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

Percentage Accurate: 93.0% → 97.1%

Time: 6.8s

Alternatives: 8

Speedup: 1.0×

• 26.2% 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: 93.0% 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: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - t) / (a / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

   1. *-commutative 93.8%

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

   2. associate-/l* 97.4%

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

3. Simplified 97.4%

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

4. Final simplification 97.4%

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

Alternative 2: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z - t\right) \cdot y\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+300}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- z t) y)))
   (if (<= t_1 -1e+300) (+ x (* y (/ z a))) (+ x (/ t_1 a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) * y;
	double tmp;
	if (t_1 <= -1e+300) {
		tmp = x + (y * (z / a));
	} else {
		tmp = x + (t_1 / 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) :: t_1
    real(8) :: tmp
    t_1 = (z - t) * y
    if (t_1 <= (-1d+300)) then
        tmp = x + (y * (z / a))
    else
        tmp = x + (t_1 / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) * y;
	double tmp;
	if (t_1 <= -1e+300) {
		tmp = x + (y * (z / a));
	} else {
		tmp = x + (t_1 / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) * y
	tmp = 0
	if t_1 <= -1e+300:
		tmp = x + (y * (z / a))
	else:
		tmp = x + (t_1 / a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) * y)
	tmp = 0.0
	if (t_1 <= -1e+300)
		tmp = Float64(x + Float64(y * Float64(z / a)));
	else
		tmp = Float64(x + Float64(t_1 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) * y;
	tmp = 0.0;
	if (t_1 <= -1e+300)
		tmp = x + (y * (z / a));
	else
		tmp = x + (t_1 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+300], N[(x + N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -1.0000000000000001e300

   1. Initial program 62.8%

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

      1. +-commutative 62.8%

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

      2. associate-*l/ 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. associate-/r/ 99.8%

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

      3. div-inv 99.9%

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

      4. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around inf 52.4%

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

      1. *-commutative 52.4%

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

      2. associate-*l/ 80.1%

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

   8. Simplified 80.1%

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

   if -1.0000000000000001e300 < (*.f64 y (-.f64 z t))

   1. Initial program 98.0%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \leq -1 \cdot 10^{+300}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \]

Alternative 3: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+53} \lor \neg \left(t \leq 4.4 \cdot 10^{+37}\right):\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.25e+53) (not (<= t 4.4e+37)))
   (- 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 <= -1.25e+53) || !(t <= 4.4e+37)) {
		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 <= (-1.25d+53)) .or. (.not. (t <= 4.4d+37))) 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 <= -1.25e+53) || !(t <= 4.4e+37)) {
		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 <= -1.25e+53) or not (t <= 4.4e+37):
		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 <= -1.25e+53) || !(t <= 4.4e+37))
		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 <= -1.25e+53) || ~((t <= 4.4e+37)))
		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, -1.25e+53], N[Not[LessEqual[t, 4.4e+37]], $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 < -1.2500000000000001e53 or 4.4000000000000001e37 < t

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. associate-*l/ 97.2%

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

      3. fma-def 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in z around 0 86.3%

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

      1. mul-1-neg 86.3%

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

      2. associate-/l* 87.9%

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

      3. associate-/r/ 84.8%

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

      4. distribute-lft-neg-in 84.8%

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

      5. cancel-sign-sub-inv 84.8%

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

   6. Simplified 84.8%

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

   if -1.2500000000000001e53 < t < 4.4000000000000001e37

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 88.6%

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

      1. associate-*l/ 93.4%

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

      2. *-commutative 93.4%

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

   4. Simplified 93.4%

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

      1. clear-num 93.4%

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

      2. un-div-inv 93.6%

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

   6. Applied egg-rr 93.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+53} \lor \neg \left(t \leq 4.4 \cdot 10^{+37}\right):\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+51}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8.2e+51)
   (- x (/ t (/ a y)))
   (if (<= t 6e+37) (+ x (/ z (/ a y))) (- x (* y (/ t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8.2e+51) {
		tmp = x - (t / (a / y));
	} else if (t <= 6e+37) {
		tmp = x + (z / (a / y));
	} else {
		tmp = x - (y * (t / 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 <= (-8.2d+51)) then
        tmp = x - (t / (a / y))
    else if (t <= 6d+37) then
        tmp = x + (z / (a / y))
    else
        tmp = x - (y * (t / 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 <= -8.2e+51) {
		tmp = x - (t / (a / y));
	} else if (t <= 6e+37) {
		tmp = x + (z / (a / y));
	} else {
		tmp = x - (y * (t / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -8.2e+51:
		tmp = x - (t / (a / y))
	elif t <= 6e+37:
		tmp = x + (z / (a / y))
	else:
		tmp = x - (y * (t / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8.2e+51)
		tmp = Float64(x - Float64(t / Float64(a / y)));
	elseif (t <= 6e+37)
		tmp = Float64(x + Float64(z / Float64(a / y)));
	else
		tmp = Float64(x - Float64(y * Float64(t / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -8.2e+51)
		tmp = x - (t / (a / y));
	elseif (t <= 6e+37)
		tmp = x + (z / (a / y));
	else
		tmp = x - (y * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8.2e+51], N[(x - N[(t / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e+37], N[(x + N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.20000000000000021e51

