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

Percentage Accurate: 100.0% → 99.7%

Time: 8.6s

Alternatives: 10

Speedup: N/A×

• 25.4% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (0.5 / t) * (x + (y - 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 = (0.5d0 / t) * (x + (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-inv N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.7

       \[\leadsto \frac{\color{blue}{0.5}}{t} \cdot \left(\left(x + y\right) - z\right) \]

   12. lift--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate--l+ N/A

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

   15. lower-+.f64 N/A

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

   16. lower--.f64 99.7

       \[\leadsto \frac{0.5}{t} \cdot \left(x + \color{blue}{\left(y - z\right)}\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]
5. Add Preprocessing

Alternative 2: 48.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.5 \cdot x}{t}\\ \mathbf{elif}\;x + y \leq 10^{-13}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-12)
   (/ (* 0.5 x) t)
   (if (<= (+ x y) 1e-13) (/ (* z -0.5) t) (/ (* 0.5 y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-12) {
		tmp = (0.5 * x) / t;
	} else if ((x + y) <= 1e-13) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (0.5 * y) / 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 + y) <= (-5d-12)) then
        tmp = (0.5d0 * x) / t
    else if ((x + y) <= 1d-13) then
        tmp = (z * (-0.5d0)) / t
    else
        tmp = (0.5d0 * y) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-12) {
		tmp = (0.5 * x) / t;
	} else if ((x + y) <= 1e-13) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (0.5 * y) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-12:
		tmp = (0.5 * x) / t
	elif (x + y) <= 1e-13:
		tmp = (z * -0.5) / t
	else:
		tmp = (0.5 * y) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-12)
		tmp = Float64(Float64(0.5 * x) / t);
	elseif (Float64(x + y) <= 1e-13)
		tmp = Float64(Float64(z * -0.5) / t);
	else
		tmp = Float64(Float64(0.5 * y) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-12)
		tmp = (0.5 * x) / t;
	elseif ((x + y) <= 1e-13)
		tmp = (z * -0.5) / t;
	else
		tmp = (0.5 * y) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-12], N[(N[(0.5 * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e-13], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -4.9999999999999997e-12

   1. Initial program 99.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   if -4.9999999999999997e-12 < (+.f64 x y) < 1e-13

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if 1e-13 < (+.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 inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot y}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 48.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.5 \cdot x}{t}\\ \mathbf{elif}\;x + y \leq 10^{-13}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-12)
   (/ (* 0.5 x) t)
   (if (<= (+ x y) 1e-13) (* z (/ -0.5 t)) (/ (* 0.5 y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-12) {
		tmp = (0.5 * x) / t;
	} else if ((x + y) <= 1e-13) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = (0.5 * y) / 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 + y) <= (-5d-12)) then
        tmp = (0.5d0 * x) / t
    else if ((x + y) <= 1d-13) then
        tmp = z * ((-0.5d0) / t)
    else
        tmp = (0.5d0 * y) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-12) {
		tmp = (0.5 * x) / t;
	} else if ((x + y) <= 1e-13) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = (0.5 * y) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-12:
		tmp = (0.5 * x) / t
	elif (x + y) <= 1e-13:
		tmp = z * (-0.5 / t)
	else:
		tmp = (0.5 * y) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-12)
		tmp = Float64(Float64(0.5 * x) / t);
	elseif (Float64(x + y) <= 1e-13)
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(Float64(0.5 * y) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-12)
		tmp = (0.5 * x) / t;
	elseif ((x + y) <= 1e-13)
		tmp = z * (-0.5 / t);
	else
		tmp = (0.5 * y) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-12], N[(N[(0.5 * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e-13], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -4.9999999999999997e-12

   1. Initial program 99.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   if -4.9999999999999997e-12 < (+.f64 x y) < 1e-13

