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

Percentage Accurate: 99.8% → 99.8%

Time: 7.0s

Alternatives: 12

Speedup: 1.0×

• 27.0% 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: 99.8% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * ((y + (z + (y + z))) + 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 + (y + z))) + t)) + (y * 5.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 2: 45.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + z\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.08 \cdot 10^{-128}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-183}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+69}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z z))))
   (if (<= z -2.4e+75)
     t_1
     (if (<= z -1.08e-128)
       (* x t)
       (if (<= z 6.1e-183) (* y 5.0) (if (<= z 6e+69) (* x t) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + z);
	double tmp;
	if (z <= -2.4e+75) {
		tmp = t_1;
	} else if (z <= -1.08e-128) {
		tmp = x * t;
	} else if (z <= 6.1e-183) {
		tmp = y * 5.0;
	} else if (z <= 6e+69) {
		tmp = x * t;
	} 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 * (z + z)
    if (z <= (-2.4d+75)) then
        tmp = t_1
    else if (z <= (-1.08d-128)) then
        tmp = x * t
    else if (z <= 6.1d-183) then
        tmp = y * 5.0d0
    else if (z <= 6d+69) then
        tmp = x * t
    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 * (z + z);
	double tmp;
	if (z <= -2.4e+75) {
		tmp = t_1;
	} else if (z <= -1.08e-128) {
		tmp = x * t;
	} else if (z <= 6.1e-183) {
		tmp = y * 5.0;
	} else if (z <= 6e+69) {
		tmp = x * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (z + z)
	tmp = 0
	if z <= -2.4e+75:
		tmp = t_1
	elif z <= -1.08e-128:
		tmp = x * t
	elif z <= 6.1e-183:
		tmp = y * 5.0
	elif z <= 6e+69:
		tmp = x * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + z))
	tmp = 0.0
	if (z <= -2.4e+75)
		tmp = t_1;
	elseif (z <= -1.08e-128)
		tmp = Float64(x * t);
	elseif (z <= 6.1e-183)
		tmp = Float64(y * 5.0);
	elseif (z <= 6e+69)
		tmp = Float64(x * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (z + z);
	tmp = 0.0;
	if (z <= -2.4e+75)
		tmp = t_1;
	elseif (z <= -1.08e-128)
		tmp = x * t;
	elseif (z <= 6.1e-183)
		tmp = y * 5.0;
	elseif (z <= 6e+69)
		tmp = x * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e+75], t$95$1, If[LessEqual[z, -1.08e-128], N[(x * t), $MachinePrecision], If[LessEqual[z, 6.1e-183], N[(y * 5.0), $MachinePrecision], If[LessEqual[z, 6e+69], N[(x * t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4e75 or 5.99999999999999967e69 < z

   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 z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 60.5

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

   5. Simplified 60.5%

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

      1. count-2 N/A

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

      2. *-commutative N/A

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

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

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

      4. +-lowering-+.f64 60.5

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

   7. Applied egg-rr 60.5%

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

   if -2.4e75 < z < -1.08e-128 or 6.1000000000000002e-183 < z < 5.99999999999999967e69

   1. Initial program 98.9%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 49.4

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

   5. Simplified 49.4%

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

   if -1.08e-128 < z < 6.1000000000000002e-183

   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 41.8

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

   5. Simplified 41.8%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+75}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \mathbf{elif}\;z \leq -1.08 \cdot 10^{-128}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-183}:\\ \;\;\;\;y \cdot 5\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+69}:\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -122:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -122.0)
     t_1
     (if (<= x 4.5e-8) (fma y 5.0 (* x (+ t (+ z z)))) 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 <= -122.0) {
		tmp = t_1;
	} else if (x <= 4.5e-8) {
		tmp = fma(y, 5.0, (x * (t + (z + z))));
	} 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 <= -122.0)
		tmp = t_1;
	elseif (x <= 4.5e-8)
		tmp = fma(y, 5.0, Float64(x * Float64(t + Float64(z + z))));
	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, -122.0], t$95$1, If[LessEqual[x, 4.5e-8], N[(y * 5.0 + N[(x * N[(t + N[(z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -122 or 4.49999999999999993e-8 < x

