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

Percentage Accurate: 92.3% → 97.9%

Time: 6.3s

Alternatives: 7

Speedup: 1.1×

• 25.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 2: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-30} \lor \neg \left(x \leq 2.05 \cdot 10^{-119}\right):\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.7e-30) (not (<= x 2.05e-119)))
   (* (- 1.0 (/ y t)) x)
   (+ x (/ (* z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e-30) || !(x <= 2.05e-119)) {
		tmp = (1.0 - (y / t)) * x;
	} else {
		tmp = x + ((z * y) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.7d-30)) .or. (.not. (x <= 2.05d-119))) then
        tmp = (1.0d0 - (y / t)) * x
    else
        tmp = x + ((z * y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e-30) || !(x <= 2.05e-119)) {
		tmp = (1.0 - (y / t)) * x;
	} else {
		tmp = x + ((z * y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.7e-30) or not (x <= 2.05e-119):
		tmp = (1.0 - (y / t)) * x
	else:
		tmp = x + ((z * y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.7e-30) || !(x <= 2.05e-119))
		tmp = Float64(Float64(1.0 - Float64(y / t)) * x);
	else
		tmp = Float64(x + Float64(Float64(z * y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.7e-30) || ~((x <= 2.05e-119)))
		tmp = (1.0 - (y / t)) * x;
	else
		tmp = x + ((z * y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.7e-30], N[Not[LessEqual[x, 2.05e-119]], $MachinePrecision]], N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999987e-30 or 2.0500000000000001e-119 < x

   1. Initial program 89.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -2.69999999999999987e-30 < x < 2.0500000000000001e-119

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-30} \lor \neg \left(x \leq 2.05 \cdot 10^{-119}\right):\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-30} \lor \neg \left(x \leq 1.65 \cdot 10^{-80}\right):\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.3e-30) (not (<= x 1.65e-80)))
   (* (- 1.0 (/ y t)) x)
   (fma (/ z t) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.3e-30) || !(x <= 1.65e-80)) {
		tmp = (1.0 - (y / t)) * x;
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.3e-30) || !(x <= 1.65e-80))
		tmp = Float64(Float64(1.0 - Float64(y / t)) * x);
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.3e-30], N[Not[LessEqual[x, 1.65e-80]], $MachinePrecision]], N[(N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.29999999999999966e-30 or 1.65e-80 < x

   1. Initial program 89.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -4.29999999999999966e-30 < x < 1.65e-80

   1. Initial program 94.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-30} \lor \neg \left(x \leq 1.65 \cdot 10^{-80}\right):\\ \;\;\;\;\left(1 - \frac{y}{t}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+175} \lor \neg \left(y \leq 2 \cdot 10^{+237}\right):\\ \;\;\;\;\frac{y}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2e+175) (not (<= y 2e+237)))
   (* (/ y t) (- x))
   (fma (/ z t) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2e+175) || !(y <= 2e+237)) {
		tmp = (y / t) * -x;
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2e+175) || !(y <= 2e+237))
		tmp = Float64(Float64(y / t) * Float64(-x));
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2e+175], N[Not[LessEqual[y, 2e+237]], $MachinePrecision]], N[(N[(y / t), $MachinePrecision] * (-x)), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9999999999999999e175 or 1.99999999999999988e237 < y

   1. Initial program 76.5%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.2%

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

   10. Applied rewrites 68.6%

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

   if -1.9999999999999999e175 < y < 1.99999999999999988e237

   1. Initial program 94.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 90.1

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 74.6

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

   7. Applied rewrites 74.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+175} \lor \neg \left(y \leq 2 \cdot 10^{+237}\right):\\ \;\;\;\;\frac{y}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+175} \lor \neg \left(y \leq 1.4 \cdot 10^{+221}\right):\\ \;\;\;\;\frac{-x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.25e+175) (not (<= y 1.4e+221)))
   (* (/ (- x) t) y)
   (fma (/ z t) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.25e+175) || !(y <= 1.4e+221)) {
		tmp = (-x / t) * y;
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.25e+175) || !(y <= 1.4e+221))
		tmp = Float64(Float64(Float64(-x) / t) * y);
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.25e+175], N[Not[LessEqual[y, 1.4e+221]], $MachinePrecision]], N[(N[((-x) / t), $MachinePrecision] * y), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.24999999999999995e175 or 1.39999999999999994e221 < y

   1. Initial program 77.4%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.5%

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

   if -2.24999999999999995e175 < y < 1.39999999999999994e221

   1. Initial program 94.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 74.8

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

   7. Applied rewrites 74.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{+175} \lor \neg \left(y \leq 1.4 \cdot 10^{+221}\right):\\ \;\;\;\;\frac{-x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 91.6

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

4. Applied rewrites 91.6%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 68.9

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

7. Applied rewrites 68.9%

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

Alternative 7: 41.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 34.1

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

5. Applied rewrites 34.1%

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

7. Applied rewrites 37.9%

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

Developer Target 1: 90.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

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

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