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

Percentage Accurate: 92.4% → 99.1%

Time: 8.0s

Alternatives: 10

Speedup: 0.8×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-38}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.1e+40)
   (- x (/ y (/ a (- z t))))
   (if (<= a 1.75e-38) (+ x (/ (* y (- t z)) a)) (+ x (* y (/ (- t z) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.1e+40) {
		tmp = x - (y / (a / (z - t)));
	} else if (a <= 1.75e-38) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = x + (y * ((t - z) / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-4.1d+40)) then
        tmp = x - (y / (a / (z - t)))
    else if (a <= 1.75d-38) then
        tmp = x + ((y * (t - z)) / a)
    else
        tmp = x + (y * ((t - z) / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.1e+40) {
		tmp = x - (y / (a / (z - t)));
	} else if (a <= 1.75e-38) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = x + (y * ((t - z) / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -4.1e+40:
		tmp = x - (y / (a / (z - t)))
	elif a <= 1.75e-38:
		tmp = x + ((y * (t - z)) / a)
	else:
		tmp = x + (y * ((t - z) / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.1e+40)
		tmp = Float64(x - Float64(y / Float64(a / Float64(z - t))));
	elseif (a <= 1.75e-38)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / a));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(t - z) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -4.1e+40)
		tmp = x - (y / (a / (z - t)));
	elseif (a <= 1.75e-38)
		tmp = x + ((y * (t - z)) / a);
	else
		tmp = x + (y * ((t - z) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.1e+40], N[(x - N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.75e-38], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.1000000000000002e40

   1. Initial program 87.5%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   if -4.1000000000000002e40 < a < 1.7500000000000001e-38

   1. Initial program 99.9%

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

   if 1.7500000000000001e-38 < a

   1. Initial program 90.1%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-38}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 2: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+42} \lor \neg \left(z \leq 1.15 \cdot 10^{-102}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.3e+42) (not (<= z 1.15e-102)))
   (+ x (* (/ y a) (- t z)))
   (+ x (* y (/ (- t z) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.3e+42) || !(z <= 1.15e-102)) {
		tmp = x + ((y / a) * (t - z));
	} else {
		tmp = x + (y * ((t - z) / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-5.3d+42)) .or. (.not. (z <= 1.15d-102))) then
        tmp = x + ((y / a) * (t - z))
    else
        tmp = x + (y * ((t - z) / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.3e+42) || !(z <= 1.15e-102)) {
		tmp = x + ((y / a) * (t - z));
	} else {
		tmp = x + (y * ((t - z) / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -5.3e+42) or not (z <= 1.15e-102):
		tmp = x + ((y / a) * (t - z))
	else:
		tmp = x + (y * ((t - z) / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.3e+42) || !(z <= 1.15e-102))
		tmp = Float64(x + Float64(Float64(y / a) * Float64(t - z)));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(t - z) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -5.3e+42) || ~((z <= 1.15e-102)))
		tmp = x + ((y / a) * (t - z));
	else
		tmp = x + (y * ((t - z) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.3e+42], N[Not[LessEqual[z, 1.15e-102]], $MachinePrecision]], N[(x + N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.30000000000000028e42 or 1.14999999999999993e-102 < z

