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

Percentage Accurate: 93.2% → 97.5%

Time: 6.5s

Alternatives: 6

Speedup: 1.0×

• 25.5% 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: 93.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

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

   1. associate-*l/ 98.1%

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

3. Simplified 98.1%

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

4. Final simplification 98.1%

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

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-19} \lor \neg \left(x \leq 18000000000\right):\\ \;\;\;\;x - x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.95e-19) (not (<= x 18000000000.0)))
   (- x (* x (/ y t)))
   (+ x (* (/ y t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.95e-19) || !(x <= 18000000000.0)) {
		tmp = x - (x * (y / t));
	} else {
		tmp = x + ((y / t) * z);
	}
	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 <= (-1.95d-19)) .or. (.not. (x <= 18000000000.0d0))) then
        tmp = x - (x * (y / t))
    else
        tmp = x + ((y / t) * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.95e-19) || !(x <= 18000000000.0)) {
		tmp = x - (x * (y / t));
	} else {
		tmp = x + ((y / t) * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.95e-19) or not (x <= 18000000000.0):
		tmp = x - (x * (y / t))
	else:
		tmp = x + ((y / t) * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.95e-19) || !(x <= 18000000000.0))
		tmp = Float64(x - Float64(x * Float64(y / t)));
	else
		tmp = Float64(x + Float64(Float64(y / t) * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.95e-19) || ~((x <= 18000000000.0)))
		tmp = x - (x * (y / t));
	else
		tmp = x + ((y / t) * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.95e-19], N[Not[LessEqual[x, 18000000000.0]], $MachinePrecision]], N[(x - N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999998e-19 or 1.8e10 < x

   1. Initial program 93.5%

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

      1. associate-/l* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in z around 0 83.4%

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

      1. associate-*r/ 83.4%

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

      2. neg-mul-1 83.4%

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

   6. Simplified 83.4%

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

      1. frac-2neg 83.4%

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

      2. div-inv 83.4%

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

      3. distribute-frac-neg 83.4%

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

      4. remove-double-neg 83.4%

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

      5. clear-num 83.4%

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

      6. add-sqr-sqrt 37.5%

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

      7. sqrt-unprod 56.6%

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

      8. sqr-neg 56.6%

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

      9. sqrt-unprod 21.9%

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

      10. add-sqr-sqrt 42.1%

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

      11. cancel-sign-sub-inv 42.1%

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

      12. clear-num 42.1%

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

      13. div-inv 42.1%

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

      14. associate-/r/ 45.8%

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

      15. *-commutative 45.8%

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

      16. add-sqr-sqrt 23.3%

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

      17. sqrt-unprod 58.7%

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

      18. sqr-neg 58.7%

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

      19. sqrt-unprod 41.8%

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

      20. add-sqr-sqrt 89.1%

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

   8. Applied egg-rr 89.1%

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

   if -1.94999999999999998e-19 < x < 1.8e10

   1. Initial program 94.4%

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

      1. associate-*l/ 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around inf 83.5%

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

      1. associate-*l/ 85.3%

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

      2. *-commutative 85.3%

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

   6. Simplified 85.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-19} \lor \neg \left(x \leq 18000000000\right):\\ \;\;\;\;x - x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \end{array} \]

Alternative 3: 49.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.9e-24) x (if (<= t 5.4e+23) (* y (/ (- x) t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.9e-24) {
		tmp = x;
	} else if (t <= 5.4e+23) {
		tmp = y * (-x / 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 <= (-1.9d-24)) then
        tmp = x
    else if (t <= 5.4d+23) then
        tmp = y * (-x / 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 <= -1.9e-24) {
		tmp = x;
	} else if (t <= 5.4e+23) {
		tmp = y * (-x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.9e-24:
		tmp = x
	elif t <= 5.4e+23:
		tmp = y * (-x / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.9e-24)
		tmp = x;
	elseif (t <= 5.4e+23)
		tmp = Float64(y * Float64(Float64(-x) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.9e-24)
		tmp = x;
	elseif (t <= 5.4e+23)
		tmp = y * (-x / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.9e-24], x, If[LessEqual[t, 5.4e+23], N[(y * N[((-x) / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -1.90000000000000013e-24 or 5.3999999999999997e23 < t

