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

Percentage Accurate: 92.7% → 97.5%

Time: 10.0s

Alternatives: 10

Speedup: 1.1×

• 24.8% 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: 92.7% 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: 97.5% 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 91.7%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. --lowering--.f64 98.7

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

4. Applied egg-rr 98.7%

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

Alternative 2: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\ t_2 := y \cdot \frac{z - x}{t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+307}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y (- z x)) t))) (t_2 (* y (/ (- z x) t))))
   (if (<= t_1 -1e+307) t_2 (if (<= t_1 5e+305) (+ x (/ (* y z) t)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((y * (z - x)) / t);
	double t_2 = y * ((z - x) / t);
	double tmp;
	if (t_1 <= -1e+307) {
		tmp = t_2;
	} else if (t_1 <= 5e+305) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x + ((y * (z - x)) / t)
    t_2 = y * ((z - x) / t)
    if (t_1 <= (-1d+307)) then
        tmp = t_2
    else if (t_1 <= 5d+305) then
        tmp = x + ((y * z) / t)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + ((y * (z - x)) / t);
	double t_2 = y * ((z - x) / t);
	double tmp;
	if (t_1 <= -1e+307) {
		tmp = t_2;
	} else if (t_1 <= 5e+305) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + ((y * (z - x)) / t)
	t_2 = y * ((z - x) / t)
	tmp = 0
	if t_1 <= -1e+307:
		tmp = t_2
	elif t_1 <= 5e+305:
		tmp = x + ((y * z) / t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(y * Float64(z - x)) / t))
	t_2 = Float64(y * Float64(Float64(z - x) / t))
	tmp = 0.0
	if (t_1 <= -1e+307)
		tmp = t_2;
	elseif (t_1 <= 5e+305)
		tmp = Float64(x + Float64(Float64(y * z) / t));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + ((y * (z - x)) / t);
	t_2 = y * ((z - x) / t);
	tmp = 0.0;
	if (t_1 <= -1e+307)
		tmp = t_2;
	elseif (t_1 <= 5e+305)
		tmp = x + ((y * z) / t);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+307], t$95$2, If[LessEqual[t$95$1, 5e+305], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -9.99999999999999986e306 or 5.00000000000000009e305 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 78.4%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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. *-lowering-*.f64 N/A

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

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

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

      4. --lowering--.f64 92.4

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

   5. Simplified 92.4%

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

   if -9.99999999999999986e306 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 5.00000000000000009e305

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

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

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

      2. *-lowering-*.f64 83.0

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

   5. Simplified 83.0%

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

Alternative 3: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= z -3.8e-33) t_1 (if (<= z 3.4e+75) (fma (/ y t) (- x) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (z <= -3.8e-33) {
		tmp = t_1;
	} else if (z <= 3.4e+75) {
		tmp = fma((y / t), -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (z <= -3.8e-33)
		tmp = t_1;
	elseif (z <= 3.4e+75)
		tmp = fma(Float64(y / t), Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -3.8e-33], t$95$1, If[LessEqual[z, 3.4e+75], N[(N[(y / t), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.79999999999999994e-33 or 3.40000000000000011e75 < z

   1. Initial program 88.1%

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

   3. Taylor expanded in z around inf

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

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

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

      2. *-lowering-*.f64 83.5

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

   5. Simplified 83.5%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. times-frac N/A

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

      4. /-rgt-identity N/A

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

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

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

      6. /-lowering-/.f64 92.9

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

   7. Applied egg-rr 92.9%

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

   if -3.79999999999999994e-33 < z < 3.40000000000000011e75

   1. Initial program 95.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. --lowering--.f64 97.8

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

   4. Applied egg-rr 97.8%

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

   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. neg-lowering-neg.f64 90.0

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

   7. Simplified 90.0%

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

Alternative 4: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 52000000000:\\ \;\;\;\;x - \frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= z -9e-34) t_1 (if (<= z 52000000000.0) (- x (/ (* y x) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (z <= -9e-34) {
		tmp = t_1;
	} else if (z <= 52000000000.0) {
		tmp = x - ((y * x) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (z <= -9e-34)
		tmp = t_1;
	elseif (z <= 52000000000.0)
		tmp = Float64(x - Float64(Float64(y * x) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -9e-34], t$95$1, If[LessEqual[z, 52000000000.0], N[(x - N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.00000000000000085e-34 or 5.2e10 < z

   1. Initial program 88.8%

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

   3. Taylor expanded in z around inf

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

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

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

      2. *-lowering-*.f64 81.2

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

   5. Simplified 81.2%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. times-frac N/A

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

      4. /-rgt-identity N/A

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

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

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

      6. /-lowering-/.f64 89.6

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

   7. Applied egg-rr 89.6%

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

   if -9.00000000000000085e-34 < z < 5.2e10

   1. Initial program 95.2%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. distribute-lft-out-- N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 88.2

