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

Percentage Accurate: 99.9% → 100.0%

Time: 10.3s

Alternatives: 11

Speedup: 1.2×

• 26.7% 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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(y * 5.0 + N[(x * N[(N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-def 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. associate-+l+ 100.0%

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

   6. *-un-lft-identity 100.0%

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

   7. +-commutative 100.0%

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

   8. *-un-lft-identity 100.0%

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

   9. distribute-rgt-out 100.0%

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

   10. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 47.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-78} \lor \neg \left(x \leq 4.4 \cdot 10^{-30}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* 2.0 (* x z))))
   (if (<= x -1e+68)
     t_1
     (if (<= x -6.2e+16)
       (* x t)
       (if (or (<= x -3.8e-78) (not (<= x 4.4e-30))) t_1 (* y 5.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 * (x * z);
	double tmp;
	if (x <= -1e+68) {
		tmp = t_1;
	} else if (x <= -6.2e+16) {
		tmp = x * t;
	} else if ((x <= -3.8e-78) || !(x <= 4.4e-30)) {
		tmp = t_1;
	} else {
		tmp = y * 5.0;
	}
	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 = 2.0d0 * (x * z)
    if (x <= (-1d+68)) then
        tmp = t_1
    else if (x <= (-6.2d+16)) then
        tmp = x * t
    else if ((x <= (-3.8d-78)) .or. (.not. (x <= 4.4d-30))) then
        tmp = t_1
    else
        tmp = y * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 * (x * z);
	double tmp;
	if (x <= -1e+68) {
		tmp = t_1;
	} else if (x <= -6.2e+16) {
		tmp = x * t;
	} else if ((x <= -3.8e-78) || !(x <= 4.4e-30)) {
		tmp = t_1;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 * (x * z)
	tmp = 0
	if x <= -1e+68:
		tmp = t_1
	elif x <= -6.2e+16:
		tmp = x * t
	elif (x <= -3.8e-78) or not (x <= 4.4e-30):
		tmp = t_1
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 * Float64(x * z))
	tmp = 0.0
	if (x <= -1e+68)
		tmp = t_1;
	elseif (x <= -6.2e+16)
		tmp = Float64(x * t);
	elseif ((x <= -3.8e-78) || !(x <= 4.4e-30))
		tmp = t_1;
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 * (x * z);
	tmp = 0.0;
	if (x <= -1e+68)
		tmp = t_1;
	elseif (x <= -6.2e+16)
		tmp = x * t;
	elseif ((x <= -3.8e-78) || ~((x <= 4.4e-30)))
		tmp = t_1;
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+68], t$95$1, If[LessEqual[x, -6.2e+16], N[(x * t), $MachinePrecision], If[Or[LessEqual[x, -3.8e-78], N[Not[LessEqual[x, 4.4e-30]], $MachinePrecision]], t$95$1, N[(y * 5.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999953e67 or -6.2e16 < x < -3.7999999999999999e-78 or 4.39999999999999967e-30 < 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 z around inf 47.2%

