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

Percentage Accurate: 99.9% → 99.3%

Time: 9.4s

Alternatives: 11

Speedup: 1.1×

• 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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -125000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -125000.0)
     t_1
     (if (<= x 2.5) (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 <= -125000.0) {
		tmp = t_1;
	} else if (x <= 2.5) {
		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 <= -125000.0)
		tmp = t_1;
	elseif (x <= 2.5)
		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, -125000.0], t$95$1, If[LessEqual[x, 2.5], 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 < -125000 or 2.5 < 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 99.9

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

   5. Simplified 99.9%

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

   if -125000 < x < 2.5

   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 99.4

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -125000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot \left(t + \left(z + z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{+125}:\\ \;\;\;\;x \cdot \left(t + \left(y + y\right)\right)\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z (+ z t)))))
   (if (<= x -2.9e+227)
     t_1
     (if (<= x -5.7e+125)
       (* x (+ t (+ y y)))
       (if (<= x -1.02e-82) t_1 (if (<= x 3.4e-54) (* y 5.0) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + (z + t));
	double tmp;
	if (x <= -2.9e+227) {
		tmp = t_1;
	} else if (x <= -5.7e+125) {
		tmp = x * (t + (y + y));
	} else if (x <= -1.02e-82) {
		tmp = t_1;
	} else if (x <= 3.4e-54) {
		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 * (z + (z + t))
    if (x <= (-2.9d+227)) then
        tmp = t_1
    else if (x <= (-5.7d+125)) then
        tmp = x * (t + (y + y))
    else if (x <= (-1.02d-82)) then
        tmp = t_1
    else if (x <= 3.4d-54) 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 * (z + (z + t));
	double tmp;
	if (x <= -2.9e+227) {
		tmp = t_1;
	} else if (x <= -5.7e+125) {
		tmp = x * (t + (y + y));
	} else if (x <= -1.02e-82) {
		tmp = t_1;
	} else if (x <= 3.4e-54) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (z + (z + t))
	tmp = 0
	if x <= -2.9e+227:
		tmp = t_1
	elif x <= -5.7e+125:
		tmp = x * (t + (y + y))
	elif x <= -1.02e-82:
		tmp = t_1
	elif x <= 3.4e-54:
		tmp = y * 5.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + Float64(z + t)))
	tmp = 0.0
	if (x <= -2.9e+227)
		tmp = t_1;
	elseif (x <= -5.7e+125)
		tmp = Float64(x * Float64(t + Float64(y + y)));
	elseif (x <= -1.02e-82)
		tmp = t_1;
	elseif (x <= 3.4e-54)
		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 * (z + (z + t));
	tmp = 0.0;
	if (x <= -2.9e+227)
		tmp = t_1;
	elseif (x <= -5.7e+125)
		tmp = x * (t + (y + y));
	elseif (x <= -1.02e-82)
		tmp = t_1;
	elseif (x <= 3.4e-54)
		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[(z + N[(z + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e+227], t$95$1, If[LessEqual[x, -5.7e+125], N[(x * N[(t + N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.02e-82], t$95$1, If[LessEqual[x, 3.4e-54], N[(y * 5.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8999999999999998e227 or -5.6999999999999996e125 < x < -1.02000000000000007e-82 or 3.39999999999999987e-54 < 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 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 78.5

