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

Percentage Accurate: 93.3% → 95.7%

Time: 11.1s

Alternatives: 12

Speedup: 1.1×

• 24.6% 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: 93.3% 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: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -9e-88) (fma (/ (- z t) a) y x) (+ x (/ (* (- z t) y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9e-88) {
		tmp = fma(((z - t) / a), y, x);
	} else {
		tmp = x + (((z - t) * y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9e-88)
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	else
		tmp = Float64(x + Float64(Float64(Float64(z - t) * y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9e-88], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.99999999999999982e-88

   1. Initial program 88.1%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   if -8.99999999999999982e-88 < a

   1. Initial program 97.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)))
   (if (<= t_1 -4e+97)
     t_1
     (if (<= t_1 2e+30)
       (+ x (/ (* z y) a))
       (if (<= t_1 2e+301) t_1 (* (/ (- z t) a) y))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_1;
	} else if (t_1 <= 2e+30) {
		tmp = x + ((z * y) / a);
	} else if (t_1 <= 2e+301) {
		tmp = t_1;
	} else {
		tmp = ((z - t) / a) * y;
	}
	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 = ((z - t) * y) / a
    if (t_1 <= (-4d+97)) then
        tmp = t_1
    else if (t_1 <= 2d+30) then
        tmp = x + ((z * y) / a)
    else if (t_1 <= 2d+301) then
        tmp = t_1
    else
        tmp = ((z - t) / a) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_1;
	} else if (t_1 <= 2e+30) {
		tmp = x + ((z * y) / a);
	} else if (t_1 <= 2e+301) {
		tmp = t_1;
	} else {
		tmp = ((z - t) / a) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) * y) / a
	tmp = 0
	if t_1 <= -4e+97:
		tmp = t_1
	elif t_1 <= 2e+30:
		tmp = x + ((z * y) / a)
	elif t_1 <= 2e+301:
		tmp = t_1
	else:
		tmp = ((z - t) / a) * y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / a)
	tmp = 0.0
	if (t_1 <= -4e+97)
		tmp = t_1;
	elseif (t_1 <= 2e+30)
		tmp = Float64(x + Float64(Float64(z * y) / a));
	elseif (t_1 <= 2e+301)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(z - t) / a) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) * y) / a;
	tmp = 0.0;
	if (t_1 <= -4e+97)
		tmp = t_1;
	elseif (t_1 <= 2e+30)
		tmp = x + ((z * y) / a);
	elseif (t_1 <= 2e+301)
		tmp = t_1;
	else
		tmp = ((z - t) / a) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+97], t$95$1, If[LessEqual[t$95$1, 2e+30], N[(x + N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], t$95$1, N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.0000000000000003e97 or 2e30 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2.00000000000000011e301

   1. Initial program 92.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 84.9

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

   5. Simplified 84.9%

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

   if -4.0000000000000003e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e30

   1. Initial program 98.4%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 93.1

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

   5. Simplified 93.1%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 84.3

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

   5. Simplified 84.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 98.1

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

   7. Applied egg-rr 98.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -4 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ t_2 := \frac{z - t}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+97}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+212}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)) (t_2 (* (/ (- z t) a) y)))
   (if (<= t_1 -4e+97) t_2 (if (<= t_1 1e+212) (+ x (/ (* z y) a)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double t_2 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_2;
	} else if (t_1 <= 1e+212) {
		tmp = x + ((z * y) / a);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = ((z - t) * y) / a
    t_2 = ((z - t) / a) * y
    if (t_1 <= (-4d+97)) then
        tmp = t_2
    else if (t_1 <= 1d+212) then
        tmp = x + ((z * y) / a)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double t_2 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_2;
	} else if (t_1 <= 1e+212) {
		tmp = x + ((z * y) / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) * y) / a
	t_2 = ((z - t) / a) * y
	tmp = 0
	if t_1 <= -4e+97:
		tmp = t_2
	elif t_1 <= 1e+212:
		tmp = x + ((z * y) / a)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / a)
	t_2 = Float64(Float64(Float64(z - t) / a) * y)
	tmp = 0.0
	if (t_1 <= -4e+97)
		tmp = t_2;
	elseif (t_1 <= 1e+212)
		tmp = Float64(x + Float64(Float64(z * y) / a));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) * y) / a;
	t_2 = ((z - t) / a) * y;
	tmp = 0.0;
	if (t_1 <= -4e+97)
		tmp = t_2;
	elseif (t_1 <= 1e+212)
		tmp = x + ((z * y) / a);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+97], t$95$2, If[LessEqual[t$95$1, 1e+212], N[(x + N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.0000000000000003e97 or 9.9999999999999991e211 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 86.4

