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

Percentage Accurate: 92.8% → 98.1%

Time: 6.9s

Alternatives: 8

Speedup: 1.1×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - z}{\frac{t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5e+18) (fma (/ (- z x) t) y x) (- x (/ (- x z) (/ t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5e+18) {
		tmp = fma(((z - x) / t), y, x);
	} else {
		tmp = x - ((x - z) / (t / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5e+18)
		tmp = fma(Float64(Float64(z - x) / t), y, x);
	else
		tmp = Float64(x - Float64(Float64(x - z) / Float64(t / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5e+18], N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], N[(x - N[(N[(x - z), $MachinePrecision] / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5e18

   1. Initial program 90.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -5e18 < y

   1. Initial program 94.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

         \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{t}{y}}{z - x}}} \]

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.9

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

   4. Applied rewrites 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - z}{\frac{t}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-134}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x)))
   (if (<= z -6.5e-25) t_1 (if (<= z 3.7e-134) (* (- 1.0 (/ y t)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z * y) / t) + x;
	double tmp;
	if (z <= -6.5e-25) {
		tmp = t_1;
	} else if (z <= 3.7e-134) {
		tmp = (1.0 - (y / t)) * x;
	} 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 * y) / t) + x
    if (z <= (-6.5d-25)) then
        tmp = t_1
    else if (z <= 3.7d-134) then
        tmp = (1.0d0 - (y / t)) * x
    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 * y) / t) + x;
	double tmp;
	if (z <= -6.5e-25) {
		tmp = t_1;
	} else if (z <= 3.7e-134) {
		tmp = (1.0 - (y / t)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((z * y) / t) + x
	tmp = 0
	if z <= -6.5e-25:
		tmp = t_1
	elif z <= 3.7e-134:
		tmp = (1.0 - (y / t)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	tmp = 0.0
	if (z <= -6.5e-25)
		tmp = t_1;
	elseif (z <= 3.7e-134)
		tmp = Float64(Float64(1.0 - Float64(y / t)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((z * y) / t) + x;
	tmp = 0.0;
	if (z <= -6.5e-25)
		tmp = t_1;
	elseif (z <= 3.7e-134)
		tmp = (1.0 - (y / t)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -6.5e-25], t$95$1, If[LessEqual[z, 3.7e-134], N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e-25 or 3.7e-134 < z

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if -6.5e-25 < z < 3.7e-134

   1. Initial program 91.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\left(1 - \frac{y}{t}\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-134}:\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7000000:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (/ y t)) x)))
   (if (<= t -5.4e-8) t_1 (if (<= t 7000000.0) (/ (* (- z x) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - (y / t)) * x;
	double tmp;
	if (t <= -5.4e-8) {
		tmp = t_1;
	} else if (t <= 7000000.0) {
		tmp = ((z - x) * y) / 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 = (1.0d0 - (y / t)) * x
    if (t <= (-5.4d-8)) then
        tmp = t_1
    else if (t <= 7000000.0d0) then
        tmp = ((z - x) * y) / 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 = (1.0 - (y / t)) * x;
	double tmp;
	if (t <= -5.4e-8) {
		tmp = t_1;
	} else if (t <= 7000000.0) {
		tmp = ((z - x) * y) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (1.0 - (y / t)) * x
	tmp = 0
	if t <= -5.4e-8:
		tmp = t_1
	elif t <= 7000000.0:
		tmp = ((z - x) * y) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - Float64(y / t)) * x)
	tmp = 0.0
	if (t <= -5.4e-8)
		tmp = t_1;
	elseif (t <= 7000000.0)
		tmp = Float64(Float64(Float64(z - x) * y) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (1.0 - (y / t)) * x;
	tmp = 0.0;
	if (t <= -5.4e-8)
		tmp = t_1;
	elseif (t <= 7000000.0)
		tmp = ((z - x) * y) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -5.4e-8], t$95$1, If[LessEqual[t, 7000000.0], N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.40000000000000005e-8 or 7e6 < t