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

   if -9.99999999999999953e67 < x < -6.2e16

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf 59.8%

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

   if -3.7999999999999999e-78 < x < 4.39999999999999967e-30

   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 66.9%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+68}:\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot t\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-78} \lor \neg \left(x \leq 4.4 \cdot 10^{-30}\right):\\ \;\;\;\;2 \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-69} \lor \neg \left(x \leq 8 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot \left(t + \left(y \cdot 2 + z \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.5e-69) (not (<= x 8e-91)))
   (* x (+ t (+ (* y 2.0) (* z 2.0))))
   (+ (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e-69) || !(x <= 8e-91)) {
		tmp = x * (t + ((y * 2.0) + (z * 2.0)));
	} else {
		tmp = (y * 5.0) + (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 <= (-4.5d-69)) .or. (.not. (x <= 8d-91))) then
        tmp = x * (t + ((y * 2.0d0) + (z * 2.0d0)))
    else
        tmp = (y * 5.0d0) + (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 <= -4.5e-69) || !(x <= 8e-91)) {
		tmp = x * (t + ((y * 2.0) + (z * 2.0)));
	} else {
		tmp = (y * 5.0) + (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.5e-69) or not (x <= 8e-91):
		tmp = x * (t + ((y * 2.0) + (z * 2.0)))
	else:
		tmp = (y * 5.0) + (x * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.5e-69) || !(x <= 8e-91))
		tmp = Float64(x * Float64(t + Float64(Float64(y * 2.0) + Float64(z * 2.0))));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.5e-69) || ~((x <= 8e-91)))
		tmp = x * (t + ((y * 2.0) + (z * 2.0)));
	else
		tmp = (y * 5.0) + (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.5e-69], N[Not[LessEqual[x, 8e-91]], $MachinePrecision]], N[(x * N[(t + N[(N[(y * 2.0), $MachinePrecision] + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000009e-69 or 8.00000000000000018e-91 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 92.2%

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

   if -4.50000000000000009e-69 < x < 8.00000000000000018e-91

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-def 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf 90.5%

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

      1. *-commutative 90.5%

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

   7. Simplified 90.5%

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

      1. fma-udef 90.5%

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

   9. Applied egg-rr 90.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-69} \lor \neg \left(x \leq 8 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot \left(t + \left(y \cdot 2 + z \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-12} \lor \neg \left(t \leq 7 \cdot 10^{+83}\right):\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot \left(y + z\right)\right) + y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.9e-12) (not (<= t 7e+83)))
   (+ (* y 5.0) (* x t))
   (+ (* 2.0 (* x (+ y z))) (* y 5.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.9e-12) || !(t <= 7e+83)) {
		tmp = (y * 5.0) + (x * t);
	} else {
		tmp = (2.0 * (x * (y + z))) + (y * 5.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.9d-12)) .or. (.not. (t <= 7d+83))) then
        tmp = (y * 5.0d0) + (x * t)
    else
        tmp = (2.0d0 * (x * (y + z))) + (y * 5.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.9e-12) || !(t <= 7e+83)) {
		tmp = (y * 5.0) + (x * t);
	} else {
		tmp = (2.0 * (x * (y + z))) + (y * 5.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.9e-12) or not (t <= 7e+83):
		tmp = (y * 5.0) + (x * t)
	else:
		tmp = (2.0 * (x * (y + z))) + (y * 5.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.9e-12) || !(t <= 7e+83))
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	else
		tmp = Float64(Float64(2.0 * Float64(x * Float64(y + z))) + Float64(y * 5.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.9e-12) || ~((t <= 7e+83)))
		tmp = (y * 5.0) + (x * t);
	else
		tmp = (2.0 * (x * (y + z))) + (y * 5.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.9e-12], N[Not[LessEqual[t, 7e+83]], $MachinePrecision]], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.89999999999999998e-12 or 6.99999999999999954e83 < t

