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

Percentage Accurate: 92.8% → 97.1%

Time: 6.4s

Alternatives: 8

Speedup: 1.1×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y * (z - t)) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y * (z - t)) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

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

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 98.0%

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

Alternative 2: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+21} \lor \neg \left(t\_1 \leq 10^{+38}\right):\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (or (<= t_1 -2e+21) (not (<= t_1 1e+38)))
     (* (- t z) (/ y a))
     (- x (/ (* z y) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -2e+21) || !(t_1 <= 1e+38)) {
		tmp = (t - z) * (y / a);
	} else {
		tmp = x - ((z * y) / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    if ((t_1 <= (-2d+21)) .or. (.not. (t_1 <= 1d+38))) then
        tmp = (t - z) * (y / a)
    else
        tmp = x - ((z * y) / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -2e+21) || !(t_1 <= 1e+38)) {
		tmp = (t - z) * (y / a);
	} else {
		tmp = x - ((z * y) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if (t_1 <= -2e+21) or not (t_1 <= 1e+38):
		tmp = (t - z) * (y / a)
	else:
		tmp = x - ((z * y) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if ((t_1 <= -2e+21) || !(t_1 <= 1e+38))
		tmp = Float64(Float64(t - z) * Float64(y / a));
	else
		tmp = Float64(x - Float64(Float64(z * y) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	tmp = 0.0;
	if ((t_1 <= -2e+21) || ~((t_1 <= 1e+38)))
		tmp = (t - z) * (y / a);
	else
		tmp = x - ((z * y) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+21], N[Not[LessEqual[t$95$1, 1e+38]], $MachinePrecision]], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2e21 or 9.99999999999999977e37 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 89.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. fp-cancel-sign-sub N/A

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

      16. mul-1-neg N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 N/A

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

      22. lower-/.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -2e21 < (/.f64 (*.f64 y (-.f64 z t)) a) < 9.99999999999999977e37

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2 \cdot 10^{+21} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 10^{+38}\right):\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+166} \lor \neg \left(t\_1 \leq 10^{+38}\right):\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (or (<= t_1 -4e+166) (not (<= t_1 1e+38)))
     (* (- t z) (/ y a))
     (fma (/ t a) y x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -4e+166) || !(t_1 <= 1e+38)) {
		tmp = (t - z) * (y / a);
	} else {
		tmp = fma((t / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if ((t_1 <= -4e+166) || !(t_1 <= 1e+38))
		tmp = Float64(Float64(t - z) * Float64(y / a));
	else
		tmp = fma(Float64(t / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+166], N[Not[LessEqual[t$95$1, 1e+38]], $MachinePrecision]], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -3.99999999999999976e166 or 9.99999999999999977e37 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. mul-1-neg N/A

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

      14. *-commutative N/A

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

      15. fp-cancel-sign-sub N/A

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

      16. mul-1-neg N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f64 N/A

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

      22. lower-/.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -3.99999999999999976e166 < (/.f64 (*.f64 y (-.f64 z t)) a) < 9.99999999999999977e37

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -4 \cdot 10^{+166} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 10^{+38}\right):\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+93} \lor \neg \left(z \leq 1.05 \cdot 10^{+160}\right):\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.3e+93) (not (<= z 1.05e+160)))
   (* (- z) (/ y a))
   (fma (/ y a) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.3e+93) || !(z <= 1.05e+160)) {
		tmp = -z * (y / a);
	} else {
		tmp = fma((y / a), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.3e+93) || !(z <= 1.05e+160))
		tmp = Float64(Float64(-z) * Float64(y / a));
	else
		tmp = fma(Float64(y / a), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.3e+93], N[Not[LessEqual[z, 1.05e+160]], $MachinePrecision]], N[((-z) * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e93 or 1.04999999999999998e160 < z

   1. Initial program 93.6%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 57.3

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

   5. Applied rewrites 57.3%

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

   7. Applied rewrites 69.6%

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

   if -1.3e93 < z < 1.04999999999999998e160

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 82.9

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

   8. Applied rewrites 82.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+93} \lor \neg \left(z \leq 1.05 \cdot 10^{+160}\right):\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

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

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in z around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l/ N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 69.7

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

8. Applied rewrites 69.7%

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

Alternative 6: 69.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in z around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. +-commutative N/A

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

   5. associate-*l/ N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 68.7

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

5. Applied rewrites 68.7%

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

Alternative 7: 35.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{y}{a} \cdot t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in t around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 32.3

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

5. Applied rewrites 32.3%

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

7. Applied rewrites 33.4%

   \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]
8. Add Preprocessing

Alternative 8: 32.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{t}{a} \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in t around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 32.3

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

5. Applied rewrites 32.3%

   \[\leadsto \color{blue}{\frac{t}{a} \cdot y} \]
6. Add Preprocessing

Developer Target 1: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{t\_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (- x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (- x (/ (* y (- z t)) a))
       (- x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x - (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x - ((y * (z - t)) / a)
    else
        tmp = x - (y / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x - (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x - ((y * (z - t)) / a)
	else:
		tmp = x - (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x - Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x - Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x - Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x - (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x - ((y * (z - t)) / a);
	else
		tmp = x - (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x - N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -430450648655599/4000000000000000000000000) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t)))))))

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