   1. Initial program 94.9%

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

      1. +-commutative 94.9%

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

      2. associate-*l/ 98.2%

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

      3. fma-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in z around 0 87.3%

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

      1. mul-1-neg 87.3%

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

      2. associate-/l* 88.8%

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

      3. associate-/r/ 81.2%

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

      4. distribute-lft-neg-in 81.2%

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

      5. cancel-sign-sub-inv 81.2%

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

   6. Simplified 81.2%

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

      1. associate-*l/ 87.3%

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

      2. associate-/l* 88.8%

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

   8. Applied egg-rr 88.8%

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

   if -8.20000000000000021e51 < t < 6.00000000000000043e37

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 88.6%

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

      1. associate-*l/ 93.4%

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

      2. *-commutative 93.4%

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

   4. Simplified 93.4%

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

      1. clear-num 93.4%

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

      2. un-div-inv 93.6%

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

   6. Applied egg-rr 93.6%

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

   if 6.00000000000000043e37 < t

   1. Initial program 89.3%

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

      1. +-commutative 89.3%

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

      2. associate-*l/ 96.2%

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

      3. fma-def 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. associate-/l* 86.9%

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

      3. associate-/r/ 88.7%

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

      4. distribute-lft-neg-in 88.7%

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

      5. cancel-sign-sub-inv 88.7%

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

   6. Simplified 88.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+51}:\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \]

Alternative 5: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+51}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.3e+51)
   (- x (* t (/ y a)))
   (if (<= t 4e+37) (+ x (/ z (/ a y))) (- x (* y (/ t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.3e+51) {
		tmp = x - (t * (y / a));
	} else if (t <= 4e+37) {
		tmp = x + (z / (a / y));
	} else {
		tmp = x - (y * (t / 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 <= (-4.3d+51)) then
        tmp = x - (t * (y / a))
    else if (t <= 4d+37) then
        tmp = x + (z / (a / y))
    else
        tmp = x - (y * (t / 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 <= -4.3e+51) {
		tmp = x - (t * (y / a));
	} else if (t <= 4e+37) {
		tmp = x + (z / (a / y));
	} else {
		tmp = x - (y * (t / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -4.3e+51:
		tmp = x - (t * (y / a))
	elif t <= 4e+37:
		tmp = x + (z / (a / y))
	else:
		tmp = x - (y * (t / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.3e+51)
		tmp = Float64(x - Float64(t * Float64(y / a)));
	elseif (t <= 4e+37)
		tmp = Float64(x + Float64(z / Float64(a / y)));
	else
		tmp = Float64(x - Float64(y * Float64(t / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -4.3e+51)
		tmp = x - (t * (y / a));
	elseif (t <= 4e+37)
		tmp = x + (z / (a / y));
	else
		tmp = x - (y * (t / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.3e+51], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e+37], N[(x + N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2999999999999997e51

   1. Initial program 94.9%

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

   2. Taylor expanded in z around 0 87.3%

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

      1. associate-*r/ 88.9%

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

      2. associate-*r* 88.9%

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

      3. neg-mul-1 88.9%

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

      4. *-commutative 88.9%

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

   4. Simplified 88.9%

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

   if -4.2999999999999997e51 < t < 3.99999999999999982e37

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 88.6%

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

      1. associate-*l/ 93.4%

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

      2. *-commutative 93.4%

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

   4. Simplified 93.4%

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

      1. clear-num 93.4%

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

      2. un-div-inv 93.6%

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

   6. Applied egg-rr 93.6%

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

   if 3.99999999999999982e37 < t

   1. Initial program 89.3%

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

      1. +-commutative 89.3%

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

      2. associate-*l/ 96.2%

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

      3. fma-def 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. associate-/l* 86.9%

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

      3. associate-/r/ 88.7%

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

      4. distribute-lft-neg-in 88.7%

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

      5. cancel-sign-sub-inv 88.7%

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

   6. Simplified 88.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+51}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+37}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \end{array} \]

Alternative 6: 71.2% 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 93.8%

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

2. Taylor expanded in z around inf 69.7%

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

   1. associate-*l/ 75.0%

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

   2. *-commutative 75.0%

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

4. Simplified 75.0%

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

5. Final simplification 75.0%

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

Alternative 7: 71.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 / (a / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

2. Taylor expanded in z around inf 69.7%

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

   1. associate-*l/ 75.0%

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

   2. *-commutative 75.0%

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

4. Simplified 75.0%

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

   1. clear-num 75.0%

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

   2. un-div-inv 75.1%

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

6. Applied egg-rr 75.1%

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

7. Final simplification 75.1%

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

Alternative 8: 38.9% 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 93.8%

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

   1. +-commutative 93.8%

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

   2. associate-*l/ 97.3%

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

   3. fma-def 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in y around 0 39.2%

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

5. Final simplification 39.2%

   \[\leadsto x \]

Developer target: 99.2% 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 2023308 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))