   1. Initial program 99.2%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 -122 < x < 4.49999999999999993e-8

   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 99.1

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

   4. Applied egg-rr 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -122:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.5e+57)
   (* x (fma 2.0 (+ y z) t))
   (if (<= z 9.2e+139)
     (fma x (fma y 2.0 t) (* y 5.0))
     (fma (+ z z) x (* y 5.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.5e+57) {
		tmp = x * fma(2.0, (y + z), t);
	} else if (z <= 9.2e+139) {
		tmp = fma(x, fma(y, 2.0, t), (y * 5.0));
	} else {
		tmp = fma((z + z), x, (y * 5.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.5e+57)
		tmp = Float64(x * fma(2.0, Float64(y + z), t));
	elseif (z <= 9.2e+139)
		tmp = fma(x, fma(y, 2.0, t), Float64(y * 5.0));
	else
		tmp = fma(Float64(z + z), x, Float64(y * 5.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.5e+57], N[(x * N[(2.0 * N[(y + z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.2e+139], N[(x * N[(y * 2.0 + t), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], N[(N[(z + z), $MachinePrecision] * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.49999999999999996e57

   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 83.9

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

   5. Simplified 83.9%

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

   if -4.49999999999999996e57 < z < 9.2e139

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 93.4

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

   5. Simplified 93.4%

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

   if 9.2e139 < z

   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. 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 94.5

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

   4. Applied egg-rr 94.5%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

         \[\leadsto x \cdot \left(2 \cdot z\right) + 5 \cdot \color{blue}{\left(y \cdot 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \left(2 \cdot z\right) + \color{blue}{\left(5 \cdot y\right) \cdot 1} \]

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

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

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

      20. metadata-eval N/A

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

      21. associate-*l* N/A

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

      22. associate-*r/ N/A

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

      23. associate-*l/ N/A

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

      24. associate-/l* N/A

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

      25. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, 2 \cdot z, \left(5 \cdot y\right) \cdot \color{blue}{1}\right) \]

   7. Simplified 93.1%

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

      1. count-2 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 93.1

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

   9. Applied egg-rr 93.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+294}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.8e-11)
   (* x (fma 2.0 y t))
   (if (<= y 1.12e+140)
     (* x (fma 2.0 z t))
     (if (<= y 4.2e+294) (* y 5.0) (* y (* x 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.8e-11) {
		tmp = x * fma(2.0, y, t);
	} else if (y <= 1.12e+140) {
		tmp = x * fma(2.0, z, t);
	} else if (y <= 4.2e+294) {
		tmp = y * 5.0;
	} else {
		tmp = y * (x * 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.8e-11)
		tmp = Float64(x * fma(2.0, y, t));
	elseif (y <= 1.12e+140)
		tmp = Float64(x * fma(2.0, z, t));
	elseif (y <= 4.2e+294)
		tmp = Float64(y * 5.0);
	else
		tmp = Float64(y * Float64(x * 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.8e-11], N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+140], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+294], N[(y * 5.0), $MachinePrecision], N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.79999999999999992e-11

   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 63.9

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

   5. Simplified 63.9%

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

   6. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 59.6

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

   8. Simplified 59.6%

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

   if -1.79999999999999992e-11 < y < 1.1199999999999999e140

   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 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 77.1

