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

Percentage Accurate: 92.5% → 97.8%

Time: 8.9s

Alternatives: 8

Speedup: 1.0×

• 26.1% 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: 92.5% 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: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{\frac{t}{y}}{z - x}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.0%

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

   1. associate-*l/ 99.0%

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

3. Simplified 99.0%

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

   1. associate-*l/ 93.0%

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

   2. clear-num 93.0%

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

   3. associate-/r* 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

   \[\leadsto x + \frac{1}{\frac{\frac{t}{y}}{z - x}} \]
8. Add Preprocessing

Alternative 2: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.32 \cdot 10^{+108} \lor \neg \left(x \leq 4.2 \cdot 10^{+49}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.32e+108) (not (<= x 4.2e+49)))
   (* x (- 1.0 (/ y t)))
   (+ x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.32e+108) || !(x <= 4.2e+49)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + (z * (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 ((x <= (-2.32d+108)) .or. (.not. (x <= 4.2d+49))) then
        tmp = x * (1.0d0 - (y / t))
    else
        tmp = x + (z * (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.32e+108) || !(x <= 4.2e+49)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.32e+108) or not (x <= 4.2e+49):
		tmp = x * (1.0 - (y / t))
	else:
		tmp = x + (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.32e+108) || !(x <= 4.2e+49))
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(x + Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.32e+108) || ~((x <= 4.2e+49)))
		tmp = x * (1.0 - (y / t));
	else
		tmp = x + (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.32e+108], N[Not[LessEqual[x, 4.2e+49]], $MachinePrecision]], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3199999999999999e108 or 4.20000000000000022e49 < x

   1. Initial program 90.9%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   7. Simplified 96.5%

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

   if -2.3199999999999999e108 < x < 4.20000000000000022e49

   1. Initial program 94.2%

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

      1. associate-*l/ 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in z around inf 86.2%

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

      1. associate-*l/ 91.1%

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

      2. *-commutative 91.1%

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

   7. Simplified 91.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.32 \cdot 10^{+108} \lor \neg \left(x \leq 4.2 \cdot 10^{+49}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{+108}:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+52}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.35e+108)
   (- x (/ x (/ t y)))
   (if (<= x 7.8e+52) (+ x (* z (/ y t))) (* x (- 1.0 (/ y t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.35e+108) {
		tmp = x - (x / (t / y));
	} else if (x <= 7.8e+52) {
		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 (x <= (-3.35d+108)) then
        tmp = x - (x / (t / y))
    else if (x <= 7.8d+52) 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 (x <= -3.35e+108) {
		tmp = x - (x / (t / y));
	} else if (x <= 7.8e+52) {
		tmp = x + (z * (y / t));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.35e+108:
		tmp = x - (x / (t / y))
	elif x <= 7.8e+52:
		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 (x <= -3.35e+108)
		tmp = Float64(x - Float64(x / Float64(t / y)));
	elseif (x <= 7.8e+52)
		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 (x <= -3.35e+108)
		tmp = x - (x / (t / y));
	elseif (x <= 7.8e+52)
		tmp = x + (z * (y / t));
	else
		tmp = x * (1.0 - (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.35e+108], N[(x - N[(x / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e+52], 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 3 regimes

2. if x < -3.35000000000000007e108

   1. Initial program 88.5%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 87.6%

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

      1. mul-1-neg 87.6%

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

      2. associate-/l* 96.6%

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

      3. distribute-neg-frac 96.6%

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

   7. Simplified 96.6%

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

   if -3.35000000000000007e108 < x < 7.7999999999999999e52

   1. Initial program 94.2%

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

      1. associate-*l/ 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in z around inf 86.2%

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

      1. associate-*l/ 91.1%

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

      2. *-commutative 91.1%

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

   7. Simplified 91.1%

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

   if 7.7999999999999999e52 < x

   1. Initial program 92.7%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   7. Simplified 96.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{+108}:\\ \;\;\;\;x - \frac{x}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+52}:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+37} \lor \neg \left(y \leq 1.85 \cdot 10^{+158}\right):\\ \;\;\;\;x \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.4e+37) (not (<= y 1.85e+158))) (* x (/ (- y) t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.4e+37) || !(y <= 1.85e+158)) {
		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 ((y <= (-4.4d+37)) .or. (.not. (y <= 1.85d+158))) 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 ((y <= -4.4e+37) || !(y <= 1.85e+158)) {
		tmp = x * (-y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4.4e+37) or not (y <= 1.85e+158):
		tmp = x * (-y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.4e+37) || !(y <= 1.85e+158))
		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 ((y <= -4.4e+37) || ~((y <= 1.85e+158)))
		tmp = x * (-y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.4e+37], N[Not[LessEqual[y, 1.85e+158]], $MachinePrecision]], N[(x * N[((-y) / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -4.4000000000000001e37 or 1.85000000000000005e158 < y

   1. Initial program 85.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{t}{z - x}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 52.7%