   1. Initial program 89.5%

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

      1. associate-*l/ 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around inf 81.2%

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

      1. associate-*l/ 85.5%

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

      2. *-commutative 85.5%

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

   6. Simplified 85.5%

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

   7. Taylor expanded in x around inf 56.2%

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

   if -1.90000000000000013e-24 < t < 5.3999999999999997e23

   1. Initial program 99.0%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   6. Simplified 57.0%

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

      1. frac-2neg 57.0%

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

      2. div-inv 57.1%

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

      3. distribute-frac-neg 57.1%

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

      4. remove-double-neg 57.1%

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

      5. clear-num 57.1%

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

      6. add-sqr-sqrt 26.0%

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

      7. sqrt-unprod 34.3%

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

      8. sqr-neg 34.3%

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

      9. sqrt-unprod 7.5%

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

      10. add-sqr-sqrt 15.2%

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

      11. cancel-sign-sub-inv 15.2%

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

      12. clear-num 15.2%

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

      13. div-inv 15.2%

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

      14. associate-/r/ 25.6%

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

      15. *-commutative 25.6%

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

      16. add-sqr-sqrt 13.0%

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

      17. sqrt-unprod 38.9%

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

      18. sqr-neg 38.9%

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

      19. sqrt-unprod 30.7%

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

      20. add-sqr-sqrt 64.2%

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

   8. Applied egg-rr 64.2%

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

   9. Taylor expanded in x around 0 62.6%

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

       1. *-commutative 62.6%

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

       2. associate-*r/ 57.1%

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

   11. Simplified 57.1%

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

   12. Taylor expanded in y around inf 48.8%

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

       1. associate-*r/ 48.8%

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

       2. *-commutative 48.8%

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

       3. neg-mul-1 48.8%

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

       4. distribute-rgt-neg-out 48.8%

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

       5. associate-*r/ 46.8%

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

   14. Simplified 46.8%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 50.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+23}:\\ \;\;\;\;-\frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.46e-21) x (if (<= t 5.4e+23) (- (/ (* x y) t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.46e-21) {
		tmp = x;
	} else if (t <= 5.4e+23) {
		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 <= (-1.46d-21)) then
        tmp = x
    else if (t <= 5.4d+23) 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 <= -1.46e-21) {
		tmp = x;
	} else if (t <= 5.4e+23) {
		tmp = -((x * y) / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.46e-21:
		tmp = x
	elif t <= 5.4e+23:
		tmp = -((x * y) / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.46e-21)
		tmp = x;
	elseif (t <= 5.4e+23)
		tmp = Float64(-Float64(Float64(x * y) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.46e-21)
		tmp = x;
	elseif (t <= 5.4e+23)
		tmp = -((x * y) / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.46e-21], x, If[LessEqual[t, 5.4e+23], (-N[(N[(x * y), $MachinePrecision] / t), $MachinePrecision]), x]]

Derivation

1. Split input into 2 regimes

2. if t < -1.46000000000000006e-21 or 5.3999999999999997e23 < t

   1. Initial program 89.5%

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

      1. associate-*l/ 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in z around inf 81.2%