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

   5. Simplified 86.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 89.1

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

   7. Applied egg-rr 89.1%

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

   if -4.0000000000000003e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 9.9999999999999991e211

   1. Initial program 98.6%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 87.4

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

   5. Simplified 87.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -4 \cdot 10^{+97}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 10^{+212}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ t_2 := \frac{z - t}{a} \cdot y\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+97}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)) (t_2 (* (/ (- z t) a) y)))
   (if (<= t_1 -4e+97) t_2 (if (<= t_1 2e+30) (fma y (/ z a) x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double t_2 = ((z - t) / a) * y;
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_2;
	} else if (t_1 <= 2e+30) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / a)
	t_2 = Float64(Float64(Float64(z - t) / a) * y)
	tmp = 0.0
	if (t_1 <= -4e+97)
		tmp = t_2;
	elseif (t_1 <= 2e+30)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+97], t$95$2, If[LessEqual[t$95$1, 2e+30], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.0000000000000003e97 or 2e30 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 90.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 84.7

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

   5. Simplified 84.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 83.2

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

   7. Applied egg-rr 83.2%

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

   if -4.0000000000000003e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e30

   1. Initial program 98.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 92.2

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

   5. Simplified 92.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -4 \cdot 10^{+97}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+97}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)))
   (if (<= t_1 -4e+97) (* z (/ y a)) (if (<= t_1 0.002) x (/ (* z y) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = z * (y / a);
	} else if (t_1 <= 0.002) {
		tmp = x;
	} else {
		tmp = (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 = ((z - t) * y) / a
    if (t_1 <= (-4d+97)) then
        tmp = z * (y / a)
    else if (t_1 <= 0.002d0) then
        tmp = x
    else
        tmp = (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 = ((z - t) * y) / a;
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = z * (y / a);
	} else if (t_1 <= 0.002) {
		tmp = x;
	} else {
		tmp = (z * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) * y) / a
	tmp = 0
	if t_1 <= -4e+97:
		tmp = z * (y / a)
	elif t_1 <= 0.002:
		tmp = x
	else:
		tmp = (z * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / a)
	tmp = 0.0
	if (t_1 <= -4e+97)
		tmp = Float64(z * Float64(y / a));
	elseif (t_1 <= 0.002)
		tmp = x;
	else
		tmp = Float64(Float64(z * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) * y) / a;
	tmp = 0.0;
	if (t_1 <= -4e+97)
		tmp = z * (y / a);
	elseif (t_1 <= 0.002)
		tmp = x;
	else
		tmp = (z * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+97], N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], x, N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.0000000000000003e97

   1. Initial program 87.7%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 38.9

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

   5. Simplified 38.9%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{y \cdot z}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{y}} \cdot z} \]

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 42.4

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

   7. Applied egg-rr 42.4%

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

   if -4.0000000000000003e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e-3

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 79.8%

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

   if 2e-3 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 93.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 47.6

