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

Percentage Accurate: 93.1% → 97.6%

Time: 9.2s

Alternatives: 8

Speedup: 1.0×

• 24.4% 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: 93.1% 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: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{z - x}{\frac{t}{y}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((z - x) / (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 = x + ((z - x) / (t / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. associate-*l/ 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative 96.6%

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

   2. clear-num 96.6%

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

   3. un-div-inv 97.3%

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

5. Applied egg-rr 97.3%

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

6. Final simplification 97.3%

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

Alternative 2: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-65} \lor \neg \left(z \leq 4.5 \cdot 10^{-99}\right):\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.2e-65) (not (<= z 4.5e-99)))
   (+ x (* z (/ y t)))
   (* x (- 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.2e-65) || !(z <= 4.5e-99)) {
		tmp = x + (z * (y / t));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-4.2d-65)) .or. (.not. (z <= 4.5d-99))) then
        tmp = x + (z * (y / t))
    else
        tmp = x * (1.0d0 - (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.2e-65) || !(z <= 4.5e-99)) {
		tmp = x + (z * (y / t));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.2e-65) or not (z <= 4.5e-99):
		tmp = x + (z * (y / t))
	else:
		tmp = x * (1.0 - (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.2e-65) || !(z <= 4.5e-99))
		tmp = Float64(x + Float64(z * Float64(y / t)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.2e-65) || ~((z <= 4.5e-99)))
		tmp = x + (z * (y / t));
	else
		tmp = x * (1.0 - (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.2e-65], N[Not[LessEqual[z, 4.5e-99]], $MachinePrecision]], N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.20000000000000006e-65 or 4.5000000000000003e-99 < z

   1. Initial program 92.2%

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

      1. associate-*l/ 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in z around inf 84.0%

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

      1. associate-*l/ 88.4%

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

      2. *-commutative 88.4%

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

   6. Simplified 88.4%

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

   if -4.20000000000000006e-65 < z < 4.5000000000000003e-99

   1. Initial program 92.5%

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

      1. associate-/l* 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in z around 0 81.3%

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

      1. associate-*r/ 81.3%

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

      2. neg-mul-1 81.3%

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

   6. Simplified 81.3%

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

   7. Taylor expanded in x around 0 84.2%

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

      1. mul-1-neg 84.2%

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

      2. sub-neg 84.2%

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

      3. *-commutative 84.2%

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

   9. Simplified 84.2%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{y}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-65} \lor \neg \left(z \leq 4.5 \cdot 10^{-99}\right):\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \]

Alternative 3: 51.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.5:\\ \;\;\;\;x \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.5e-51) x (if (<= t 1.5) (* x (/ (- y) t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.5e-51) {
		tmp = x;
	} else if (t <= 1.5) {
		tmp = x * (-y / 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 (t <= (-4.5d-51)) then
        tmp = x
    else if (t <= 1.5d0) then
        tmp = x * (-y / 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 (t <= -4.5e-51) {
		tmp = x;
	} else if (t <= 1.5) {
		tmp = x * (-y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.5e-51:
		tmp = x
	elif t <= 1.5:
		tmp = x * (-y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.5e-51)
		tmp = x;
	elseif (t <= 1.5)
		tmp = Float64(x * Float64(Float64(-y) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4.5e-51)
		tmp = x;
	elseif (t <= 1.5)
		tmp = x * (-y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.5e-51], x, If[LessEqual[t, 1.5], N[(x * N[((-y) / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -4.49999999999999974e-51 or 1.5 < t

   1. Initial program 87.4%

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

      1. associate-/l* 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in z around 0 67.1%

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

      1. associate-*r/ 67.1%

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

      2. neg-mul-1 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in y around 0 56.7%

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

   if -4.49999999999999974e-51 < t < 1.5

   1. Initial program 98.2%

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

      1. associate-/l* 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in z around 0 48.7%

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

      1. associate-*r/ 48.7%

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

      2. neg-mul-1 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in x around 0 52.0%

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

      1. mul-1-neg 52.0%

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

      2. sub-neg 52.0%

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

      3. *-commutative 52.0%

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

   9. Simplified 52.0%

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

   10. Taylor expanded in y around inf 46.3%

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

       1. mul-1-neg 46.3%

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

       2. associate-*l/ 45.5%

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

       3. distribute-rgt-neg-in 45.5%

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

   12. Simplified 45.5%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.5:\\ \;\;\;\;x \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 51.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.6:\\ \;\;\;\;\frac{-x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.8e-51) x (if (<= t 2.6) (/ (- x) (/ t y)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.8e-51) {
		tmp = x;
	} else if (t <= 2.6) {
		tmp = -x / (t / y);
	} 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 <= (-7.8d-51)) then
        tmp = x
    else if (t <= 2.6d0) then
        tmp = -x / (t / y)
    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 <= -7.8e-51) {
		tmp = x;
	} else if (t <= 2.6) {
		tmp = -x / (t / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7.8e-51:
		tmp = x
	elif t <= 2.6:
		tmp = -x / (t / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.8e-51)
		tmp = x;
	elseif (t <= 2.6)
		tmp = Float64(Float64(-x) / Float64(t / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7.8e-51)
		tmp = x;
	elseif (t <= 2.6)
		tmp = -x / (t / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.8e-51], x, If[LessEqual[t, 2.6], N[((-x) / N[(t / y), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -7.7999999999999995e-51 or 2.60000000000000009 < t

