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

Percentage Accurate: 92.9% → 97.4%

Time: 7.6s

Alternatives: 9

Speedup: 1.1×

• 25.6% 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: 92.9% 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: 97.4% 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 93.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 97.7

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

4. Applied rewrites 97.7%

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

Alternative 2: 50.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-45} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+128}\right):\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (or (<= t_1 -2e-45) (not (<= t_1 5e+128)))
     (* (/ y a) z)
     (* (/ x y) y))))

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-45) || !(t_1 <= 5e+128)) {
		tmp = (y / a) * z;
	} else {
		tmp = (x / y) * 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 = (y * (z - t)) / a
    if ((t_1 <= (-2d-45)) .or. (.not. (t_1 <= 5d+128))) then
        tmp = (y / a) * z
    else
        tmp = (x / y) * 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 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -2e-45) || !(t_1 <= 5e+128)) {
		tmp = (y / a) * z;
	} else {
		tmp = (x / y) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if (t_1 <= -2e-45) or not (t_1 <= 5e+128):
		tmp = (y / a) * z
	else:
		tmp = (x / y) * y
	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-45) || !(t_1 <= 5e+128))
		tmp = Float64(Float64(y / a) * z);
	else
		tmp = Float64(Float64(x / y) * y);
	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-45) || ~((t_1 <= 5e+128)))
		tmp = (y / a) * z;
	else
		tmp = (x / y) * y;
	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-45], N[Not[LessEqual[t$95$1, 5e+128]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * z), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.99999999999999997e-45 or 5e128 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.6%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 52.6

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

   5. Applied rewrites 52.6%

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

   if -1.99999999999999997e-45 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5e128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.2%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2 \cdot 10^{-45} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 5 \cdot 10^{+128}\right):\\ \;\;\;\;\frac{y}{a} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-45} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+128}\right):\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (or (<= t_1 -2e-45) (not (<= t_1 5e+128)))
     (* (/ z a) y)
     (* (/ x y) y))))

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-45) || !(t_1 <= 5e+128)) {
		tmp = (z / a) * y;
	} else {
		tmp = (x / y) * 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 = (y * (z - t)) / a
    if ((t_1 <= (-2d-45)) .or. (.not. (t_1 <= 5d+128))) then
        tmp = (z / a) * y
    else
        tmp = (x / y) * 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 = (y * (z - t)) / a;
	double tmp;
	if ((t_1 <= -2e-45) || !(t_1 <= 5e+128)) {
		tmp = (z / a) * y;
	} else {
		tmp = (x / y) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if (t_1 <= -2e-45) or not (t_1 <= 5e+128):
		tmp = (z / a) * y
	else:
		tmp = (x / y) * y
	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-45) || !(t_1 <= 5e+128))
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = Float64(Float64(x / y) * y);
	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-45) || ~((t_1 <= 5e+128)))
		tmp = (z / a) * y;
	else
		tmp = (x / y) * y;
	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-45], N[Not[LessEqual[t$95$1, 5e+128]], $MachinePrecision]], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -1.99999999999999997e-45 or 5e128 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 45.0%

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

   if -1.99999999999999997e-45 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5e128

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.2%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2 \cdot 10^{-45} \lor \neg \left(\frac{y \cdot \left(z - t\right)}{a} \leq 5 \cdot 10^{+128}\right):\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-49} \lor \neg \left(z \leq 1.2 \cdot 10^{+78}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, -t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.9e-49) (not (<= z 1.2e+78)))
   (fma (/ y a) z x)
   (fma (/ y a) (- t) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.9e-49) || !(z <= 1.2e+78)) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = fma((y / a), -t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.9e-49) || !(z <= 1.2e+78))
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = fma(Float64(y / a), Float64(-t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.9e-49], N[Not[LessEqual[z, 1.2e+78]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * (-t) + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.90000000000000011e-49 or 1.1999999999999999e78 < z

   1. Initial program 92.7%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -3.90000000000000011e-49 < z < 1.1999999999999999e78

   1. Initial program 93.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   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. lower-neg.f64 90.4

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

   7. Applied rewrites 90.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-49} \lor \neg \left(z \leq 1.2 \cdot 10^{+78}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, -t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-49} \lor \neg \left(z \leq 2.3 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.5e-49) (not (<= z 2.3e-42)))
   (fma (/ y a) z x)
   (- x (* (/ t a) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.5e-49) || !(z <= 2.3e-42)) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = x - ((t / a) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.5e-49) || !(z <= 2.3e-42))
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(x - Float64(Float64(t / a) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.5e-49], N[Not[LessEqual[z, 2.3e-42]], $MachinePrecision]], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(x - N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000006e-49 or 2.30000000000000004e-42 < z

   1. Initial program 93.0%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -3.50000000000000006e-49 < z < 2.30000000000000004e-42

   1. Initial program 93.9%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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-sign-sub-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. lower--.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-49} \lor \neg \left(z \leq 2.3 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{a} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+269} \lor \neg \left(t \leq 1.2 \cdot 10^{+194}\right):\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -3.1e+269) (not (<= t 1.2e+194)))
   (* (- y) (/ t a))
   (fma (/ y a) z x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -3.1e+269) || !(t <= 1.2e+194)) {
		tmp = -y * (t / a);
	} else {
		tmp = fma((y / a), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -3.1e+269) || !(t <= 1.2e+194))
		tmp = Float64(Float64(-y) * Float64(t / a));
	else
		tmp = fma(Float64(y / a), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -3.1e+269], N[Not[LessEqual[t, 1.2e+194]], $MachinePrecision]], N[((-y) * N[(t / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.10000000000000003e269 or 1.2e194 < t

   1. Initial program 89.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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-sign-sub-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. lower--.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.6%

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

   if -3.10000000000000003e269 < t < 1.2e194

   1. Initial program 93.9%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.5

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

   5. Applied rewrites 80.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+269} \lor \neg \left(t \leq 1.2 \cdot 10^{+194}\right):\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+269}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.1e+269)
   (* (- y) (/ t a))
   (if (<= t 1.2e+194) (fma (/ y a) z x) (/ (* (- t) y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.1e+269) {
		tmp = -y * (t / a);
	} else if (t <= 1.2e+194) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = (-t * y) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.1e+269)
		tmp = Float64(Float64(-y) * Float64(t / a));
	elseif (t <= 1.2e+194)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(Float64(Float64(-t) * y) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.1e+269], N[((-y) * N[(t / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e+194], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[((-t) * y), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.10000000000000003e269

   1. Initial program 85.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   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-sign-sub-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. lower--.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.8%

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

   if -3.10000000000000003e269 < t < 1.2e194

   1. Initial program 93.9%

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

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if 1.2e194 < t

   1. Initial program 91.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 76.9

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

   7. Applied rewrites 76.9%

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

   9. Applied rewrites 81.9%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 77.4%

       \[\leadsto \frac{\left(-t\right) \cdot y}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+269}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.0% 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 93.3%

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

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 74.3

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

5. Applied rewrites 74.3%

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

6. Final simplification 74.3%

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

Alternative 9: 29.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(x / y), $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 y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate--l+ N/A

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

   4. div-sub N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 81.2

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

5. Applied rewrites 81.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 29.4%

   \[\leadsto \frac{x}{y} \cdot y \]

9. Final simplification 29.4%

   \[\leadsto \frac{x}{y} \cdot y \]
10. Add Preprocessing

Developer Target 1: 99.2% 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 2024329 
(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)))