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

      1. associate-*r/ 52.7%

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

      2. neg-mul-1 52.7%

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

   7. Simplified 52.7%

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

      1. frac-2neg 52.7%

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

      2. div-inv 52.7%

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

      3. distribute-frac-neg 52.7%

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

      4. remove-double-neg 52.7%

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

      5. clear-num 52.8%

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

      6. add-sqr-sqrt 30.2%

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

      7. sqrt-unprod 35.5%

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

      8. sqr-neg 35.5%

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

      9. sqrt-unprod 6.5%

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

      10. add-sqr-sqrt 13.9%

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

      11. cancel-sign-sub-inv 13.9%

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

      12. add-sqr-sqrt 6.5%

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

      13. sqrt-unprod 35.5%

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

      14. sqr-neg 35.5%

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

      15. sqrt-unprod 30.2%

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

      16. add-sqr-sqrt 52.8%

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

   9. Applied egg-rr 52.8%

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

   10. Taylor expanded in y around inf 41.9%

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

       1. mul-1-neg 41.9%

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

       2. *-commutative 41.9%

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

       3. associate-/l* 47.0%

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

       4. associate-/r/ 49.3%

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

   12. Simplified 49.3%

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

   if -4.4000000000000001e37 < y < 1.85000000000000005e158

   1. Initial program 97.1%

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

      1. associate-*l/ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 57.6%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+37} \lor \neg \left(y \leq 1.85 \cdot 10^{+158}\right):\\ \;\;\;\;x \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \frac{-y}{t}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+158}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5e+37) (* x (/ (- y) t)) (if (<= y 1.8e+158) x (/ (- x) (/ t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5e+37) {
		tmp = x * (-y / t);
	} else if (y <= 1.8e+158) {
		tmp = x;
	} else {
		tmp = -x / (t / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5d+37)) then
        tmp = x * (-y / t)
    else if (y <= 1.8d+158) then
        tmp = x
    else
        tmp = -x / (t / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5e+37) {
		tmp = x * (-y / t);
	} else if (y <= 1.8e+158) {
		tmp = x;
	} else {
		tmp = -x / (t / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -5e+37:
		tmp = x * (-y / t)
	elif y <= 1.8e+158:
		tmp = x
	else:
		tmp = -x / (t / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5e+37)
		tmp = Float64(x * Float64(Float64(-y) / t));
	elseif (y <= 1.8e+158)
		tmp = x;
	else
		tmp = Float64(Float64(-x) / Float64(t / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -5e+37)
		tmp = x * (-y / t);
	elseif (y <= 1.8e+158)
		tmp = x;
	else
		tmp = -x / (t / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5e+37], N[(x * N[((-y) / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+158], x, N[((-x) / N[(t / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999989e37

   1. Initial program 90.5%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{t}{z - x}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 50.7%

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

      1. associate-*r/ 50.7%

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

      2. neg-mul-1 50.7%

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

   7. Simplified 50.7%

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

      1. frac-2neg 50.7%

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

      2. div-inv 50.7%

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

      3. distribute-frac-neg 50.7%

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

      4. remove-double-neg 50.7%

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

      5. clear-num 50.7%

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

      6. add-sqr-sqrt 23.8%

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

      7. sqrt-unprod 34.6%

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

      8. sqr-neg 34.6%

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

      9. sqrt-unprod 9.5%

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

      10. add-sqr-sqrt 17.2%

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

      11. cancel-sign-sub-inv 17.2%

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

      12. add-sqr-sqrt 9.5%

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

      13. sqrt-unprod 34.6%

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

      14. sqr-neg 34.6%

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

      15. sqrt-unprod 23.8%

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

      16. add-sqr-sqrt 50.7%

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

   9. Applied egg-rr 50.7%

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

   10. Taylor expanded in y around inf 40.8%

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

       1. mul-1-neg 40.8%

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

       2. *-commutative 40.8%

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

       3. associate-/l* 43.7%

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

       4. associate-/r/ 45.5%

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

   12. Simplified 45.5%

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

   if -4.99999999999999989e37 < y < 1.79999999999999994e158

   1. Initial program 97.1%

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

      1. associate-*l/ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 57.6%

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

   if 1.79999999999999994e158 < y

   1. Initial program 75.5%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{t}{z - x}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 56.5%

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

      1. associate-*r/ 56.5%

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

      2. neg-mul-1 56.5%

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

   7. Simplified 56.5%

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

      1. frac-2neg 56.5%

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

      2. div-inv 56.4%

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

      3. distribute-frac-neg 56.4%

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

      4. remove-double-neg 56.4%

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

      5. clear-num 56.6%

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

      6. add-sqr-sqrt 42.0%

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

      7. sqrt-unprod 37.3%

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

      8. sqr-neg 37.3%

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

      9. sqrt-unprod 1.0%

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

      10. add-sqr-sqrt 7.8%

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

      11. cancel-sign-sub-inv 7.8%

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

      12. add-sqr-sqrt 1.0%

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

      13. sqrt-unprod 37.3%

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

      14. sqr-neg 37.3%

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

      15. sqrt-unprod 42.0%

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

      16. add-sqr-sqrt 56.6%

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

   9. Applied egg-rr 56.6%

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

   10. Taylor expanded in y around inf 43.8%

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

       1. mul-1-neg 43.8%

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

       2. *-commutative 43.8%

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

       3. associate-/l* 53.2%

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

       4. associate-/r/ 56.4%

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

   12. Simplified 56.4%

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

       1. *-commutative 56.4%

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

       2. clear-num 56.4%

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

       3. un-div-inv 56.5%

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

   14. Applied egg-rr 56.5%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \frac{-y}{t}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+158}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{t}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% 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 93.0%

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

   1. associate-*l/ 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 7: 64.9% 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 93.0%

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

   1. associate-*l/ 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in x around inf 62.3%

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

   1. mul-1-neg 62.3%

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

   2. unsub-neg 62.3%

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

7. Simplified 62.3%

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

8. Final simplification 62.3%

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

Alternative 8: 38.0% 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 93.0%

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

   1. associate-*l/ 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in y around 0 40.6%

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

6. Final simplification 40.6%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 90.8% 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 2024014 
(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)))