * t + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5000000000000006e-45 or 3.05000000000000016e147 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

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

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      8. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      9. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \]

      10. neg-sub0 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \]

      12. metadata-eval N/A

          \[\leadsto y \cdot \left(\color{blue}{2} \cdot x + 5\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{x \cdot 2} + 5\right) \]

      14. accelerator-lowering-fma.f64 84.0

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, 2, 5\right)} \]

   5. Simplified 84.0%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, 2, 5\right)} \]

   if -7.5000000000000006e-45 < y < 5.5000000000000001e-23

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot z + t\right)} \]

      3. accelerator-lowering-fma.f64 85.8

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, z, t\right)} \]

   5. Simplified 85.8%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, z, t\right)} \]

   if 5.5000000000000001e-23 < y < 3.05000000000000016e147

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

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

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right)\right) \]

      7. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      9. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      10. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right)\right) \]

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

      13. +-lowering-+.f64 85.3

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

   4. Applied egg-rr 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(z + z\right) + t\right)\right)} \]

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

      2. *-lowering-*.f64 76.8

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

   7. Simplified 76.8%

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot t + y \cdot 5} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, t, y \cdot 5\right)} \]

      3. *-lowering-*.f64 76.8

         \[\leadsto \mathsf{fma}\left(x, t, \color{blue}{y \cdot 5}\right) \]

   9. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, t, y \cdot 5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+23}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(\left(z + z\right) + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, 2, y + \left(y + t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.4e+23)
   (* x (fma 2.0 (+ y z) t))
   (if (<= x 1.8e-5)
     (fma y 5.0 (* x (+ (+ z z) t)))
     (* x (fma z 2.0 (+ y (+ y t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.4e+23) {
		tmp = x * fma(2.0, (y + z), t);
	} else if (x <= 1.8e-5) {
		tmp = fma(y, 5.0, (x * ((z + z) + t)));
	} else {
		tmp = x * fma(z, 2.0, (y + (y + t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.4e+23)
		tmp = Float64(x * fma(2.0, Float64(y + z), t));
	elseif (x <= 1.8e-5)
		tmp = fma(y, 5.0, Float64(x * Float64(Float64(z + z) + t)));
	else
		tmp = Float64(x * fma(z, 2.0, Float64(y + Float64(y + t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.4e+23], N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e-5], N[(y * 5.0 + N[(x * N[(N[(z + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(z * 2.0 + N[(y + N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3999999999999997e23

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. +-lowering-+.f64 100.0

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   if -5.3999999999999997e23 < x < 1.80000000000000005e-5

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

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

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right)\right) \]

      7. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      9. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      10. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right)\right) \]

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

      13. +-lowering-+.f64 99.0

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

   4. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(z + z\right) + t\right)\right)} \]

   if 1.80000000000000005e-5 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. +-lowering-+.f64 99.9

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(z + y\right)} + t\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(2 \cdot z + 2 \cdot y\right)} + t\right) \]

      3. count-2 N/A

         \[\leadsto x \cdot \left(\left(2 \cdot z + \color{blue}{\left(y + y\right)}\right) + t\right) \]

      4. associate-+l+ N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(2 \cdot z + y\right) + y\right)} + t\right) \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(y + 2 \cdot z\right)} + y\right) + t\right) \]

      6. count-2 N/A

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

      7. associate-+r+ N/A

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

      8. count-2 N/A

         \[\leadsto x \cdot \left(\left(y + \color{blue}{2 \cdot z}\right) + \left(y + t\right)\right) \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(2 \cdot z + y\right)} + \left(y + t\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto x \cdot \color{blue}{\left(2 \cdot z + \left(y + \left(y + t\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{z \cdot 2} + \left(y + \left(y + t\right)\right)\right) \]

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

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(z, 2, y + \left(y + t\right)\right)} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(z, 2, \color{blue}{y + \left(y + t\right)}\right) \]

      14. +-lowering-+.f64 99.9

          \[\leadsto x \cdot \mathsf{fma}\left(z, 2, y + \color{blue}{\left(y + t\right)}\right) \]

   7. Applied egg-rr 99.9%

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

Alternative 6: 88.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, 2, y + \left(y + t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.7e-15)
   (* x (fma 2.0 (+ y z) t))
   (if (<= x 2.9e-8) (fma y 5.0 (* x t)) (* x (fma z 2.0 (+ y (+ y t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.7e-15) {
		tmp = x * fma(2.0, (y + z), t);
	} else if (x <= 2.9e-8) {
		tmp = fma(y, 5.0, (x * t));
	} else {
		tmp = x * fma(z, 2.0, (y + (y + t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.7e-15)
		tmp = Float64(x * fma(2.0, Float64(y + z), t));
	elseif (x <= 2.9e-8)
		tmp = fma(y, 5.0, Float64(x * t));
	else
		tmp = Float64(x * fma(z, 2.0, Float64(y + Float64(y + t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.7e-15], N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-8], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], N[(x * N[(z * 2.0 + N[(y + N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6999999999999999e-15