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

      1. associate-*r/ N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.4

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

   7. Applied rewrites 74.4%

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

   if 1e-13 < (+.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 inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.5 \cdot x}{t}\\ \mathbf{elif}\;x + y \leq 10^{-13}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot -0.5}{t}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+175}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+124}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z -0.5) t)))
   (if (<= z -4.9e+175) t_1 (if (<= z 1.85e+124) (* (/ 0.5 t) (+ x y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * -0.5) / t;
	double tmp;
	if (z <= -4.9e+175) {
		tmp = t_1;
	} else if (z <= 1.85e+124) {
		tmp = (0.5 / t) * (x + y);
	} 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 = (z * (-0.5d0)) / t
    if (z <= (-4.9d+175)) then
        tmp = t_1
    else if (z <= 1.85d+124) then
        tmp = (0.5d0 / t) * (x + y)
    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 = (z * -0.5) / t;
	double tmp;
	if (z <= -4.9e+175) {
		tmp = t_1;
	} else if (z <= 1.85e+124) {
		tmp = (0.5 / t) * (x + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * -0.5) / t
	tmp = 0
	if z <= -4.9e+175:
		tmp = t_1
	elif z <= 1.85e+124:
		tmp = (0.5 / t) * (x + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * -0.5) / t)
	tmp = 0.0
	if (z <= -4.9e+175)
		tmp = t_1;
	elseif (z <= 1.85e+124)
		tmp = Float64(Float64(0.5 / t) * Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * -0.5) / t;
	tmp = 0.0;
	if (z <= -4.9e+175)
		tmp = t_1;
	elseif (z <= 1.85e+124)
		tmp = (0.5 / t) * (x + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -4.9e+175], t$95$1, If[LessEqual[z, 1.85e+124], N[(N[(0.5 / t), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.90000000000000001e175 or 1.85000000000000004e124 < 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -4.90000000000000001e175 < z < 1.85000000000000004e124

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.6

          \[\leadsto \frac{\color{blue}{0.5}}{t} \cdot \left(\left(x + y\right) - z\right) \]

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 99.6

          \[\leadsto \frac{0.5}{t} \cdot \left(x + \color{blue}{\left(y - z\right)}\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{1}{2}}{t} \cdot \color{blue}{\left(x + y\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 85.1

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(x + y\right)} \]

   7. Applied rewrites 85.1%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(x + y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -2e-154) (* (/ 0.5 t) (- x z)) (/ (- y z) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -2e-154) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (y - z) / (t * 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 + y) <= (-2d-154)) then
        tmp = (0.5d0 / t) * (x - z)
    else
        tmp = (y - z) / (t * 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 + y) <= -2e-154) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -2e-154:
		tmp = (0.5 / t) * (x - z)
	else:
		tmp = (y - z) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -2e-154)
		tmp = Float64(Float64(0.5 / t) * Float64(x - z));
	else
		tmp = Float64(Float64(y - z) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -2e-154)
		tmp = (0.5 / t) * (x - z);
	else
		tmp = (y - z) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -2e-154], N[(N[(0.5 / t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.9999999999999999e-154

   1. Initial program 99.2%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 67.1

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

   5. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 66.9

          \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x - z\right)} \]

   7. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x - z\right)} \]

   if -1.9999999999999999e-154 < (+.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 68.1

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

   5. Applied rewrites 68.1%

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

Alternative 6: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 10^{-13}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) 1e-13) (* (/ 0.5 t) (- x z)) (/ (+ x y) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 1e-13) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (x + y) / (t * 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 + y) <= 1d-13) then
        tmp = (0.5d0 / t) * (x - z)
    else
        tmp = (x + y) / (t * 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 + y) <= 1e-13) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (x + y) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= 1e-13:
		tmp = (0.5 / t) * (x - z)
	else:
		tmp = (x + y) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= 1e-13)
		tmp = Float64(Float64(0.5 / t) * Float64(x - z));
	else
		tmp = Float64(Float64(x + y) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= 1e-13)
		tmp = (0.5 / t) * (x - z);
	else
		tmp = (x + y) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], 1e-13], N[(N[(0.5 / t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 1e-13

   1. Initial program 99.4%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 70.5

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

   5. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 70.3

          \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x - z\right)} \]