   1. Initial program 87.4%

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

      1. associate-/l* 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in z around 0 67.1%

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

      1. associate-*r/ 67.1%

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

      2. neg-mul-1 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in y around 0 56.7%

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

   if -7.7999999999999995e-51 < t < 2.60000000000000009

   1. Initial program 98.2%

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

      1. associate-/l* 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in z around 0 48.7%

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

      1. associate-*r/ 48.7%

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

      2. neg-mul-1 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in x around 0 52.0%

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

      1. mul-1-neg 52.0%

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

      2. sub-neg 52.0%

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

      3. *-commutative 52.0%

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

   9. Simplified 52.0%

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

   10. Taylor expanded in y around inf 46.3%

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

       1. mul-1-neg 46.3%

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

       2. *-commutative 46.3%

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

       3. associate-/l* 47.0%

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

       4. distribute-neg-frac 47.0%

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

   12. Simplified 47.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.6:\\ \;\;\;\;\frac{-x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 38.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-233}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6.6e-233) x (if (<= t 9.5e-81) (/ x (/ t y)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.6e-233) {
		tmp = x;
	} else if (t <= 9.5e-81) {
		tmp = x / (t / y);
	} 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 <= (-6.6d-233)) then
        tmp = x
    else if (t <= 9.5d-81) then
        tmp = x / (t / y)
    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 <= -6.6e-233) {
		tmp = x;
	} else if (t <= 9.5e-81) {
		tmp = x / (t / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6.6e-233:
		tmp = x
	elif t <= 9.5e-81:
		tmp = x / (t / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6.6e-233)
		tmp = x;
	elseif (t <= 9.5e-81)
		tmp = Float64(x / Float64(t / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -6.6e-233)
		tmp = x;
	elseif (t <= 9.5e-81)
		tmp = x / (t / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6.6e-233], x, If[LessEqual[t, 9.5e-81], N[(x / N[(t / y), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -6.6000000000000001e-233 or 9.49999999999999917e-81 < t

   1. Initial program 90.0%

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

      1. associate-/l* 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in z around 0 63.7%

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

      1. associate-*r/ 63.7%

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

      2. neg-mul-1 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in y around 0 44.6%

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

   if -6.6000000000000001e-233 < t < 9.49999999999999917e-81

   1. Initial program 99.8%

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

      1. associate-/l* 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in z around 0 42.5%

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

      1. associate-*r/ 42.5%

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

      2. neg-mul-1 42.5%

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

   6. Simplified 42.5%

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

   7. Taylor expanded in x around 0 44.1%

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

      1. mul-1-neg 44.1%

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

      2. sub-neg 44.1%

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

      3. *-commutative 44.1%

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

   9. Simplified 44.1%

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

   10. Taylor expanded in y around inf 44.3%

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

       1. mul-1-neg 44.3%

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

       2. associate-*l/ 42.8%

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

       3. distribute-rgt-neg-in 42.8%

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

   12. Simplified 42.8%

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

       1. clear-num 42.8%

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

       2. associate-*l/ 44.5%

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

       3. *-un-lft-identity 44.5%

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

       4. add-sqr-sqrt 19.6%

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

       5. sqrt-unprod 30.2%

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

       6. sqr-neg 30.2%

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

       7. sqrt-unprod 17.3%

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

       8. add-sqr-sqrt 24.6%

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

   14. Applied egg-rr 24.6%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{-233}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. associate-*l/ 96.6%

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

3. Simplified 96.6%

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

4. Final simplification 96.6%

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

Alternative 7: 65.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \frac{y}{t}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- 1.0 (/ y t))))

C

double code(double x, double y, double z, double t) {
	return x * (1.0 - (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 * (1.0d0 - (y / t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * (1.0 - (y / t));
}

Python

def code(x, y, z, t):
	return x * (1.0 - (y / t))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(1.0 - Float64(y / t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (1.0 - (y / t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.3%

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

   1. associate-/l* 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in z around 0 58.7%

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

   1. associate-*r/ 58.7%

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

   2. neg-mul-1 58.7%

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

6. Simplified 58.7%

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

7. Taylor expanded in x around 0 61.0%

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

   1. mul-1-neg 61.0%

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

   2. sub-neg 61.0%

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

   3. *-commutative 61.0%

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

9. Simplified 61.0%

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

10. Final simplification 61.0%

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

Alternative 8: 39.2% accurate, 9.0× 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 92.3%

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

   1. associate-/l* 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in z around 0 58.7%

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

   1. associate-*r/ 58.7%

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

   2. neg-mul-1 58.7%

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

6. Simplified 58.7%

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

7. Taylor expanded in y around 0 34.9%

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

8. Final simplification 34.9%

   \[\leadsto x \]

Developer target: 91.0% 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 2023274 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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