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

   5. Simplified 47.6%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -4 \cdot 10^{+97}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot y}{a}\\ t_2 := z \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+97}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) y) a)) (t_2 (* z (/ y a))))
   (if (<= t_1 -4e+97) t_2 (if (<= t_1 0.002) x t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double t_2 = z * (y / a);
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = ((z - t) * y) / a
    t_2 = z * (y / a)
    if (t_1 <= (-4d+97)) then
        tmp = t_2
    else if (t_1 <= 0.002d0) then
        tmp = x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) * y) / a;
	double t_2 = z * (y / a);
	double tmp;
	if (t_1 <= -4e+97) {
		tmp = t_2;
	} else if (t_1 <= 0.002) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((z - t) * y) / a
	t_2 = z * (y / a)
	tmp = 0
	if t_1 <= -4e+97:
		tmp = t_2
	elif t_1 <= 0.002:
		tmp = x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) * y) / a)
	t_2 = Float64(z * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -4e+97)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = x;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z - t) * y) / a;
	t_2 = z * (y / a);
	tmp = 0.0;
	if (t_1 <= -4e+97)
		tmp = t_2;
	elseif (t_1 <= 0.002)
		tmp = x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+97], t$95$2, If[LessEqual[t$95$1, 0.002], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.0000000000000003e97 or 2e-3 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 44.2

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

   5. Simplified 44.2%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{y \cdot z}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{y}} \cdot z} \]

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 45.0

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

   7. Applied egg-rr 45.0%

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

   if -4.0000000000000003e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e-3

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 79.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} \leq -4 \cdot 10^{+97}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, -t, x\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-18}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) (- t) x)))
   (if (<= t -1e+70) t_1 (if (<= t 7e-18) (+ x (/ (* z y) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), -t, x);
	double tmp;
	if (t <= -1e+70) {
		tmp = t_1;
	} else if (t <= 7e-18) {
		tmp = x + ((z * y) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), Float64(-t), x)
	tmp = 0.0
	if (t <= -1e+70)
		tmp = t_1;
	elseif (t <= 7e-18)
		tmp = Float64(x + Float64(Float64(z * y) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * (-t) + x), $MachinePrecision]}, If[LessEqual[t, -1e+70], t$95$1, If[LessEqual[t, 7e-18], N[(x + N[(N[(z * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.00000000000000007e70 or 6.9999999999999997e-18 < t

   1. Initial program 88.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 89.7

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

   7. Simplified 89.7%

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

   if -1.00000000000000007e70 < t < 6.9999999999999997e-18

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 89.3

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

   5. Simplified 89.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, -t, x\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-18}:\\ \;\;\;\;x + \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, -t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -t \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq -1.16 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* t (/ y a)))))
   (if (<= t -1.16e+94) t_1 (if (<= t 1.5e+185) (fma (/ y a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -(t * (y / a));
	double tmp;
	if (t <= -1.16e+94) {
		tmp = t_1;
	} else if (t <= 1.5e+185) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(-Float64(t * Float64(y / a)))
	tmp = 0.0
	if (t <= -1.16e+94)
		tmp = t_1;
	elseif (t <= 1.5e+185)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = (-N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[t, -1.16e+94], t$95$1, If[LessEqual[t, 1.5e+185], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1599999999999999e94 or 1.49999999999999997e185 < t

   1. Initial program 86.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 68.7

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

   5. Simplified 68.7%

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

   if -1.1599999999999999e94 < t < 1.49999999999999997e185

   1. Initial program 97.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 82.1

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

   5. Simplified 82.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 83.1

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

   7. Applied egg-rr 83.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{+94}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 96.1

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

4. Applied egg-rr 96.1%

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

Alternative 10: 71.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 70.1

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

5. Simplified 70.1%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. associate-*r* N/A

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

   4. div-inv N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 71.2

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

7. Applied egg-rr 71.2%

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

Alternative 11: 68.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 70.1

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

5. Simplified 70.1%

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

Alternative 12: 39.3% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 42.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 99.3% 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 2024198 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))