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

Percentage Accurate: 92.1% → 97.7%

Time: 9.4s

Alternatives: 9

Speedup: 1.1×

• 24.1% 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: 92.1% 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: 97.7% 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 91.9%

   \[x + \frac{y \cdot \left(z - x\right)}{t} \]
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 97.4

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

4. Applied egg-rr 97.4%

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

Alternative 2: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= z -4e-51) t_1 (if (<= z 1.55e-20) (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 <= -4e-51) {
		tmp = t_1;
	} else if (z <= 1.55e-20) {
		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 <= -4e-51)
		tmp = t_1;
	elseif (z <= 1.55e-20)
		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, -4e-51], t$95$1, If[LessEqual[z, 1.55e-20], N[(N[(y / t), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4e-51 or 1.55e-20 < z

   1. Initial program 87.5%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 97.4

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

   4. Applied egg-rr 97.4%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 89.7%

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

   if -4e-51 < z < 1.55e-20

   1. Initial program 97.5%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 97.4

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

   4. Applied egg-rr 97.4%

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

   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 91.8

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

   7. Simplified 91.8%

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

Alternative 3: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-43}:\\ \;\;\;\;x - \frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= z -1.7e-54) t_1 (if (<= z 5.1e-43) (- 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 <= -1.7e-54) {
		tmp = t_1;
	} else if (z <= 5.1e-43) {
		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 <= -1.7e-54)
		tmp = t_1;
	elseif (z <= 5.1e-43)
		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, -1.7e-54], t$95$1, If[LessEqual[z, 5.1e-43], N[(x - N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999994e-54 or 5.0999999999999997e-43 < z

   1. Initial program 88.0%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 97.5

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

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 88.9%

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

   if -1.69999999999999994e-54 < z < 5.0999999999999997e-43

   1. Initial program 97.3%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 89.2

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

   5. Simplified 89.2%

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

Alternative 4: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -7.5e-69) t_1 (if (<= t 1.6e-46) (* y (/ (- z x) 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 <= -7.5e-69) {
		tmp = t_1;
	} else if (t <= 1.6e-46) {
		tmp = y * ((z - x) / 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 <= -7.5e-69)
		tmp = t_1;
	elseif (t <= 1.6e-46)
		tmp = Float64(y * Float64(Float64(z - x) / 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, -7.5e-69], t$95$1, If[LessEqual[t, 1.6e-46], N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.5e-69 or 1.6e-46 < t

   1. Initial program 87.0%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 97.6

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 87.6%

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

      1. associate-*l/ N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 89.9

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

   9. Applied egg-rr 89.9%

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

   if -7.5e-69 < t < 1.6e-46

   1. Initial program 98.9%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 79.9

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

   5. Simplified 79.9%

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

Alternative 5: 54.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.6e-63) x (if (<= t 5.5e-54) (* (/ y t) z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.6e-63) {
		tmp = x;
	} else if (t <= 5.5e-54) {
		tmp = (y / t) * z;
	} else {
		tmp = x;
	}
	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 (t <= (-3.6d-63)) then
        tmp = x
    else if (t <= 5.5d-54) then
        tmp = (y / t) * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.6e-63) {
		tmp = x;
	} else if (t <= 5.5e-54) {
		tmp = (y / t) * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.6e-63:
		tmp = x
	elif t <= 5.5e-54:
		tmp = (y / t) * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.6e-63)
		tmp = x;
	elseif (t <= 5.5e-54)
		tmp = Float64(Float64(y / t) * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.6e-63)
		tmp = x;
	elseif (t <= 5.5e-54)
		tmp = (y / t) * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.6e-63], x, If[LessEqual[t, 5.5e-54], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -3.60000000000000008e-63 or 5.50000000000000046e-54 < t

   1. Initial program 87.2%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 62.2%

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

   if -3.60000000000000008e-63 < t < 5.50000000000000046e-54

   1. Initial program 98.9%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 79.5

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

   5. Simplified 79.5%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 45.1%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

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

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

      4. /-lowering-/.f64 53.5

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

   10. Applied egg-rr 53.5%

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

Alternative 6: 51.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7e-28) x (if (<= x 3.7e+15) (* y (/ z t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7e-28) {
		tmp = x;
	} else if (x <= 3.7e+15) {
		tmp = y * (z / t);
	} else {
		tmp = x;
	}
	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 <= (-7d-28)) then
        tmp = x
    else if (x <= 3.7d+15) then
        tmp = y * (z / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7e-28) {
		tmp = x;
	} else if (x <= 3.7e+15) {
		tmp = y * (z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -7e-28:
		tmp = x
	elif x <= 3.7e+15:
		tmp = y * (z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7e-28)
		tmp = x;
	elseif (x <= 3.7e+15)
		tmp = Float64(y * Float64(z / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -7e-28)
		tmp = x;
	elseif (x <= 3.7e+15)
		tmp = y * (z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7e-28], x, If[LessEqual[x, 3.7e+15], N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999999e-28 or 3.7e15 < x

   1. Initial program 91.5%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 56.2%

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

   if -6.9999999999999999e-28 < x < 3.7e15

   1. Initial program 92.2%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 66.2

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

   5. Simplified 66.2%

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

   6. Taylor expanded in z around inf

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

   8. Simplified 55.1%

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

Alternative 7: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.3 \cdot 10^{+240}:\\ \;\;\;\;x \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.3e+240) (* x (/ y (- t))) (fma (/ y t) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.3e+240) {
		tmp = x * (y / -t);
	} else {
		tmp = fma((y / t), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.3e+240)
		tmp = Float64(x * Float64(y / Float64(-t)));
	else
		tmp = fma(Float64(y / t), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.3e+240], N[(x * N[(y / (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.2999999999999997e240

   1. Initial program 79.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

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

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

      9. neg-lowering-neg.f64 84.4

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

   8. Simplified 84.4%

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

   if -6.2999999999999997e240 < y

   1. Initial program 92.6%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 97.6

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 78.6%

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

Alternative 8: 76.9% 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 91.9%

   \[x + \frac{y \cdot \left(z - x\right)}{t} \]
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 97.4

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

4. Applied egg-rr 97.4%

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

5. Taylor expanded in z around inf

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

7. Simplified 76.3%

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

Alternative 9: 39.3% 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 91.9%

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

3. Taylor expanded in y around 0

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

5. Simplified 41.3%

   \[\leadsto \color{blue}{x} \]
6. 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 2024199 
(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)))