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf 86.6%

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

      1. *-commutative 86.6%

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

   7. Simplified 86.6%

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

      1. fma-udef 86.5%

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

   9. Applied egg-rr 86.5%

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

   if -1.89999999999999998e-12 < t < 6.99999999999999954e83

   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 99.9%

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

      2. fma-def 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around 0 93.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-12} \lor \neg \left(t \leq 7 \cdot 10^{+83}\right):\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot \left(y + z\right)\right) + y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+36} \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;x \cdot \left(t + \left(y \cdot 2 + z \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + z \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4e+36) (not (<= x 2.5)))
   (* x (+ t (+ (* y 2.0) (* z 2.0))))
   (+ (* y 5.0) (* x (+ t (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4e+36) || !(x <= 2.5)) {
		tmp = x * (t + ((y * 2.0) + (z * 2.0)));
	} else {
		tmp = (y * 5.0) + (x * (t + (z * 2.0)));
	}
	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 <= (-4d+36)) .or. (.not. (x <= 2.5d0))) then
        tmp = x * (t + ((y * 2.0d0) + (z * 2.0d0)))
    else
        tmp = (y * 5.0d0) + (x * (t + (z * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4e+36) || !(x <= 2.5)) {
		tmp = x * (t + ((y * 2.0) + (z * 2.0)));
	} else {
		tmp = (y * 5.0) + (x * (t + (z * 2.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4e+36) or not (x <= 2.5):
		tmp = x * (t + ((y * 2.0) + (z * 2.0)))
	else:
		tmp = (y * 5.0) + (x * (t + (z * 2.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4e+36) || !(x <= 2.5))
		tmp = Float64(x * Float64(t + Float64(Float64(y * 2.0) + Float64(z * 2.0))));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(x * Float64(t + Float64(z * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4e+36) || ~((x <= 2.5)))
		tmp = x * (t + ((y * 2.0) + (z * 2.0)));
	else
		tmp = (y * 5.0) + (x * (t + (z * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4e+36], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(x * N[(t + N[(N[(y * 2.0), $MachinePrecision] + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * N[(t + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000017e36 or 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 99.3%

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

   if -4.00000000000000017e36 < 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. Taylor expanded in y around 0 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+36} \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;x \cdot \left(t + \left(y \cdot 2 + z \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot \left(t + z \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-90} \lor \neg \left(x \leq 5.6 \cdot 10^{-99}\right):\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.2e-90) (not (<= x 5.6e-99)))
   (* x (+ t (* z 2.0)))
   (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.2e-90) || !(x <= 5.6e-99)) {
		tmp = x * (t + (z * 2.0));
	} else {
		tmp = y * 5.0;
	}
	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 <= (-4.2d-90)) .or. (.not. (x <= 5.6d-99))) then
        tmp = x * (t + (z * 2.0d0))
    else
        tmp = y * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.2e-90) || !(x <= 5.6e-99)) {
		tmp = x * (t + (z * 2.0));
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.2e-90) or not (x <= 5.6e-99):
		tmp = x * (t + (z * 2.0))
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.2e-90) || !(x <= 5.6e-99))
		tmp = Float64(x * Float64(t + Float64(z * 2.0)));
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.2e-90) || ~((x <= 5.6e-99)))
		tmp = x * (t + (z * 2.0));
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.2e-90], N[Not[LessEqual[x, 5.6e-99]], $MachinePrecision]], N[(x * N[(t + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999998e-90 or 5.6000000000000001e-99 < 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 75.8%

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

   if -4.1999999999999998e-90 < x < 5.6000000000000001e-99

   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 70.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-90} \lor \neg \left(x \leq 5.6 \cdot 10^{-99}\right):\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+100} \lor \neg \left(y \leq 3.1 \cdot 10^{+85}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -6e+100) (not (<= y 3.1e+85)))
   (* y (+ 5.0 (* x 2.0)))
   (* x (+ t (* z 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6e+100) || !(y <= 3.1e+85)) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = x * (t + (z * 2.0));
	}
	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 ((y <= (-6d+100)) .or. (.not. (y <= 3.1d+85))) then
        tmp = y * (5.0d0 + (x * 2.0d0))
    else
        tmp = x * (t + (z * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -6e+100) || !(y <= 3.1e+85)) {
		tmp = y * (5.0 + (x * 2.0));
	} else {
		tmp = x * (t + (z * 2.0));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -6e+100) or not (y <= 3.1e+85):
		tmp = y * (5.0 + (x * 2.0))
	else:
		tmp = x * (t + (z * 2.0))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -6e+100) || !(y <= 3.1e+85))
		tmp = Float64(y * Float64(5.0 + Float64(x * 2.0)));
	else
		tmp = Float64(x * Float64(t + Float64(z * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -6e+100) || ~((y <= 3.1e+85)))
		tmp = y * (5.0 + (x * 2.0));
	else
		tmp = x * (t + (z * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -6e+100], N[Not[LessEqual[y, 3.1e+85]], $MachinePrecision]], N[(y * N[(5.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(t + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999971e100 or 3.10000000000000011e85 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.0%

