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

Percentage Accurate: 92.2% → 97.6%

Time: 9.0s

Alternatives: 9

Speedup: 1.1×

• 25.3% 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: 92.2% 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: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.2%

   \[x + \frac{y \cdot \left(z - x\right)}{t} \]
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 97.2

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

4. Applied rewrites 97.2%

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

Alternative 2: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, -x, x\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- x) x)))
   (if (<= x -4e-26) t_1 (if (<= x 4.4e+78) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), -x, x);
	double tmp;
	if (x <= -4e-26) {
		tmp = t_1;
	} else if (x <= 4.4e+78) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), Float64(-x), x)
	tmp = 0.0
	if (x <= -4e-26)
		tmp = t_1;
	elseif (x <= 4.4e+78)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * (-x) + x), $MachinePrecision]}, If[LessEqual[x, -4e-26], t$95$1, If[LessEqual[x, 4.4e+78], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000002e-26 or 4.40000000000000028e78 < x

   1. Initial program 90.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.6

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

   7. Applied rewrites 91.6%

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

   if -4.0000000000000002e-26 < x < 4.40000000000000028e78

   1. Initial program 95.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. 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 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 88.7

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

   7. Applied rewrites 88.7%

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

Alternative 3: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{x \cdot y}{t}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (* x y) t))))
   (if (<= x -4e-26) t_1 (if (<= x 6.2e+78) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((x * y) / t);
	double tmp;
	if (x <= -4e-26) {
		tmp = t_1;
	} else if (x <= 6.2e+78) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(x * y) / t))
	tmp = 0.0
	if (x <= -4e-26)
		tmp = t_1;
	elseif (x <= 6.2e+78)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(x * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e-26], t$95$1, If[LessEqual[x, 6.2e+78], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000002e-26 or 6.2e78 < x

   1. Initial program 90.7%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -4.0000000000000002e-26 < x < 6.2e78

   1. Initial program 95.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. 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 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 88.7

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

   7. Applied rewrites 88.7%

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

Alternative 4: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - x}{t} \cdot y\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (- z x) t) y)))
   (if (<= y -2.1e+86) t_1 (if (<= y 1.35e-34) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z - x) / t) * y;
	double tmp;
	if (y <= -2.1e+86) {
		tmp = t_1;
	} else if (y <= 1.35e-34) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z - x) / t) * y)
	tmp = 0.0
	if (y <= -2.1e+86)
		tmp = t_1;
	elseif (y <= 1.35e-34)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.1e+86], t$95$1, If[LessEqual[y, 1.35e-34], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0999999999999999e86 or 1.35000000000000008e-34 < y

   1. Initial program 87.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 76.0

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

   5. Applied rewrites 76.0%

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

   7. Applied rewrites 81.9%

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

   if -2.0999999999999999e86 < y < 1.35000000000000008e-34

   1. Initial program 98.4%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 86.6

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

   4. Applied rewrites 86.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 82.3

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

   7. Applied rewrites 82.3%

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

Alternative 5: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -5.3 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.7 \cdot 10^{-303}:\\ \;\;\;\;\left(-x\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -5.3e-83) t_1 (if (<= t 8.7e-303) (* (- x) (/ y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (t <= -5.3e-83) {
		tmp = t_1;
	} else if (t <= 8.7e-303) {
		tmp = -x * (y / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (t <= -5.3e-83)
		tmp = t_1;
	elseif (t <= 8.7e-303)
		tmp = Float64(Float64(-x) * Float64(y / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -5.3e-83], t$95$1, If[LessEqual[t, 8.7e-303], N[((-x) * N[(y / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.3e-83 or 8.70000000000000009e-303 < t

   1. Initial program 91.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 94.5

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

   4. Applied rewrites 94.5%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 79.5

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

   7. Applied rewrites 79.5%

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

   if -5.3e-83 < t < 8.70000000000000009e-303

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 91.1

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.9%

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

   10. Applied rewrites 68.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 8.7 \cdot 10^{-303}:\\ \;\;\;\;\left(-x\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-116}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\frac{x \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.6e-116)
   (* z (/ y t))
   (if (<= y 6e-35) (/ (* x t) t) (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.6e-116) {
		tmp = z * (y / t);
	} else if (y <= 6e-35) {
		tmp = (x * t) / t;
	} else {
		tmp = (z / t) * y;
	}
	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 (y <= (-3.6d-116)) then
        tmp = z * (y / t)
    else if (y <= 6d-35) then
        tmp = (x * t) / t
    else
        tmp = (z / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.6e-116) {
		tmp = z * (y / t);
	} else if (y <= 6e-35) {
		tmp = (x * t) / t;
	} else {
		tmp = (z / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3.6e-116:
		tmp = z * (y / t)
	elif y <= 6e-35:
		tmp = (x * t) / t
	else:
		tmp = (z / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.6e-116)
		tmp = Float64(z * Float64(y / t));
	elseif (y <= 6e-35)
		tmp = Float64(Float64(x * t) / t);
	else
		tmp = Float64(Float64(z / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -3.6e-116)
		tmp = z * (y / t);
	elseif (y <= 6e-35)
		tmp = (x * t) / t;
	else
		tmp = (z / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.6e-116], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-35], N[(N[(x * t), $MachinePrecision] / t), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.59999999999999975e-116

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 48.6

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

   7. Applied rewrites 48.6%

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

   if -3.59999999999999975e-116 < y < 5.99999999999999978e-35

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 83.4

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

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 83.3

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 88.4

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

   10. Applied rewrites 88.4%

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

   11. Taylor expanded in t around inf

       \[\leadsto \frac{t \cdot x}{t} \]
   12. Step-by-step derivation

   13. Applied rewrites 54.1%

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

   if 5.99999999999999978e-35 < y

   1. Initial program 86.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   7. Applied rewrites 49.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-116}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\frac{x \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-194}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.05e-194) (* z (/ y t)) (* (/ z t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.05e-194) {
		tmp = z * (y / t);
	} else {
		tmp = (z / t) * y;
	}
	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 <= (-2.05d-194)) then
        tmp = z * (y / t)
    else
        tmp = (z / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.05e-194) {
		tmp = z * (y / t);
	} else {
		tmp = (z / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.05e-194:
		tmp = z * (y / t)
	else:
		tmp = (z / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.05e-194)
		tmp = Float64(z * Float64(y / t));
	else
		tmp = Float64(Float64(z / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.05e-194)
		tmp = z * (y / t);
	else
		tmp = (z / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.05e-194], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0500000000000001e-194

   1. Initial program 93.2%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 30.3

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

   7. Applied rewrites 30.3%

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

   if -2.0500000000000001e-194 < x

   1. Initial program 93.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
   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 42.3

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

   5. Applied rewrites 42.3%

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

   7. Applied rewrites 46.4%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-194}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% 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 93.2%

   \[x + \frac{y \cdot \left(z - x\right)}{t} \]
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 92.1

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

4. Applied rewrites 92.1%

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

5. Taylor expanded in z around inf

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

   1. lower-/.f64 73.6

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

7. Applied rewrites 73.6%

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

Alternative 9: 37.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = (z / t) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.2%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
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 36.9

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

5. Applied rewrites 36.9%

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

7. Applied rewrites 38.3%

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

8. Final simplification 38.3%

   \[\leadsto \frac{z}{t} \cdot y \]
9. Add Preprocessing

Developer Target 1: 90.9% 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 2024235 
(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)))