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

   5. Simplified 77.1%

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

   if 1.1199999999999999e140 < y < 4.2e294

   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 60.1

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

   5. Simplified 60.1%

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

   if 4.2e294 < y

   1. Initial program 83.3%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

      \[\leadsto y \cdot \color{blue}{\left(x \cdot 2\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+294}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+272}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \mathbf{elif}\;x \leq -122:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* x 2.0))))
   (if (<= x -1.35e+272)
     (* x (+ z z))
     (if (<= x -122.0) t_1 (if (<= x 4.5e-8) (* y 5.0) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (x * 2.0);
	double tmp;
	if (x <= -1.35e+272) {
		tmp = x * (z + z);
	} else if (x <= -122.0) {
		tmp = t_1;
	} else if (x <= 4.5e-8) {
		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 = y * (x * 2.0d0)
    if (x <= (-1.35d+272)) then
        tmp = x * (z + z)
    else if (x <= (-122.0d0)) then
        tmp = t_1
    else if (x <= 4.5d-8) 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 = y * (x * 2.0);
	double tmp;
	if (x <= -1.35e+272) {
		tmp = x * (z + z);
	} else if (x <= -122.0) {
		tmp = t_1;
	} else if (x <= 4.5e-8) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (x * 2.0)
	tmp = 0
	if x <= -1.35e+272:
		tmp = x * (z + z)
	elif x <= -122.0:
		tmp = t_1
	elif x <= 4.5e-8:
		tmp = y * 5.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(x * 2.0))
	tmp = 0.0
	if (x <= -1.35e+272)
		tmp = Float64(x * Float64(z + z));
	elseif (x <= -122.0)
		tmp = t_1;
	elseif (x <= 4.5e-8)
		tmp = Float64(y * 5.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (x * 2.0);
	tmp = 0.0;
	if (x <= -1.35e+272)
		tmp = x * (z + z);
	elseif (x <= -122.0)
		tmp = t_1;
	elseif (x <= 4.5e-8)
		tmp = y * 5.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+272], N[(x * N[(z + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -122.0], t$95$1, If[LessEqual[x, 4.5e-8], N[(y * 5.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000006e272

   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 z around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 64.0

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

   5. Simplified 64.0%

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

      1. count-2 N/A

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

      2. *-commutative N/A

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

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

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

      4. +-lowering-+.f64 64.0

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

   7. Applied egg-rr 64.0%

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

   if -1.35000000000000006e272 < x < -122 or 4.49999999999999993e-8 < x

   1. Initial program 99.2%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 50.4

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

   5. Simplified 50.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 49.5

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

   8. Simplified 49.5%

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

   if -122 < x < 4.49999999999999993e-8

   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 55.0

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

   5. Simplified 55.0%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+272}:\\ \;\;\;\;x \cdot \left(z + z\right)\\ \mathbf{elif}\;x \leq -122:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z + z, x, y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -1.85e-19) t_1 (if (<= x 1.6e-8) (fma (+ z z) x (* y 5.0)) 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 <= -1.85e-19) {
		tmp = t_1;
	} else if (x <= 1.6e-8) {
		tmp = fma((z + z), x, (y * 5.0));
	} 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 <= -1.85e-19)
		tmp = t_1;
	elseif (x <= 1.6e-8)
		tmp = fma(Float64(z + z), x, Float64(y * 5.0));
	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, -1.85e-19], t$95$1, If[LessEqual[x, 1.6e-8], N[(N[(z + z), $MachinePrecision] * x + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85000000000000003e-19 or 1.6000000000000001e-8 < x

   1. Initial program 99.2%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 98.4

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

   5. Simplified 98.4%

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

   if -1.85000000000000003e-19 < x < 1.6000000000000001e-8

   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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

         \[\leadsto x \cdot \left(2 \cdot z\right) + 5 \cdot \color{blue}{\left(y \cdot 1\right)} \]

      5. associate-*r* N/A

         \[\leadsto x \cdot \left(2 \cdot z\right) + \color{blue}{\left(5 \cdot y\right) \cdot 1} \]

      6. *-inverses N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

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

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

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

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

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

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

      20. metadata-eval N/A

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

      21. associate-*l* N/A

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

      22. associate-*r/ N/A

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

      23. associate-*l/ N/A

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

      24. associate-/l* N/A

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

      25. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, 2 \cdot z, \left(5 \cdot y\right) \cdot \color{blue}{1}\right) \]

