Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 14

Speedup: 1.0×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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) / (t * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (t * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + x) - z) / (2.0d0 * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{\left(y + x\right) - z}{2 \cdot t} \]
4. Add Preprocessing

Alternative 2: 50.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y + x \leq 10^{-33}:\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) -2e+82)
   (* 0.5 (/ x t))
   (if (<= (+ y x) 1e-33) (/ (* -0.5 z) t) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e+82) {
		tmp = 0.5 * (x / t);
	} else if ((y + x) <= 1e-33) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = (y / t) * 0.5;
	}
	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 + x) <= (-2d+82)) then
        tmp = 0.5d0 * (x / t)
    else if ((y + x) <= 1d-33) then
        tmp = ((-0.5d0) * z) / t
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e+82) {
		tmp = 0.5 * (x / t);
	} else if ((y + x) <= 1e-33) {
		tmp = (-0.5 * z) / t;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= -2e+82:
		tmp = 0.5 * (x / t)
	elif (y + x) <= 1e-33:
		tmp = (-0.5 * z) / t
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= -2e+82)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (Float64(y + x) <= 1e-33)
		tmp = Float64(Float64(-0.5 * z) / t);
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= -2e+82)
		tmp = 0.5 * (x / t);
	elseif ((y + x) <= 1e-33)
		tmp = (-0.5 * z) / t;
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], -2e+82], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + x), $MachinePrecision], 1e-33], N[(N[(-0.5 * z), $MachinePrecision] / t), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -1.9999999999999999e82

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 47.3

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   if -1.9999999999999999e82 < (+.f64 x y) < 1.0000000000000001e-33

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 73.2

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.4%

      \[\leadsto \frac{-0.5 \cdot z}{\color{blue}{t}} \]

   if 1.0000000000000001e-33 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. div-add-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot \frac{1}{2} \]

      7. lower-+.f64 82.0

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot 0.5 \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{y + x}{t} \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto \frac{y}{t} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y + x \leq 10^{-33}:\\ \;\;\;\;\frac{-0.5 \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y + x \leq 10^{-33}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) -2e+82)
   (* 0.5 (/ x t))
   (if (<= (+ y x) 1e-33) (* (/ -0.5 t) z) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e+82) {
		tmp = 0.5 * (x / t);
	} else if ((y + x) <= 1e-33) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	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 + x) <= (-2d+82)) then
        tmp = 0.5d0 * (x / t)
    else if ((y + x) <= 1d-33) then
        tmp = ((-0.5d0) / t) * z
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e+82) {
		tmp = 0.5 * (x / t);
	} else if ((y + x) <= 1e-33) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= -2e+82:
		tmp = 0.5 * (x / t)
	elif (y + x) <= 1e-33:
		tmp = (-0.5 / t) * z
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= -2e+82)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (Float64(y + x) <= 1e-33)
		tmp = Float64(Float64(-0.5 / t) * z);
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= -2e+82)
		tmp = 0.5 * (x / t);
	elseif ((y + x) <= 1e-33)
		tmp = (-0.5 / t) * z;
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], -2e+82], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + x), $MachinePrecision], 1e-33], N[(N[(-0.5 / t), $MachinePrecision] * z), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -1.9999999999999999e82

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 47.3

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]

   if -1.9999999999999999e82 < (+.f64 x y) < 1.0000000000000001e-33

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 73.2

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]

   if 1.0000000000000001e-33 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. div-add-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot \frac{1}{2} \]

      7. lower-+.f64 82.0

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot 0.5 \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{y + x}{t} \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto \frac{y}{t} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{+82}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;y + x \leq 10^{-33}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\frac{0.5}{t} \cdot x\\ \mathbf{elif}\;y + x \leq 10^{-33}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) -2e+82)
   (* (/ 0.5 t) x)
   (if (<= (+ y x) 1e-33) (* (/ -0.5 t) z) (* (/ y t) 0.5))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e+82) {
		tmp = (0.5 / t) * x;
	} else if ((y + x) <= 1e-33) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	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 + x) <= (-2d+82)) then
        tmp = (0.5d0 / t) * x
    else if ((y + x) <= 1d-33) then
        tmp = ((-0.5d0) / t) * z
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e+82) {
		tmp = (0.5 / t) * x;
	} else if ((y + x) <= 1e-33) {
		tmp = (-0.5 / t) * z;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= -2e+82:
		tmp = (0.5 / t) * x
	elif (y + x) <= 1e-33:
		tmp = (-0.5 / t) * z
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= -2e+82)
		tmp = Float64(Float64(0.5 / t) * x);
	elseif (Float64(y + x) <= 1e-33)
		tmp = Float64(Float64(-0.5 / t) * z);
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= -2e+82)
		tmp = (0.5 / t) * x;
	elseif ((y + x) <= 1e-33)
		tmp = (-0.5 / t) * z;
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], -2e+82], N[(N[(0.5 / t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(y + x), $MachinePrecision], 1e-33], N[(N[(-0.5 / t), $MachinePrecision] * z), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -1.9999999999999999e82