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

   5. Simplified 78.5%

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

      1. +-commutative N/A

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

      2. count-2 N/A

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

      3. associate-+r+ N/A

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

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

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

      5. +-lowering-+.f64 78.5

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

   7. Applied egg-rr 78.5%

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

   if -2.8999999999999998e227 < x < -5.6999999999999996e125

   1. Initial program 95.8%

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

   3. Taylor expanded in 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 83.1

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

   5. Simplified 83.1%

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

   6. Taylor expanded in x around inf

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

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

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

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

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

      3. count-2 N/A

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

      4. +-lowering-+.f64 87.3

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

   8. Simplified 87.3%

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

   if -1.02000000000000007e-82 < x < 3.39999999999999987e-54

   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 64.1

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

   5. Simplified 64.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+227}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{+125}:\\ \;\;\;\;x \cdot \left(t + \left(y + y\right)\right)\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + \left(z + t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(t + \left(y + y\right)\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+228}:\\ \;\;\;\;x \cdot \left(z + t\right)\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-49}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ t (+ y y)))))
   (if (<= x -7.5e+228)
     (* x (+ z t))
     (if (<= x -1.12e-11) t_1 (if (<= x 6e-49) (* y 5.0) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t + (y + y));
	double tmp;
	if (x <= -7.5e+228) {
		tmp = x * (z + t);
	} else if (x <= -1.12e-11) {
		tmp = t_1;
	} else if (x <= 6e-49) {
		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 * (t + (y + y))
    if (x <= (-7.5d+228)) then
        tmp = x * (z + t)
    else if (x <= (-1.12d-11)) then
        tmp = t_1
    else if (x <= 6d-49) 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 * (t + (y + y));
	double tmp;
	if (x <= -7.5e+228) {
		tmp = x * (z + t);
	} else if (x <= -1.12e-11) {
		tmp = t_1;
	} else if (x <= 6e-49) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t + (y + y))
	tmp = 0
	if x <= -7.5e+228:
		tmp = x * (z + t)
	elif x <= -1.12e-11:
		tmp = t_1
	elif x <= 6e-49:
		tmp = y * 5.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t + Float64(y + y)))
	tmp = 0.0
	if (x <= -7.5e+228)
		tmp = Float64(x * Float64(z + t));
	elseif (x <= -1.12e-11)
		tmp = t_1;
	elseif (x <= 6e-49)
		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 * (t + (y + y));
	tmp = 0.0;
	if (x <= -7.5e+228)
		tmp = x * (z + t);
	elseif (x <= -1.12e-11)
		tmp = t_1;
	elseif (x <= 6e-49)
		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[(t + N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e+228], N[(x * N[(z + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.12e-11], t$95$1, If[LessEqual[x, 6e-49], N[(y * 5.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999994e228

   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 89.9

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

   5. Simplified 89.9%

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

      1. +-commutative N/A

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

      2. count-2 N/A

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

      3. associate-+r+ N/A

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

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

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

      5. +-lowering-+.f64 89.9

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

   7. Applied egg-rr 89.9%

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

   8. Taylor expanded in t around inf

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

   10. Simplified 86.4%

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

   if -7.49999999999999994e228 < x < -1.1200000000000001e-11 or 6e-49 < x

   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 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 67.5

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

   5. Simplified 67.5%

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

   6. Taylor expanded in x around inf

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

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

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

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

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

      3. count-2 N/A

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

      4. +-lowering-+.f64 66.7

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

   8. Simplified 66.7%

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

   if -1.1200000000000001e-11 < x < 6e-49

   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 60.3

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

   5. Simplified 60.3%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+228}:\\ \;\;\;\;x \cdot \left(z + t\right)\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(t + \left(y + y\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-49}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + \left(y + y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -4.9 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 20000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -4.9e-55)
     t_1
     (if (<= x 20000000000000.0) (fma x (fma y 2.0 t) (* 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 <= -4.9e-55) {
		tmp = t_1;
	} else if (x <= 20000000000000.0) {
		tmp = fma(x, fma(y, 2.0, t), (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 <= -4.9e-55)
		tmp = t_1;
	elseif (x <= 20000000000000.0)
		tmp = fma(x, fma(y, 2.0, t), 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, -4.9e-55], t$95$1, If[LessEqual[x, 20000000000000.0], N[(x * N[(y * 2.0 + t), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.90000000000000035e-55 or 2e13 < 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 97.6

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

   5. Simplified 97.6%

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

   if -4.90000000000000035e-55 < x < 2e13

   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 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 83.6

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

   5. Simplified 83.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-55}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{elif}\;x \leq 20000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, 2, t\right), y \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{fma}\left(2, y + z, t\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.35 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(y, 5, x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (fma 2.0 (+ y z) t))))
   (if (<= x -2e-53) t_1 (if (<= x 3.35e-21) (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 <= -2e-53) {
		tmp = t_1;
	} else if (x <= 3.35e-21) {
		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 <= -2e-53)
		tmp = t_1;
	elseif (x <= 3.35e-21)
		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, -2e-53], t$95$1, If[LessEqual[x, 3.35e-21], N[(y * 5.0 + N[(x * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000006e-53 or 3.3499999999999999e-21 < 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 97.0

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

   5. Simplified 97.0%

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

   if -2.00000000000000006e-53 < x < 3.3499999999999999e-21

   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 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 83.8

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

   5. Simplified 83.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. accelerator-lowering-fma.f64 83.8

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

   7. Applied egg-rr 83.8%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 83.8%

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

Alternative 6: 99.9% 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 7: 78.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(x, 2, 5\right)\\ \mathbf{if}\;y \leq -390:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(2, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (fma x 2.0 5.0))))
   (if (<= y -390.0) t_1 (if (<= y 4.5e+36) (* 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 <= -390.0) {
		tmp = t_1;
	} else if (y <= 4.5e+36) {
		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 <= -390.0)
		tmp = t_1;
	elseif (y <= 4.5e+36)
		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, -390.0], t$95$1, If[LessEqual[y, 4.5e+36], N[(x * N[(2.0 * z + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -390 or 4.49999999999999997e36 < y