   7. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x - z\right)} \]

   if 1e-13 < (+.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

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

      1. +-commutative N/A

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

      2. lower-+.f64 84.5

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

   5. Applied rewrites 84.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 10^{-13}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 10^{-13}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t} \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) 1e-13) (* (/ 0.5 t) (- x z)) (* (/ 0.5 t) (+ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 1e-13) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (0.5 / t) * (x + y);
	}
	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 + y) <= 1d-13) then
        tmp = (0.5d0 / t) * (x - z)
    else
        tmp = (0.5d0 / t) * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= 1e-13) {
		tmp = (0.5 / t) * (x - z);
	} else {
		tmp = (0.5 / t) * (x + y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= 1e-13:
		tmp = (0.5 / t) * (x - z)
	else:
		tmp = (0.5 / t) * (x + y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= 1e-13)
		tmp = Float64(Float64(0.5 / t) * Float64(x - z));
	else
		tmp = Float64(Float64(0.5 / t) * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= 1e-13)
		tmp = (0.5 / t) * (x - z);
	else
		tmp = (0.5 / t) * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], 1e-13], N[(N[(0.5 / t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / t), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 1e-13

   1. Initial program 99.4%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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 70.5

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

   5. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 70.3

          \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x - z\right)} \]

   7. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x - z\right)} \]

   if 1e-13 < (+.f64 x y)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval 99.7

          \[\leadsto \frac{\color{blue}{0.5}}{t} \cdot \left(\left(x + y\right) - z\right) \]

      12. lift--.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate--l+ N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 99.7

          \[\leadsto \frac{0.5}{t} \cdot \left(x + \color{blue}{\left(y - z\right)}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{1}{2}}{t} \cdot \color{blue}{\left(x + y\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 84.2

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(x + y\right)} \]

   7. Applied rewrites 84.2%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(x + y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 42.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.5 \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-12) (/ (* 0.5 x) t) (* z (/ -0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-12) {
		tmp = (0.5 * x) / t;
	} else {
		tmp = z * (-0.5 / 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 + y) <= (-5d-12)) then
        tmp = (0.5d0 * x) / t
    else
        tmp = z * ((-0.5d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-12) {
		tmp = (0.5 * x) / t;
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-12:
		tmp = (0.5 * x) / t
	else:
		tmp = z * (-0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-12)
		tmp = Float64(Float64(0.5 * x) / t);
	else
		tmp = Float64(z * Float64(-0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-12)
		tmp = (0.5 * x) / t;
	else
		tmp = z * (-0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-12], N[(N[(0.5 * x), $MachinePrecision] / t), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999997e-12

   1. Initial program 99.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   if -4.9999999999999997e-12 < (+.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 inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 41.4

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

   5. Applied rewrites 41.4%

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

      1. associate-*r/ N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 41.3

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

   7. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.5 \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.5}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x y) -5e-12) (* (/ 0.5 t) x) (* z (/ -0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-12) {
		tmp = (0.5 / t) * x;
	} else {
		tmp = z * (-0.5 / 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 + y) <= (-5d-12)) then
        tmp = (0.5d0 / t) * x
    else
        tmp = z * ((-0.5d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + y) <= -5e-12) {
		tmp = (0.5 / t) * x;
	} else {
		tmp = z * (-0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + y) <= -5e-12:
		tmp = (0.5 / t) * x
	else:
		tmp = z * (-0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + y) <= -5e-12)
		tmp = Float64(Float64(0.5 / t) * x);
	else
		tmp = Float64(z * Float64(-0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + y) <= -5e-12)
		tmp = (0.5 / t) * x;
	else
		tmp = z * (-0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-12], N[(N[(0.5 / t), $MachinePrecision] * x), $MachinePrecision], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.9999999999999997e-12

   1. Initial program 99.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   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. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

      1. associate-*l/ N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 46.1

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

   7. Applied rewrites 46.1%

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

   if -4.9999999999999997e-12 < (+.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 inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 41.4

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

   5. Applied rewrites 41.4%

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

      1. associate-*r/ N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 41.3

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

   7. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{-0.5}{t} \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.5}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return z * (-0.5 / 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 * ((-0.5d0) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
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. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 32.5

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

5. Applied rewrites 32.5%

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

   1. associate-*r/ N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 32.4

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

7. Applied rewrites 32.4%

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

8. Final simplification 32.4%

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

Reproduce

herbie shell --seed 2024214 
(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)))