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 47.3

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.2%

      \[\leadsto x \cdot \color{blue}{\frac{0.5}{t}} \]

   if -1.9999999999999999e82 < (+.f64 x y) < 1.0000000000000001e-33

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 73.2

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]

   if 1.0000000000000001e-33 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. div-add-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot \frac{1}{2} \]

      7. lower-+.f64 82.0

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot 0.5 \]

   5. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{y + x}{t} \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto \frac{y}{t} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\frac{0.5}{t} \cdot x\\ \mathbf{elif}\;y + x \leq 10^{-33}:\\ \;\;\;\;\frac{-0.5}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5 \cdot z}{t}\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+110}:\\ \;\;\;\;\frac{y + x}{t + t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* -0.5 z) t)))
   (if (<= z -2.5e+84) t_1 (if (<= z 2.3e+110) (/ (+ y x) (+ t t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-0.5 * z) / t;
	double tmp;
	if (z <= -2.5e+84) {
		tmp = t_1;
	} else if (z <= 2.3e+110) {
		tmp = (y + x) / (t + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-0.5d0) * z) / t
    if (z <= (-2.5d+84)) then
        tmp = t_1
    else if (z <= 2.3d+110) then
        tmp = (y + x) / (t + t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-0.5 * z) / t;
	double tmp;
	if (z <= -2.5e+84) {
		tmp = t_1;
	} else if (z <= 2.3e+110) {
		tmp = (y + x) / (t + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-0.5 * z) / t
	tmp = 0
	if z <= -2.5e+84:
		tmp = t_1
	elif z <= 2.3e+110:
		tmp = (y + x) / (t + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5 * z) / t)
	tmp = 0.0
	if (z <= -2.5e+84)
		tmp = t_1;
	elseif (z <= 2.3e+110)
		tmp = Float64(Float64(y + x) / Float64(t + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-0.5 * z) / t;
	tmp = 0.0;
	if (z <= -2.5e+84)
		tmp = t_1;
	elseif (z <= 2.3e+110)
		tmp = (y + x) / (t + t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-0.5 * z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -2.5e+84], t$95$1, If[LessEqual[z, 2.3e+110], N[(N[(y + x), $MachinePrecision] / N[(t + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e84 or 2.3e110 < z

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

      6. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

      7. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

      9. lower-/.f64 83.4

         \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.7%

      \[\leadsto \frac{-0.5 \cdot z}{\color{blue}{t}} \]

   if -2.5e84 < z < 2.3e110

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. lower--.f64 60.4

         \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   5. Applied rewrites 60.4%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - z}{\color{blue}{t \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x - z}{\color{blue}{2 \cdot t}} \]

      3. count-2 N/A

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

      4. lift-+.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{x + y}}{t + t} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 84.8

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

   10. Applied rewrites 84.8%

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

Alternative 6: 37.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y + x\right) - z \leq -2 \cdot 10^{-191}:\\ \;\;\;\;\frac{0.5}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (+ y x) z) -2e-191) (* (/ 0.5 t) x) (* (/ y t) 0.5)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y + x) - z) <= -2e-191) {
		tmp = (0.5 / t) * x;
	} else {
		tmp = (y / t) * 0.5;
	}
	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 + x) - z) <= (-2d-191)) then
        tmp = (0.5d0 / t) * x
    else
        tmp = (y / t) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((y + x) - z) <= -2e-191) {
		tmp = (0.5 / t) * x;
	} else {
		tmp = (y / t) * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y + x) - z) <= -2e-191:
		tmp = (0.5 / t) * x
	else:
		tmp = (y / t) * 0.5
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(y + x) - z) <= -2e-191)
		tmp = Float64(Float64(0.5 / t) * x);
	else
		tmp = Float64(Float64(y / t) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((y + x) - z) <= -2e-191)
		tmp = (0.5 / t) * x;
	else
		tmp = (y / t) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision], -2e-191], N[(N[(0.5 / t), $MachinePrecision] * x), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) z) < -2e-191