   7. Simplified 79.3%

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

      1. count-2 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 79.3

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

   9. Applied egg-rr 79.3%

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

Alternative 8: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -1.12e-35) t_1 (if (<= x 1.25e-20) (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 <= -1.12e-35) {
		tmp = t_1;
	} else if (x <= 1.25e-20) {
		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 <= -1.12e-35)
		tmp = t_1;
	elseif (x <= 1.25e-20)
		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, -1.12e-35], t$95$1, If[LessEqual[x, 1.25e-20], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.12e-35 or 1.25e-20 < x

   1. Initial program 99.3%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 97.2

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

   5. Simplified 97.2%

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

   if -1.12e-35 < x < 1.25e-20

   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 99.9

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

   4. Applied egg-rr 99.9%

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

   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 78.6

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

   7. Simplified 78.6%

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

Alternative 9: 79.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -8 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+68}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -8e+37) t_1 (if (<= y 3e+68) (* 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+37) {
		tmp = t_1;
	} else if (y <= 3e+68) {
		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+37)
		tmp = t_1;
	elseif (y <= 3e+68)
		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+37], t$95$1, If[LessEqual[y, 3e+68], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.99999999999999963e37 or 3.0000000000000002e68 < y

   1. Initial program 99.1%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 87.0

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

   5. Simplified 87.0%

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

   if -7.99999999999999963e37 < y < 3.0000000000000002e68

   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 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 81.5

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

   5. Simplified 81.5%

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

Alternative 10: 63.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 y t))))
   (if (<= x -3.1e-28) t_1 (if (<= x 1.6e-8) (* y 5.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * fma(2.0, y, t);
	double tmp;
	if (x <= -3.1e-28) {
		tmp = t_1;
	} else if (x <= 1.6e-8) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * fma(2.0, y, t))
	tmp = 0.0
	if (x <= -3.1e-28)
		tmp = t_1;
	elseif (x <= 1.6e-8)
		tmp = Float64(y * 5.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(2.0 * y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.1e-28], t$95$1, If[LessEqual[x, 1.6e-8], N[(y * 5.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.09999999999999992e-28 or 1.6000000000000001e-8 < x

   1. Initial program 99.3%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 74.9

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

   8. Simplified 74.9%

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

   if -3.09999999999999992e-28 < x < 1.6000000000000001e-8

   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 57.4

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

   5. Simplified 57.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.2e-28) (* x t) (if (<= x 1.75e-8) (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.2e-28) {
		tmp = x * t;
	} else if (x <= 1.75e-8) {
		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 (x <= (-2.2d-28)) then
        tmp = x * t
    else if (x <= 1.75d-8) 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 (x <= -2.2e-28) {
		tmp = x * t;
	} else if (x <= 1.75e-8) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.2e-28:
		tmp = x * t
	elif x <= 1.75e-8:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.2e-28)
		tmp = Float64(x * t);
	elseif (x <= 1.75e-8)
		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 (x <= -2.2e-28)
		tmp = x * t;
	elseif (x <= 1.75e-8)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.2e-28], N[(x * t), $MachinePrecision], If[LessEqual[x, 1.75e-8], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.19999999999999996e-28 or 1.75000000000000012e-8 < x

   1. Initial program 99.3%

      \[x \cdot \left(\left(\left(\left(y + z\right) + z\right) + y\right) + t\right) + y \cdot 5 \]
   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 36.7

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

   5. Simplified 36.7%

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

   if -2.19999999999999996e-28 < x < 1.75000000000000012e-8

   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 57.4

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

   5. Simplified 57.4%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-8}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 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 99.5%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 28.3

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

5. Simplified 28.3%

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

6. Final simplification 28.3%

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

Reproduce

herbie shell --seed 2024198 
(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)))