   1. Initial program 92.3%

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

      1. associate-*l/ 98.4%

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

   3. Simplified 98.4%

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

   if -5.30000000000000028e42 < z < 1.14999999999999993e-102

   1. Initial program 97.3%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+42} \lor \neg \left(z \leq 1.15 \cdot 10^{-102}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 3: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+135} \lor \neg \left(t \leq 10^{-59}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -4.5e+135) (not (<= t 1e-59)))
   (+ x (* y (/ t a)))
   (- x (/ z (/ a y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.5e+135) || !(t <= 1e-59)) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x - (z / (a / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-4.5d+135)) .or. (.not. (t <= 1d-59))) then
        tmp = x + (y * (t / a))
    else
        tmp = x - (z / (a / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -4.5e+135) || !(t <= 1e-59)) {
		tmp = x + (y * (t / a));
	} else {
		tmp = x - (z / (a / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -4.5e+135) or not (t <= 1e-59):
		tmp = x + (y * (t / a))
	else:
		tmp = x - (z / (a / y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -4.5e+135) || !(t <= 1e-59))
		tmp = Float64(x + Float64(y * Float64(t / a)));
	else
		tmp = Float64(x - Float64(z / Float64(a / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -4.5e+135) || ~((t <= 1e-59)))
		tmp = x + (y * (t / a));
	else
		tmp = x - (z / (a / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -4.5e+135], N[Not[LessEqual[t, 1e-59]], $MachinePrecision]], N[(x + N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.50000000000000007e135 or 1e-59 < t

   1. Initial program 95.8%

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

      1. associate-*r/ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in z around 0 82.9%

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

      1. neg-mul-1 82.9%

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

      2. distribute-neg-frac 82.9%

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

   6. Simplified 82.9%

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

   if -4.50000000000000007e135 < t < 1e-59

   1. Initial program 93.7%

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

      1. associate-/l* 93.2%

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

   3. Simplified 93.2%

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

      1. add-cube-cbrt 92.8%

         \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\frac{a}{z - t}} \]

      2. div-inv 92.8%

         \[\leadsto x - \frac{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}{\color{blue}{a \cdot \frac{1}{z - t}}} \]

      3. times-frac 95.9%

         \[\leadsto x - \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}}} \]

      4. pow2 95.9%

         \[\leadsto x - \frac{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}} \]

   5. Applied egg-rr 95.9%

      \[\leadsto x - \color{blue}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}}} \]

   6. Taylor expanded in z around inf 84.6%

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

      1. *-commutative 84.6%

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

      2. associate-/l* 89.9%

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

   8. Simplified 89.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+135} \lor \neg \left(t \leq 10^{-59}\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+135} \lor \neg \left(t \leq 1.6 \cdot 10^{-42}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.1e+135) (not (<= t 1.6e-42)))
   (+ x (* t (/ y a)))
   (- x (/ z (/ a y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.1e+135) || !(t <= 1.6e-42)) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x - (z / (a / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-2.1d+135)) .or. (.not. (t <= 1.6d-42))) then
        tmp = x + (t * (y / a))
    else
        tmp = x - (z / (a / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.1e+135) || !(t <= 1.6e-42)) {
		tmp = x + (t * (y / a));
	} else {
		tmp = x - (z / (a / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -2.1e+135) or not (t <= 1.6e-42):
		tmp = x + (t * (y / a))
	else:
		tmp = x - (z / (a / y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.1e+135) || !(t <= 1.6e-42))
		tmp = Float64(x + Float64(t * Float64(y / a)));
	else
		tmp = Float64(x - Float64(z / Float64(a / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -2.1e+135) || ~((t <= 1.6e-42)))
		tmp = x + (t * (y / a));
	else
		tmp = x - (z / (a / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.1e+135], N[Not[LessEqual[t, 1.6e-42]], $MachinePrecision]], N[(x + N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.1000000000000001e135 or 1.60000000000000012e-42 < t

   1. Initial program 95.7%

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

      1. associate-*r/ 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in z around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. associate-*l/ 87.0%

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

      3. distribute-rgt-neg-out 87.0%

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

   6. Simplified 87.0%

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

   if -2.1000000000000001e135 < t < 1.60000000000000012e-42

   1. Initial program 93.7%

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

      1. associate-/l* 93.3%

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

   3. Simplified 93.3%

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

      1. add-cube-cbrt 92.9%

         \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\frac{a}{z - t}} \]

      2. div-inv 92.9%

         \[\leadsto x - \frac{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}{\color{blue}{a \cdot \frac{1}{z - t}}} \]

      3. times-frac 95.9%

         \[\leadsto x - \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}}} \]

      4. pow2 95.9%

         \[\leadsto x - \frac{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}} \]