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

      3. lower-/.f64 38.0

         \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 37.9%

      \[\leadsto x \cdot \color{blue}{\frac{0.5}{t}} \]

   if -2e-191 < (-.f64 (+.f64 x y) z)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. div-add-rev N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x + y}{t}} \cdot \frac{1}{2} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot \frac{1}{2} \]

      7. lower-+.f64 67.2

         \[\leadsto \frac{\color{blue}{y + x}}{t} \cdot 0.5 \]

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{y + x}{t} \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{y}{t} \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.8%

      \[\leadsto \frac{y}{t} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - z \leq -2 \cdot 10^{-191}:\\ \;\;\;\;\frac{0.5}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\frac{x - z}{t + t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t + t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) -2e-202) (/ (- x z) (+ t t)) (/ (- y z) (+ t t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e-202) {
		tmp = (x - z) / (t + t);
	} else {
		tmp = (y - z) / (t + 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 ((y + x) <= (-2d-202)) then
        tmp = (x - z) / (t + t)
    else
        tmp = (y - z) / (t + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= -2e-202) {
		tmp = (x - z) / (t + t);
	} else {
		tmp = (y - z) / (t + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= -2e-202:
		tmp = (x - z) / (t + t)
	else:
		tmp = (y - z) / (t + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= -2e-202)
		tmp = Float64(Float64(x - z) / Float64(t + t));
	else
		tmp = Float64(Float64(y - z) / Float64(t + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= -2e-202)
		tmp = (x - z) / (t + t);
	else
		tmp = (y - z) / (t + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], -2e-202], N[(N[(x - z), $MachinePrecision] / N[(t + t), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(t + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.0000000000000001e-202

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. lower--.f64 76.0

         \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   5. Applied rewrites 76.0%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - z}{\color{blue}{t \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x - z}{\color{blue}{2 \cdot t}} \]

      3. count-2 N/A

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

      4. lift-+.f64 76.0

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

   7. Applied rewrites 76.0%

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

   if -2.0000000000000001e-202 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. lower--.f64 74.7

         \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

   5. Applied rewrites 74.7%

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{y - z}{\color{blue}{t \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y - z}{\color{blue}{2 \cdot t}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

      4. lower-+.f64 74.7

         \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

   7. Applied rewrites 74.7%

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\frac{x - z}{t + t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t + t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq 10^{-33}:\\ \;\;\;\;\frac{x - z}{t + t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{t + t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ y x) 1e-33) (/ (- x z) (+ t t)) (/ (+ y x) (+ t t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= 1e-33) {
		tmp = (x - z) / (t + t);
	} else {
		tmp = (y + x) / (t + 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 ((y + x) <= 1d-33) then
        tmp = (x - z) / (t + t)
    else
        tmp = (y + x) / (t + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y + x) <= 1e-33) {
		tmp = (x - z) / (t + t);
	} else {
		tmp = (y + x) / (t + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y + x) <= 1e-33:
		tmp = (x - z) / (t + t)
	else:
		tmp = (y + x) / (t + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y + x) <= 1e-33)
		tmp = Float64(Float64(x - z) / Float64(t + t));
	else
		tmp = Float64(Float64(y + x) / Float64(t + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y + x) <= 1e-33)
		tmp = (x - z) / (t + t);
	else
		tmp = (y + x) / (t + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y + x), $MachinePrecision], 1e-33], N[(N[(x - z), $MachinePrecision] / N[(t + t), $MachinePrecision]), $MachinePrecision], N[(N[(y + x), $MachinePrecision] / N[(t + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 1.0000000000000001e-33

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. lower--.f64 78.7

         \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   5. Applied rewrites 78.7%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - z}{\color{blue}{t \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x - z}{\color{blue}{2 \cdot t}} \]

      3. count-2 N/A

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

      4. lift-+.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if 1.0000000000000001e-33 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. lower--.f64 62.1

         \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   5. Applied rewrites 62.1%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x - z}{\color{blue}{t \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x - z}{\color{blue}{2 \cdot t}} \]

      3. count-2 N/A

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

      4. lift-+.f64 62.1

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

   7. Applied rewrites 62.1%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{x + y}}{t + t} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 82.0