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

      1. associate-*l/ 85.5%

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

      2. *-commutative 85.5%

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

   6. Simplified 85.5%

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

   7. Taylor expanded in x around inf 56.2%

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

   if -1.46000000000000006e-21 < t < 5.3999999999999997e23

   1. Initial program 99.0%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   6. Simplified 57.0%

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

      1. frac-2neg 57.0%

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

      2. div-inv 57.1%

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

      3. distribute-frac-neg 57.1%

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

      4. remove-double-neg 57.1%

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

      5. clear-num 57.1%

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

      6. add-sqr-sqrt 26.0%

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

      7. sqrt-unprod 34.3%

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

      8. sqr-neg 34.3%

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

      9. sqrt-unprod 7.5%

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

      10. add-sqr-sqrt 15.2%

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

      11. cancel-sign-sub-inv 15.2%

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

      12. clear-num 15.2%

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

      13. div-inv 15.2%

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

      14. associate-/r/ 25.6%

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

      15. *-commutative 25.6%

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

      16. add-sqr-sqrt 13.0%

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

      17. sqrt-unprod 38.9%

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

      18. sqr-neg 38.9%

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

      19. sqrt-unprod 30.7%

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

      20. add-sqr-sqrt 64.2%

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

   8. Applied egg-rr 64.2%

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

   9. Taylor expanded in x around 0 62.6%

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

       1. *-commutative 62.6%

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

       2. associate-*r/ 57.1%

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

   11. Simplified 57.1%

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

   12. Taylor expanded in y around inf 48.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+23}:\\ \;\;\;\;-\frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+247}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.3e+247) (+ x (* (/ y t) z)) (* y (/ (- x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.3e+247) {
		tmp = x + ((y / t) * z);
	} else {
		tmp = y * (-x / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 2.3d+247) then
        tmp = x + ((y / t) * z)
    else
        tmp = y * (-x / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.3e+247) {
		tmp = x + ((y / t) * z);
	} else {
		tmp = y * (-x / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.3e+247:
		tmp = x + ((y / t) * z)
	else:
		tmp = y * (-x / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.3e+247)
		tmp = Float64(x + Float64(Float64(y / t) * z));
	else
		tmp = Float64(y * Float64(Float64(-x) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.3e+247)
		tmp = x + ((y / t) * z);
	else
		tmp = y * (-x / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.3e+247], N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(y * N[((-x) / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.29999999999999991e247

   1. Initial program 94.0%

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

      1. associate-*l/ 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in z around inf 72.4%

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

      1. associate-*l/ 75.5%

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

      2. *-commutative 75.5%

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

   6. Simplified 75.5%

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

   if 2.29999999999999991e247 < x

   1. Initial program 93.4%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. clear-num 100.0%

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

      6. add-sqr-sqrt 46.2%

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

      7. sqrt-unprod 53.9%

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

      8. sqr-neg 53.9%

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

      9. sqrt-unprod 15.4%

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

      10. add-sqr-sqrt 30.8%

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

      11. cancel-sign-sub-inv 30.8%

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

      12. clear-num 30.8%

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

      13. div-inv 30.8%

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

      14. associate-/r/ 30.8%

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

      15. *-commutative 30.8%

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

      16. add-sqr-sqrt 15.4%

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

      17. sqrt-unprod 53.9%

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

      18. sqr-neg 53.9%

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

      19. sqrt-unprod 46.2%

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

      20. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 93.4%

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

       1. *-commutative 93.4%

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

       2. associate-*r/ 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in y around inf 63.5%

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

       1. associate-*r/ 63.5%

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

       2. *-commutative 63.5%

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

       3. neg-mul-1 63.5%

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

       4. distribute-rgt-neg-out 63.5%

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

       5. associate-*r/ 67.7%

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

   14. Simplified 67.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+247}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{t}\\ \end{array} \]

Alternative 6: 38.1% 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 93.9%

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

   1. associate-*l/ 98.1%

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

3. Simplified 98.1%

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

4. Taylor expanded in z around inf 70.3%

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

   1. associate-*l/ 73.4%

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

   2. *-commutative 73.4%

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

6. Simplified 73.4%

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

7. Taylor expanded in x around inf 37.6%

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

8. Final simplification 37.6%

   \[\leadsto x \]

Developer target: 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 2023298 
(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)))