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

   5. Simplified 86.0%

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

   if -5.99999999999999971e100 < y < 3.10000000000000011e85

   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 77.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+100} \lor \neg \left(y \leq 3.1 \cdot 10^{+85}\right):\\ \;\;\;\;y \cdot \left(5 + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-78} \lor \neg \left(x \leq 5.6 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.5e-78) (not (<= x 5.6e-91)))
   (* x (+ t (* z 2.0)))
   (+ (* y 5.0) (* x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.5e-78) || !(x <= 5.6e-91)) {
		tmp = x * (t + (z * 2.0));
	} else {
		tmp = (y * 5.0) + (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 <= (-3.5d-78)) .or. (.not. (x <= 5.6d-91))) then
        tmp = x * (t + (z * 2.0d0))
    else
        tmp = (y * 5.0d0) + (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 <= -3.5e-78) || !(x <= 5.6e-91)) {
		tmp = x * (t + (z * 2.0));
	} else {
		tmp = (y * 5.0) + (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.5e-78) or not (x <= 5.6e-91):
		tmp = x * (t + (z * 2.0))
	else:
		tmp = (y * 5.0) + (x * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.5e-78) || !(x <= 5.6e-91))
		tmp = Float64(x * Float64(t + Float64(z * 2.0)));
	else
		tmp = Float64(Float64(y * 5.0) + Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.5e-78) || ~((x <= 5.6e-91)))
		tmp = x * (t + (z * 2.0));
	else
		tmp = (y * 5.0) + (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.5e-78], N[Not[LessEqual[x, 5.6e-91]], $MachinePrecision]], N[(x * N[(t + N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * 5.0), $MachinePrecision] + N[(x * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4999999999999999e-78 or 5.6e-91 < 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 75.7%

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

   if -3.4999999999999999e-78 < x < 5.6e-91

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-def 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. associate-+l+ 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in t around inf 91.4%

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

      1. *-commutative 91.4%

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

   7. Simplified 91.4%

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

      1. fma-udef 91.3%

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

   9. Applied egg-rr 91.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-78} \lor \neg \left(x \leq 5.6 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot \left(t + z \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5 + x \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+76} \lor \neg \left(t \leq 10^{+54}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.25e+76) (not (<= t 1e+54))) (* x t) (* y 5.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.25e+76) || !(t <= 1e+54)) {
		tmp = x * t;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.25d+76)) .or. (.not. (t <= 1d+54))) then
        tmp = x * t
    else
        tmp = y * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.25e+76) || !(t <= 1e+54)) {
		tmp = x * t;
	} else {
		tmp = y * 5.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.25e+76) or not (t <= 1e+54):
		tmp = x * t
	else:
		tmp = y * 5.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.25e+76) || !(t <= 1e+54))
		tmp = Float64(x * t);
	else
		tmp = Float64(y * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.25e+76) || ~((t <= 1e+54)))
		tmp = x * t;
	else
		tmp = y * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.25e+76], N[Not[LessEqual[t, 1e+54]], $MachinePrecision]], N[(x * t), $MachinePrecision], N[(y * 5.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.24999999999999998e76 or 1.0000000000000001e54 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf 67.8%

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

   if -1.24999999999999998e76 < t < 1.0000000000000001e54

   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 39.7%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+76} \lor \neg \left(t \leq 10^{+54}\right):\\ \;\;\;\;x \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot 5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * (((y + z) * 2.0) + 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) * 2.0d0) + t)) + (y * 5.0d0)
end function

Java

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

Python

def code(x, y, z, t):
	return (x * (((y + z) * 2.0) + t)) + (y * 5.0)

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * (((y + z) * 2.0) + t)) + (y * 5.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[(N[(N[(y + z), $MachinePrecision] * 2.0), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + N[(y * 5.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. +-commutative 99.9%

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

   4. *-un-lft-identity 99.9%

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

   5. distribute-rgt-out 99.9%

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

   6. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 11: 30.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 31.3%

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

4. Final simplification 31.3%

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

Reproduce

herbie shell --seed 2024021 
(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)))