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

   10. Applied rewrites 82.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq 10^{-33}:\\ \;\;\;\;\frac{x - z}{t + t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{t + t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{t} \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ 0.5 t) x))

C

double code(double x, double y, double z, double t) {
	return (0.5 / t) * x;
}

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 = (0.5d0 / t) * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return (0.5 / t) * x;
}

Python

def code(x, y, z, t):
	return (0.5 / t) * x

Julia

function code(x, y, z, t)
	return Float64(Float64(0.5 / t) * x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (0.5 / t) * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(0.5 / t), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{t}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{1}{2}} \]

   3. lower-/.f64 36.2

      \[\leadsto \color{blue}{\frac{x}{t}} \cdot 0.5 \]

5. Applied rewrites 36.2%

   \[\leadsto \color{blue}{\frac{x}{t} \cdot 0.5} \]
6. Step-by-step derivation

7. Applied rewrites 36.2%

   \[\leadsto x \cdot \color{blue}{\frac{0.5}{t}} \]

8. Final simplification 36.2%

   \[\leadsto \frac{0.5}{t} \cdot x \]
9. Add Preprocessing

Alternative 10: 3.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (-2.0 * t) * (z - y);
}

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 = ((-2.0d0) * t) * (z - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
4. Step-by-step derivation

   1. lower--.f64 70.8

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

5. Applied rewrites 70.8%

   \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{y - z}{\color{blue}{t \cdot 2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{y - z}{\color{blue}{2 \cdot t}} \]

   3. count-2-rev N/A

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

   4. lower-+.f64 70.8

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

7. Applied rewrites 70.8%

   \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y - z}{t + t}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(t + t\right)\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(t + t\right)}\right)} \]

   4. count-2 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(t \cdot 2\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{t \cdot 2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{2 \cdot t}\right)} \]

   10. count-2 N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\left(t + t\right)}\right)} \]

   11. flip-+ N/A

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

   12. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{\color{blue}{0}}{t - t}\right)} \]

   13. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{0}{\color{blue}{0}}\right)} \]

   14. distribute-neg-frac N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{0}}} \]

   15. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\frac{\color{blue}{0}}{0}} \]

   16. clear-num-rev N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \color{blue}{\frac{0}{0}} \]

   17. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{\color{blue}{t \cdot t - t \cdot t}}{0} \]

   18. +-inverses N/A

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

   19. flip-+ N/A

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

   20. count-2 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f64 N/A

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

   23. lower-*.f64 N/A

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

9. Applied rewrites 3.6%

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

10. Taylor expanded in x around 0

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

    1. lower--.f64 3.6

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

12. Applied rewrites 3.6%

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

13. Final simplification 3.6%

    \[\leadsto \left(-2 \cdot t\right) \cdot \left(z - y\right) \]
14. Add Preprocessing

Alternative 11: 3.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (z - x) * (-2.0 * t);
}

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 = (z - x) * ((-2.0d0) * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
4. Step-by-step derivation

   1. lower--.f64 70.8

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

5. Applied rewrites 70.8%

   \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{y - z}{\color{blue}{t \cdot 2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{y - z}{\color{blue}{2 \cdot t}} \]

   3. count-2-rev N/A

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

   4. lower-+.f64 70.8

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

7. Applied rewrites 70.8%

   \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y - z}{t + t}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(t + t\right)\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(t + t\right)}\right)} \]

   4. count-2 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(t \cdot 2\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{t \cdot 2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{2 \cdot t}\right)} \]

   10. count-2 N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\left(t + t\right)}\right)} \]

   11. flip-+ N/A

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

   12. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{\color{blue}{0}}{t - t}\right)} \]

   13. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{0}{\color{blue}{0}}\right)} \]

   14. distribute-neg-frac N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{0}}} \]

   15. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\frac{\color{blue}{0}}{0}} \]

   16. clear-num-rev N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \color{blue}{\frac{0}{0}} \]

   17. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{\color{blue}{t \cdot t - t \cdot t}}{0} \]

   18. +-inverses N/A

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

   19. flip-+ N/A

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

   20. count-2 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f64 N/A

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

   23. lower-*.f64 N/A

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

9. Applied rewrites 3.6%

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

10. Taylor expanded in y around 0

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

    1. lower--.f64 3.7

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

12. Applied rewrites 3.7%

    \[\leadsto \color{blue}{\left(z - x\right)} \cdot \left(-2 \cdot t\right) \]
13. Add Preprocessing