   5. Applied egg-rr 95.9%

      \[\leadsto x - \color{blue}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}}} \]

   6. Taylor expanded in z around inf 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-/l* 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+135} \lor \neg \left(t \leq 1.6 \cdot 10^{-42}\right):\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 5: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.8e+135)
   (+ x (/ (* y t) a))
   (if (<= t 2.3e-42) (- x (/ z (/ a y))) (+ x (* t (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.8e+135) {
		tmp = x + ((y * t) / a);
	} else if (t <= 2.3e-42) {
		tmp = x - (z / (a / y));
	} else {
		tmp = x + (t * (y / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-2.8d+135)) then
        tmp = x + ((y * t) / a)
    else if (t <= 2.3d-42) then
        tmp = x - (z / (a / y))
    else
        tmp = x + (t * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.8e+135) {
		tmp = x + ((y * t) / a);
	} else if (t <= 2.3e-42) {
		tmp = x - (z / (a / y));
	} else {
		tmp = x + (t * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -2.8e+135:
		tmp = x + ((y * t) / a)
	elif t <= 2.3e-42:
		tmp = x - (z / (a / y))
	else:
		tmp = x + (t * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.8e+135)
		tmp = Float64(x + Float64(Float64(y * t) / a));
	elseif (t <= 2.3e-42)
		tmp = Float64(x - Float64(z / Float64(a / y)));
	else
		tmp = Float64(x + Float64(t * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -2.8e+135)
		tmp = x + ((y * t) / a);
	elseif (t <= 2.3e-42)
		tmp = x - (z / (a / y));
	else
		tmp = x + (t * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.8e+135], N[(x + N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e-42], N[(x - N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.80000000000000002e135

   1. Initial program 96.6%

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

   2. Taylor expanded in z around 0 89.9%

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

      1. mul-1-neg 89.9%

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

      2. distribute-rgt-neg-out 89.9%

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

   4. Simplified 89.9%

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

   if -2.80000000000000002e135 < t < 2.30000000000000004e-42

   1. Initial program 93.7%

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

      1. associate-/l* 93.3%

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

   3. Simplified 93.3%

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

      1. add-cube-cbrt 92.9%

         \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\frac{a}{z - t}} \]

      2. div-inv 92.9%

         \[\leadsto x - \frac{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}{\color{blue}{a \cdot \frac{1}{z - t}}} \]

      3. times-frac 95.9%

         \[\leadsto x - \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}}} \]

      4. pow2 95.9%

         \[\leadsto x - \frac{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}} \]

   5. Applied egg-rr 95.9%

      \[\leadsto x - \color{blue}{\frac{{\left(\sqrt[3]{y}\right)}^{2}}{a} \cdot \frac{\sqrt[3]{y}}{\frac{1}{z - t}}} \]

   6. Taylor expanded in z around inf 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-/l* 89.5%

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

   8. Simplified 89.5%

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

   if 2.30000000000000004e-42 < t

   1. Initial program 95.3%

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

      1. associate-*r/ 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in z around 0 82.1%

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

      1. mul-1-neg 82.1%

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

      2. associate-*l/ 85.6%

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

      3. distribute-rgt-neg-out 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+135}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;x - \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 6: 93.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+127}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 3e+127) (+ x (* y (/ (- t z) a))) (- x (* z (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 3e+127) {
		tmp = x + (y * ((t - z) / a));
	} else {
		tmp = x - (z * (y / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 3d+127) then
        tmp = x + (y * ((t - z) / a))
    else
        tmp = x - (z * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 3e+127) {
		tmp = x + (y * ((t - z) / a));
	} else {
		tmp = x - (z * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= 3e+127:
		tmp = x + (y * ((t - z) / a))
	else:
		tmp = x - (z * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 3e+127)
		tmp = Float64(x + Float64(y * Float64(Float64(t - z) / a)));
	else
		tmp = Float64(x - Float64(z * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 3e+127)
		tmp = x + (y * ((t - z) / a));
	else
		tmp = x - (z * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 3e+127], N[(x + N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.0000000000000002e127