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

   5. Simplified 88.2%

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

Alternative 5: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - x}{t}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ (- z x) t))))
   (if (<= y -4.6e+174) t_1 (if (<= y 6.4e+51) (fma (/ y t) z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * ((z - x) / t);
	double tmp;
	if (y <= -4.6e+174) {
		tmp = t_1;
	} else if (y <= 6.4e+51) {
		tmp = fma((y / t), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(Float64(z - x) / t))
	tmp = 0.0
	if (y <= -4.6e+174)
		tmp = t_1;
	elseif (y <= 6.4e+51)
		tmp = fma(Float64(y / t), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+174], t$95$1, If[LessEqual[y, 6.4e+51], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5999999999999996e174 or 6.4000000000000005e51 < y

   1. Initial program 80.9%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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. *-lowering-*.f64 N/A

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

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

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

      4. --lowering--.f64 91.0

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

   5. Simplified 91.0%

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

   if -4.5999999999999996e174 < y < 6.4000000000000005e51

   1. Initial program 97.0%

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

   3. Taylor expanded in z around inf

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

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

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

      2. *-lowering-*.f64 81.2

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

   5. Simplified 81.2%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. times-frac N/A

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

      4. /-rgt-identity N/A

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

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

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

      6. /-lowering-/.f64 84.1

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

   7. Applied egg-rr 84.1%

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

Alternative 6: 53.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t} \cdot z\\ \mathbf{if}\;z \leq -3 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ y t) z)))
   (if (<= z -3e-32) t_1 (if (<= z 2.75e+76) x t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / t) * z;
	double tmp;
	if (z <= -3e-32) {
		tmp = t_1;
	} else if (z <= 2.75e+76) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (y / t) * z
    if (z <= (-3d-32)) then
        tmp = t_1
    else if (z <= 2.75d+76) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / t) * z;
	double tmp;
	if (z <= -3e-32) {
		tmp = t_1;
	} else if (z <= 2.75e+76) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / t) * z
	tmp = 0
	if z <= -3e-32:
		tmp = t_1
	elif z <= 2.75e+76:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / t) * z)
	tmp = 0.0
	if (z <= -3e-32)
		tmp = t_1;
	elseif (z <= 2.75e+76)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / t) * z;
	tmp = 0.0;
	if (z <= -3e-32)
		tmp = t_1;
	elseif (z <= 2.75e+76)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3e-32], t$95$1, If[LessEqual[z, 2.75e+76], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3e-32 or 2.75e76 < z

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

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

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

      2. *-lowering-*.f64 65.2

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

   5. Simplified 65.2%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 71.6

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

   7. Applied egg-rr 71.6%

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

   if -3e-32 < z < 2.75e76

   1. Initial program 95.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 45.5%

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

Alternative 7: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.5e+174) (* (/ y t) (- x)) (fma (/ y t) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.5e+174) {
		tmp = (y / t) * -x;
	} else {
		tmp = fma((y / t), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.5e+174)
		tmp = Float64(Float64(y / t) * Float64(-x));
	else
		tmp = fma(Float64(y / t), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.5e+174], N[(N[(y / t), $MachinePrecision] * (-x)), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000042e174

   1. Initial program 75.3%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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. *-lowering-*.f64 N/A

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

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

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

      4. --lowering--.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 71.2

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

   8. Simplified 71.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

      5. associate-/r/ N/A

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

      6. frac-2neg N/A

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

      7. remove-double-neg N/A

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

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

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

      9. frac-2neg N/A

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

      10. remove-double-neg N/A

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

      11. /-lowering-/.f64 N/A

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

      12. neg-lowering-neg.f64 76.7

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

   10. Applied egg-rr 76.7%

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

   if -4.50000000000000042e174 < y

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

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

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

      2. *-lowering-*.f64 75.1

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

   5. Simplified 75.1%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. times-frac N/A

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

      4. /-rgt-identity N/A

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

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

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

      6. /-lowering-/.f64 80.3

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

   7. Applied egg-rr 80.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{y}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+175}:\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.45e+175) (* y (/ (- x) t)) (fma (/ y t) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.45e+175) {
		tmp = y * (-x / t);
	} else {
		tmp = fma((y / t), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.45e+175)
		tmp = Float64(y * Float64(Float64(-x) / t));
	else
		tmp = fma(Float64(y / t), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.45e+175], N[(y * N[((-x) / t), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e175

   1. Initial program 75.3%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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. *-lowering-*.f64 N/A

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

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

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

      4. --lowering--.f64 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 71.2

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

   8. Simplified 71.2%

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

   if -1.45e175 < y

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

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

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

      2. *-lowering-*.f64 75.1

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

   5. Simplified 75.1%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. times-frac N/A

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

      4. /-rgt-identity N/A

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

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

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

      6. /-lowering-/.f64 80.3

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

   7. Applied egg-rr 80.3%

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

Alternative 9: 77.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in z around inf

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

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

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

   2. *-lowering-*.f64 69.0

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

5. Simplified 69.0%

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. times-frac N/A

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

   4. /-rgt-identity N/A

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

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

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

   6. /-lowering-/.f64 75.7

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

7. Applied egg-rr 75.7%

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

Alternative 10: 38.5% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

5. Simplified 35.1%

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

Developer Target 1: 91.1% 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 2024198 
(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)))