   1. Initial program 100.0%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. +-lowering-+.f64 98.3

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   if -4.6999999999999999e-15 < x < 2.9000000000000002e-8

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

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

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right)\right) \]

      7. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      9. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      10. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right)\right) \]

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

      13. +-lowering-+.f64 99.4

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(z + z\right) + t\right)\right)} \]

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

      2. *-lowering-*.f64 82.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

   7. Simplified 82.0%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

   if 2.9000000000000002e-8 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. +-lowering-+.f64 99.9

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(z + y\right)} + t\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(2 \cdot z + 2 \cdot y\right)} + t\right) \]

      3. count-2 N/A

         \[\leadsto x \cdot \left(\left(2 \cdot z + \color{blue}{\left(y + y\right)}\right) + t\right) \]

      4. associate-+l+ N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\left(2 \cdot z + y\right) + y\right)} + t\right) \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(y + 2 \cdot z\right)} + y\right) + t\right) \]

      6. count-2 N/A

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

      7. associate-+r+ N/A

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

      8. count-2 N/A

         \[\leadsto x \cdot \left(\left(y + \color{blue}{2 \cdot z}\right) + \left(y + t\right)\right) \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(2 \cdot z + y\right)} + \left(y + t\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto x \cdot \color{blue}{\left(2 \cdot z + \left(y + \left(y + t\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{z \cdot 2} + \left(y + \left(y + t\right)\right)\right) \]

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

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(z, 2, y + \left(y + t\right)\right)} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(z, 2, \color{blue}{y + \left(y + t\right)}\right) \]

      14. +-lowering-+.f64 99.9

          \[\leadsto x \cdot \mathsf{fma}\left(z, 2, y + \color{blue}{\left(y + t\right)}\right) \]

   7. Applied egg-rr 99.9%

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

Alternative 7: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -3e-11) t_1 (if (<= x 3e-8) (fma y 5.0 (* x t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, (y + z), t);
	double tmp;
	if (x <= -3e-11) {
		tmp = t_1;
	} else if (x <= 3e-8) {
		tmp = fma(y, 5.0, (x * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, Float64(y + z), t))
	tmp = 0.0
	if (x <= -3e-11)
		tmp = t_1;
	elseif (x <= 3e-8)
		tmp = fma(y, 5.0, Float64(x * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e-11], t$95$1, If[LessEqual[x, 3e-8], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3e-11 or 2.99999999999999973e-8 < x

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + \left(2 \cdot y + 2 \cdot z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(2 \cdot y + 2 \cdot z\right) + t\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto x \cdot \left(\color{blue}{2 \cdot \left(y + z\right)} + t\right) \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, y + z, t\right)} \]

      5. +-lowering-+.f64 99.1

         \[\leadsto x \cdot \mathsf{fma}\left(2, \color{blue}{y + z}, t\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(2, y + z, t\right)} \]

   if -3e-11 < x < 2.99999999999999973e-8

   1. Initial program 99.8%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot 5 + x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)\right)} \]

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

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

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

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \color{blue}{\left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right)}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(\left(y + z\right) + \left(z + y\right)\right)} + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\left(\left(y + z\right) + \color{blue}{\left(y + z\right)}\right) + t\right)\right) \]

      7. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\frac{\left(y + z\right) \cdot \left(y + z\right) - \left(y + z\right) \cdot \left(y + z\right)}{\left(y + z\right) - \left(y + z\right)}} + t\right)\right) \]

      8. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{0}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      9. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{\color{blue}{z \cdot z - z \cdot z}}{\left(y + z\right) - \left(y + z\right)} + t\right)\right) \]

      10. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{0}} + t\right)\right) \]

      11. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + t\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

      13. +-lowering-+.f64 99.4

          \[\leadsto \mathsf{fma}\left(y, 5, x \cdot \left(\color{blue}{\left(z + z\right)} + t\right)\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 5, x \cdot \left(\left(z + z\right) + t\right)\right)} \]

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

      2. *-lowering-*.f64 82.0

         \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]