   1. Initial program 88.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -5.40000000000000005e-8 < t < 7e6

   1. Initial program 98.4%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

Alternative 4: 72.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{if}\;x \leq -5.4 \cdot 10^{-170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-213}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (/ y t)) x)))
   (if (<= x -5.4e-170) t_1 (if (<= x 4.1e-213) (/ (* z y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - (y / t)) * x;
	double tmp;
	if (x <= -5.4e-170) {
		tmp = t_1;
	} else if (x <= 4.1e-213) {
		tmp = (z * y) / 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 = (1.0d0 - (y / t)) * x
    if (x <= (-5.4d-170)) then
        tmp = t_1
    else if (x <= 4.1d-213) then
        tmp = (z * y) / 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 = (1.0 - (y / t)) * x;
	double tmp;
	if (x <= -5.4e-170) {
		tmp = t_1;
	} else if (x <= 4.1e-213) {
		tmp = (z * y) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (1.0 - (y / t)) * x
	tmp = 0
	if x <= -5.4e-170:
		tmp = t_1
	elif x <= 4.1e-213:
		tmp = (z * y) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - Float64(y / t)) * x)
	tmp = 0.0
	if (x <= -5.4e-170)
		tmp = t_1;
	elseif (x <= 4.1e-213)
		tmp = Float64(Float64(z * y) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (1.0 - (y / t)) * x;
	tmp = 0.0;
	if (x <= -5.4e-170)
		tmp = t_1;
	elseif (x <= 4.1e-213)
		tmp = (z * y) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.4e-170], t$95$1, If[LessEqual[x, 4.1e-213], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.3999999999999997e-170 or 4.09999999999999975e-213 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -5.3999999999999997e-170 < x < 4.09999999999999975e-213

   1. Initial program 93.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

Alternative 5: 55.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{+45}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;t \leq 7500000:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.65e+45) (* 1.0 x) (if (<= t 7500000.0) (* (/ y t) z) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.65e+45) {
		tmp = 1.0 * x;
	} else if (t <= 7500000.0) {
		tmp = (y / t) * z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.65d+45)) then
        tmp = 1.0d0 * x
    else if (t <= 7500000.0d0) then
        tmp = (y / t) * z
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.65e+45) {
		tmp = 1.0 * x;
	} else if (t <= 7500000.0) {
		tmp = (y / t) * z;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.65e+45:
		tmp = 1.0 * x
	elif t <= 7500000.0:
		tmp = (y / t) * z
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.65e+45)
		tmp = Float64(1.0 * x);
	elseif (t <= 7500000.0)
		tmp = Float64(Float64(y / t) * z);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.65e+45)
		tmp = 1.0 * x;
	elseif (t <= 7500000.0)
		tmp = (y / t) * z;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.65e+45], N[(1.0 * x), $MachinePrecision], If[LessEqual[t, 7500000.0], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.64999999999999996e45 or 7.5e6 < t

   1. Initial program 87.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 81.5

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 69.0%

       \[\leadsto 1 \cdot x \]

   if -2.64999999999999996e45 < t < 7.5e6

   1. Initial program 98.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

   7. Applied rewrites 56.7%

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

Alternative 6: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5000.0) (fma (/ (- z x) t) y x) (fma (/ y t) (- z x) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5000.0) {
		tmp = fma(((z - x) / t), y, x);
	} else {
		tmp = fma((y / t), (z - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5000.0)
		tmp = fma(Float64(Float64(z - x) / t), y, x);
	else
		tmp = fma(Float64(y / t), Float64(z - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5000.0], N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5e3

   1. Initial program 90.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -5e3 < y

   1. Initial program 94.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 7: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{t}, z - x, x\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return fma((y / t), (z - x), x);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 96.6

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

4. Applied rewrites 96.6%

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

Alternative 8: 38.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 * x;
}

Python

def code(x, y, z, t):
	return 1.0 * x

Julia

function code(x, y, z, t)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. div-inv N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. neg-mul-1 N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 96.5

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

4. Applied rewrites 96.5%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 64.9

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

7. Applied rewrites 64.9%

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

8. Taylor expanded in y around 0

   \[\leadsto 1 \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 39.5%

    \[\leadsto 1 \cdot x \]
11. Add Preprocessing

Developer Target 1: 90.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

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

  :alt
  (! :herbie-platform default (- x (+ (* x (/ y t)) (* (- z) (/ y t)))))

  (+ x (/ (* y (- z x)) t)))