   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 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 84.5

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

   5. Simplified 84.5%

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

   if -390 < y < 4.49999999999999997e36

   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 82.9

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

   5. Simplified 82.9%

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

Alternative 8: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y + \mathsf{fma}\left(z, 2, y + t\right), x, y \cdot 5\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (+ y (fma z 2.0 (+ y t))) x (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	return fma((y + fma(z, 2.0, (y + t))), x, (y * 5.0));
}

Julia

function code(x, y, z, t)
	return fma(Float64(y + fma(z, 2.0, Float64(y + t))), x, Float64(y * 5.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y + N[(z * 2.0 + N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 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. 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. distribute-rgt-in N/A

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

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

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

   4. associate-+l+ N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

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

   7. +-inverses N/A

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

   8. +-inverses N/A

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

   9. +-inverses N/A

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

   10. flip-+ N/A

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

   11. +-commutative N/A

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

   12. associate-+l+ N/A

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

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

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

   14. associate-+l+ N/A

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

   15. +-commutative N/A

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

   16. flip-+ N/A

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

   17. +-inverses N/A

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

   18. +-inverses N/A

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

   19. +-inverses N/A

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

   20. +-inverses N/A

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

   21. flip-+ N/A

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

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

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

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

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

4. Applied egg-rr 98.0%

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

   1. distribute-rgt-out N/A

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

   2. *-commutative N/A

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

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

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

   4. count-2 N/A

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

   5. associate-+l+ N/A

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-lowering-*.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 9: 57.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(z + t\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-50}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (+ z t))))
   (if (<= x -1.8e-80) t_1 (if (<= x 3.9e-50) (* y 5.0) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (z + t);
	double tmp;
	if (x <= -1.8e-80) {
		tmp = t_1;
	} else if (x <= 3.9e-50) {
		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 * (z + t)
    if (x <= (-1.8d-80)) then
        tmp = t_1
    else if (x <= 3.9d-50) 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 * (z + t);
	double tmp;
	if (x <= -1.8e-80) {
		tmp = t_1;
	} else if (x <= 3.9e-50) {
		tmp = y * 5.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (z + t)
	tmp = 0
	if x <= -1.8e-80:
		tmp = t_1
	elif x <= 3.9e-50:
		tmp = y * 5.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(z + t))
	tmp = 0.0
	if (x <= -1.8e-80)
		tmp = t_1;
	elseif (x <= 3.9e-50)
		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 * (z + t);
	tmp = 0.0;
	if (x <= -1.8e-80)
		tmp = t_1;
	elseif (x <= 3.9e-50)
		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[(z + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e-80], t$95$1, If[LessEqual[x, 3.9e-50], N[(y * 5.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e-80 or 3.90000000000000021e-50 < 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 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 74.3

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

   5. Simplified 74.3%

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

      1. +-commutative N/A

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

      2. count-2 N/A

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

      3. associate-+r+ N/A

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

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

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

      5. +-lowering-+.f64 74.3

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

   7. Applied egg-rr 74.3%

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

   8. Taylor expanded in t around inf

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

   10. Simplified 60.6%

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

   if -1.8e-80 < x < 3.90000000000000021e-50

   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 64.1

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

   5. Simplified 64.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \left(z + t\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-50}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-13}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-49}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.1e-13) (* x t) (if (<= x 5.8e-49) (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.1e-13) {
		tmp = x * t;
	} else if (x <= 5.8e-49) {
		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 <= (-4.1d-13)) then
        tmp = x * t
    else if (x <= 5.8d-49) 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 <= -4.1e-13) {
		tmp = x * t;
	} else if (x <= 5.8e-49) {
		tmp = y * 5.0;
	} else {
		tmp = x * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.1e-13:
		tmp = x * t
	elif x <= 5.8e-49:
		tmp = y * 5.0
	else:
		tmp = x * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.1e-13)
		tmp = Float64(x * t);
	elseif (x <= 5.8e-49)
		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 <= -4.1e-13)
		tmp = x * t;
	elseif (x <= 5.8e-49)
		tmp = y * 5.0;
	else
		tmp = x * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.1e-13], N[(x * t), $MachinePrecision], If[LessEqual[x, 5.8e-49], N[(y * 5.0), $MachinePrecision], N[(x * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1000000000000002e-13 or 5.8e-49 < 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 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 42.2

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

   5. Simplified 42.2%

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

   if -4.1000000000000002e-13 < x < 5.8e-49

   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 60.3

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

   5. Simplified 60.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-13}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-49}:\\ \;\;\;\;y \cdot 5\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.7% 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 29.3

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

5. Simplified 29.3%

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

6. Final simplification 29.3%

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

Reproduce

herbie shell --seed 2024204 
(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)))