Alternative 12: 3.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(-2 \cdot t\right) \cdot z \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (-2.0 * t) * z;
}

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 = ((-2.0d0) * t) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(-2.0 * t), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
4. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{1 \cdot z}}{t} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{1}{t} \cdot z\right)} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{1}{t}\right) \cdot z} \]

   4. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{t}\right) \cdot z \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{t}\right) \cdot z} \]

   6. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\frac{-1}{2}} \cdot \frac{1}{t}\right) \cdot z \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot 1}{t}} \cdot z \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{2}}}{t} \cdot z \]

   9. lower-/.f64 43.1

      \[\leadsto \color{blue}{\frac{-0.5}{t}} \cdot z \]

5. Applied rewrites 43.1%

   \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]
6. Step-by-step derivation

7. Applied rewrites 43.3%

   \[\leadsto \frac{-0.5 \cdot z}{\color{blue}{t}} \]
8. Step-by-step derivation

9. Applied rewrites 3.2%

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

10. Final simplification 3.2%

    \[\leadsto \left(-2 \cdot t\right) \cdot z \]
11. Add Preprocessing

Alternative 13: 3.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 = (t * y) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
4. Step-by-step derivation

   1. lower--.f64 70.8

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

5. Applied rewrites 70.8%

   \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{y - z}{\color{blue}{t \cdot 2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{y - z}{\color{blue}{2 \cdot t}} \]

   3. count-2-rev N/A

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

   4. lower-+.f64 70.8

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

7. Applied rewrites 70.8%

   \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y - z}{t + t}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(t + t\right)\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(t + t\right)}\right)} \]

   4. count-2 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(t \cdot 2\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{t \cdot 2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{2 \cdot t}\right)} \]

   10. count-2 N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\left(t + t\right)}\right)} \]

   11. flip-+ N/A

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

   12. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{\color{blue}{0}}{t - t}\right)} \]

   13. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{0}{\color{blue}{0}}\right)} \]

   14. distribute-neg-frac N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{0}}} \]

   15. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\frac{\color{blue}{0}}{0}} \]

   16. clear-num-rev N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \color{blue}{\frac{0}{0}} \]

   17. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{\color{blue}{t \cdot t - t \cdot t}}{0} \]

   18. +-inverses N/A

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

   19. flip-+ N/A

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

   20. count-2 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f64 N/A

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

   23. lower-*.f64 N/A

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

9. Applied rewrites 3.6%

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

10. Taylor expanded in y around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 2.9

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

12. Applied rewrites 2.9%

    \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot 2} \]
13. Add Preprocessing

Alternative 14: 3.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 = (2.0d0 * t) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
4. Step-by-step derivation

   1. lower--.f64 70.8

      \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]

5. Applied rewrites 70.8%

   \[\leadsto \frac{\color{blue}{y - z}}{t \cdot 2} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{y - z}{\color{blue}{t \cdot 2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{y - z}{\color{blue}{2 \cdot t}} \]

   3. count-2-rev N/A

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

   4. lower-+.f64 70.8

      \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]

7. Applied rewrites 70.8%

   \[\leadsto \frac{y - z}{\color{blue}{t + t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y - z}{t + t}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(t + t\right)\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(t + t\right)}\right)} \]

   4. count-2 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(t \cdot 2\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{t \cdot 2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{2 \cdot t}\right)} \]

   10. count-2 N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\left(t + t\right)}\right)} \]

   11. flip-+ N/A

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

   12. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{\color{blue}{0}}{t - t}\right)} \]

   13. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{0}{\color{blue}{0}}\right)} \]

   14. distribute-neg-frac N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{0}}} \]

   15. metadata-eval N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{1}{\frac{\color{blue}{0}}{0}} \]

   16. clear-num-rev N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \color{blue}{\frac{0}{0}} \]

   17. +-inverses N/A

       \[\leadsto \left(\mathsf{neg}\left(\left(y - z\right)\right)\right) \cdot \frac{\color{blue}{t \cdot t - t \cdot t}}{0} \]

   18. +-inverses N/A

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

   19. flip-+ N/A

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

   20. count-2 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f64 N/A

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

   23. lower-*.f64 N/A

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

9. Applied rewrites 3.6%

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

10. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 3.0

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

12. Applied rewrites 3.0%

    \[\leadsto \color{blue}{\left(2 \cdot t\right) \cdot x} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))