   7. Simplified 82.0%

      \[\leadsto \mathsf{fma}\left(y, 5, \color{blue}{x \cdot t}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 77.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -8 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -8e-45) t_1 (if (<= y 6e-16) (* x (fma 2.0 z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * fma(x, 2.0, 5.0);
	double tmp;
	if (y <= -8e-45) {
		tmp = t_1;
	} else if (y <= 6e-16) {
		tmp = x * fma(2.0, z, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * fma(x, 2.0, 5.0))
	tmp = 0.0
	if (y <= -8e-45)
		tmp = t_1;
	elseif (y <= 6e-16)
		tmp = Float64(x * fma(2.0, z, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0 + 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e-45], t$95$1, If[LessEqual[y, 6e-16], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.99999999999999987e-45 or 5.99999999999999987e-16 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(5 + 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(2 \cdot x + 5\right)} \]

      2. metadata-eval N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x + 5\right) \]

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

         \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

      4. neg-sub0 N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(0 - -2 \cdot x\right)} + 5\right) \]

      5. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

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

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(-2 \cdot x - 5\right)\right)\right)} \]

      8. neg-sub0 N/A

         \[\leadsto y \cdot \color{blue}{\left(0 - \left(-2 \cdot x - 5\right)\right)} \]

      9. associate--r- N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(0 - -2 \cdot x\right) + 5\right)} \]

      10. neg-sub0 N/A

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} + 5\right) \]

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

          \[\leadsto y \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot x} + 5\right) \]

      12. metadata-eval N/A

          \[\leadsto y \cdot \left(\color{blue}{2} \cdot x + 5\right) \]

      13. *-commutative N/A

          \[\leadsto y \cdot \left(\color{blue}{x \cdot 2} + 5\right) \]

      14. accelerator-lowering-fma.f64 79.9

          \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(x, 2, 5\right)} \]

   5. Simplified 79.9%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(x, 2, 5\right)} \]

   if -7.99999999999999987e-45 < y < 5.99999999999999987e-16

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot z + t\right)} \]

      3. accelerator-lowering-fma.f64 85.3

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, z, t\right)} \]

   5. Simplified 85.3%

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

Alternative 9: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+33}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.2e+33)
   (* y 5.0)
   (if (<= y 4.1e-15) (* x (fma 2.0 z t)) (* y 5.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.2e+33) {
		tmp = y * 5.0;
	} else if (y <= 4.1e-15) {
		tmp = x * fma(2.0, z, t);
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.2e+33)
		tmp = Float64(y * 5.0);
	elseif (y <= 4.1e-15)
		tmp = Float64(x * fma(2.0, z, t));
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.2e+33], N[(y * 5.0), $MachinePrecision], If[LessEqual[y, 4.1e-15], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e33 or 4.10000000000000036e-15 < y

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 54.0

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 54.0%

      \[\leadsto \color{blue}{5 \cdot y} \]

   if -1.2e33 < y < 4.10000000000000036e-15

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(t + 2 \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot z + t\right)} \]

      3. accelerator-lowering-fma.f64 79.4

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(2, z, t\right)} \]

   5. Simplified 79.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+33}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+87}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+76}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.8e+87) (* x t) (if (<= t 4.6e+76) (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.8e+87) {
		tmp = x * t;
	} else if (t <= 4.6e+76) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.8d+87)) then
        tmp = x * t
    else if (t <= 4.6d+76) then
        tmp = y * 5.0d0
    else
        tmp = x * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.8e+87) {
		tmp = x * t;
	} else if (t <= 4.6e+76) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.8e+87:
		tmp = x * t
	elif t <= 4.6e+76:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.8e+87)
		tmp = Float64(x * t);
	elseif (t <= 4.6e+76)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(x * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.8e+87)
		tmp = x * t;
	elseif (t <= 4.6e+76)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.8e+87], N[(x * t), $MachinePrecision], If[LessEqual[t, 4.6e+76], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.80000000000000015e87 or 4.60000000000000002e76 < t

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 68.4

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

   5. Simplified 68.4%

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

   if -2.80000000000000015e87 < t < 4.60000000000000002e76

   1. Initial program 99.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 39.8

         \[\leadsto \color{blue}{5 \cdot y} \]

   5. Simplified 39.8%

      \[\leadsto \color{blue}{5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+87}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+76}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ y \cdot 5 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y 5.0))

C

double code(double x, double y, double z, double t) {
	return y * 5.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * 5.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * 5.0;
}

Python

def code(x, y, z, t):
	return y * 5.0

Julia

function code(x, y, z, t)
	return Float64(y * 5.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y * 5.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot y} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 34.1

      \[\leadsto \color{blue}{5 \cdot y} \]

5. Simplified 34.1%

   \[\leadsto \color{blue}{5 \cdot y} \]

6. Final simplification 34.1%

   \[\leadsto y \cdot 5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, B"
  :precision binary64
  (+ (* x (+ (+ (+ (+ y z) z) y) t)) (* y 5.0)))