   1. Initial program 95.9%

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

      1. associate-*r/ 94.7%

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

   3. Simplified 94.7%

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

   if 3.0000000000000002e127 < z

   1. Initial program 87.3%

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

      1. associate-*r/ 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in z around inf 80.3%

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

      1. associate-*l/ 90.8%

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

      2. *-commutative 90.8%

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

   6. Simplified 90.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+127}:\\ \;\;\;\;x + y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 7: 40.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + y \cdot \frac{z}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. associate-*r/ 92.2%

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

3. Simplified 92.2%

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

4. Taylor expanded in z around inf 70.0%

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

   1. *-commutative 70.0%

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

   2. frac-2neg 70.0%

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

   3. associate-*l/ 70.7%

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

   4. add-sqr-sqrt 33.7%

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

   5. sqrt-unprod 49.4%

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

   6. sqr-neg 49.4%

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

   7. sqrt-unprod 22.0%

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

   8. add-sqr-sqrt 44.3%

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

6. Applied egg-rr 44.3%

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

   1. div-inv 44.3%

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

   2. distribute-lft-neg-out 44.3%

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

   3. cancel-sign-sub 44.3%

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

   4. associate-*r* 45.0%

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

   5. add-sqr-sqrt 25.0%

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

   6. sqrt-unprod 50.2%

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

   7. sqr-neg 50.2%

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

   8. sqrt-unprod 34.3%

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

   9. add-sqr-sqrt 74.6%

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

   10. associate-*l* 70.6%

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

   11. *-commutative 70.6%

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

   12. associate-*l* 70.0%

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

   13. add-sqr-sqrt 33.7%

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

   14. sqrt-unprod 50.0%

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

   15. sqr-neg 50.0%

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

   16. sqrt-unprod 24.6%

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

   17. add-sqr-sqrt 44.6%

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

   18. div-inv 44.6%

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

8. Applied egg-rr 44.6%

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

9. Final simplification 44.6%

   \[\leadsto x + y \cdot \frac{z}{a} \]

Alternative 8: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - y \cdot \frac{z}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. associate-*r/ 92.2%

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

3. Simplified 92.2%

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

4. Taylor expanded in z around inf 70.0%

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

5. Final simplification 70.0%

   \[\leadsto x - y \cdot \frac{z}{a} \]

Alternative 9: 71.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - z \cdot \frac{y}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. associate-*r/ 92.2%

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

3. Simplified 92.2%

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

4. Taylor expanded in z around inf 70.7%

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

   1. associate-*l/ 74.6%

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

   2. *-commutative 74.6%

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

6. Simplified 74.6%

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

7. Final simplification 74.6%

   \[\leadsto x - z \cdot \frac{y}{a} \]

Alternative 10: 39.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. associate-*r/ 92.2%

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

3. Simplified 92.2%

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

4. Taylor expanded in z around inf 70.0%

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

   1. *-commutative 70.0%

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

   2. frac-2neg 70.0%

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

   3. associate-*l/ 70.7%

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

   4. add-sqr-sqrt 33.7%

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

   5. sqrt-unprod 49.4%

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

   6. sqr-neg 49.4%

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

   7. sqrt-unprod 22.0%

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

   8. add-sqr-sqrt 44.3%

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

6. Applied egg-rr 44.3%

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

7. Taylor expanded in x around inf 42.4%

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

8. Final simplification 42.4%

   \[\leadsto x \]

Developer target: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{t_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (- x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (- x (/ (* y (- z t)) a))
       (- x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x - (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x - ((y * (z - t)) / a)
    else
        tmp = x - (y / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x - (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x - ((y * (z - t)) / a);
	} else {
		tmp = x - (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x - (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x - ((y * (z - t)) / a)
	else:
		tmp = x - (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x - Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x - Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x - Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x - (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x - ((y * (z - t)) / a);
	else
		tmp = x - (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x - N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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