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

Percentage Accurate: 93.1% → 98.9%

Time: 8.9s

Alternatives: 7

Speedup: 1.0×

• 25.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y \cdot \left(x - z\right)}{t}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+303}\right):\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (* y (- x z)) t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+303)))
     (+ x (* y (/ (- z x) t)))
     t_1)))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y * (x - z)) / t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+303)) {
		tmp = x + (y * ((z - x) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - ((y * (x - z)) / t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e+303)) {
		tmp = x + (y * ((z - x) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - ((y * (x - z)) / t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e+303):
		tmp = x + (y * ((z - x) / t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y * Float64(x - z)) / t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+303))
		tmp = Float64(x + Float64(y * Float64(Float64(z - x) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - ((y * (x - z)) / t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e+303)))
		tmp = x + (y * ((z - x) / t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+303]], $MachinePrecision]], N[(x + N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -inf.0 or 1e303 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 82.6%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 1e303

   1. Initial program 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{y \cdot \left(x - z\right)}{t} \leq -\infty \lor \neg \left(x - \frac{y \cdot \left(x - z\right)}{t} \leq 10^{+303}\right):\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(x - z\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-114} \lor \neg \left(y \leq 5.6 \cdot 10^{-126}\right):\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.8e-114) (not (<= y 5.6e-126)))
   (+ x (* y (/ (- z x) t)))
   (+ x (/ (* z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.8e-114) || !(y <= 5.6e-126)) {
		tmp = x + (y * ((z - x) / 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 ((y <= (-3.8d-114)) .or. (.not. (y <= 5.6d-126))) then
        tmp = x + (y * ((z - x) / 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 ((y <= -3.8e-114) || !(y <= 5.6e-126)) {
		tmp = x + (y * ((z - x) / t));
	} else {
		tmp = x + ((z * y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -3.8e-114) or not (y <= 5.6e-126):
		tmp = x + (y * ((z - x) / t))
	else:
		tmp = x + ((z * y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.8e-114) || !(y <= 5.6e-126))
		tmp = Float64(x + Float64(y * Float64(Float64(z - x) / t)));
	else
		tmp = Float64(x + Float64(Float64(z * y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -3.8e-114) || ~((y <= 5.6e-126)))
		tmp = x + (y * ((z - x) / t));
	else
		tmp = x + ((z * y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.8e-114], N[Not[LessEqual[y, 5.6e-126]], $MachinePrecision]], N[(x + N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998e-114 or 5.59999999999999983e-126 < y

   1. Initial program 91.6%

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

      1. associate-/l* 97.6%

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

   3. Simplified 97.6%

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

   if -3.7999999999999998e-114 < y < 5.59999999999999983e-126

   1. Initial program 99.2%

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

      1. associate-/l* 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in z around inf 94.5%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-114} \lor \neg \left(y \leq 5.6 \cdot 10^{-126}\right):\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.9 \cdot 10^{-45} \lor \neg \left(z \leq 3 \cdot 10^{+14}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8.9e-45) (not (<= z 3e+14)))
   (+ x (* y (/ z t)))
   (* x (- 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.9e-45) || !(z <= 3e+14)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-8.9d-45)) .or. (.not. (z <= 3d+14))) then
        tmp = x + (y * (z / t))
    else
        tmp = x * (1.0d0 - (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8.9e-45) || !(z <= 3e+14)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8.9e-45) or not (z <= 3e+14):
		tmp = x + (y * (z / t))
	else:
		tmp = x * (1.0 - (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8.9e-45) || !(z <= 3e+14))
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -8.9e-45) || ~((z <= 3e+14)))
		tmp = x + (y * (z / t));
	else
		tmp = x * (1.0 - (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8.9e-45], N[Not[LessEqual[z, 3e+14]], $MachinePrecision]], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.90000000000000034e-45 or 3e14 < z

   1. Initial program 93.1%

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

      1. associate-/l* 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in z around inf 85.4%

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

      1. associate-/l* 84.2%

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

   7. Simplified 84.2%

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

   if -8.90000000000000034e-45 < z < 3e14

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-/l* 94.3%

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

      4. fma-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in z around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. *-rgt-identity 86.8%

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

      3. associate-/l* 89.9%

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

      4. distribute-rgt-neg-in 89.9%

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

      5. mul-1-neg 89.9%

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

      6. distribute-lft-in 89.9%

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

      7. mul-1-neg 89.9%

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

      8. unsub-neg 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.9 \cdot 10^{-45} \lor \neg \left(z \leq 3 \cdot 10^{+14}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{-44} \lor \neg \left(z \leq 4.4 \cdot 10^{+15}\right):\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -9.6e-44) (not (<= z 4.4e+15)))
   (+ x (/ z (/ t y)))
   (* x (- 1.0 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.6e-44) || !(z <= 4.4e+15)) {
		tmp = x + (z / (t / y));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-9.6d-44)) .or. (.not. (z <= 4.4d+15))) then
        tmp = x + (z / (t / y))
    else
        tmp = x * (1.0d0 - (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.6e-44) || !(z <= 4.4e+15)) {
		tmp = x + (z / (t / y));
	} else {
		tmp = x * (1.0 - (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -9.6e-44) or not (z <= 4.4e+15):
		tmp = x + (z / (t / y))
	else:
		tmp = x * (1.0 - (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -9.6e-44) || !(z <= 4.4e+15))
		tmp = Float64(x + Float64(z / Float64(t / y)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -9.6e-44) || ~((z <= 4.4e+15)))
		tmp = x + (z / (t / y));
	else
		tmp = x * (1.0 - (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -9.6e-44], N[Not[LessEqual[z, 4.4e+15]], $MachinePrecision]], N[(x + N[(z / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.60000000000000035e-44 or 4.4e15 < z

   1. Initial program 93.1%

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

      1. associate-/l* 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in z around inf 85.4%

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

      1. associate-/l* 84.2%

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

   7. Simplified 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-/r/ 90.4%

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

   9. Applied egg-rr 90.4%

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

   if -9.60000000000000035e-44 < z < 4.4e15

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-/l* 94.3%

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

      4. fma-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in z around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. *-rgt-identity 86.8%

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

      3. associate-/l* 89.9%

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

      4. distribute-rgt-neg-in 89.9%

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

      5. mul-1-neg 89.9%

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

      6. distribute-lft-in 89.9%

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

      7. mul-1-neg 89.9%

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

      8. unsub-neg 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{-44} \lor \neg \left(z \leq 4.4 \cdot 10^{+15}\right):\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. associate-/l* 91.0%

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

3. Simplified 91.0%

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

   1. associate-*r/ 94.5%

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

   2. *-commutative 94.5%

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

   3. associate-*r/ 97.1%

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

   4. clear-num 97.1%

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

   5. un-div-inv 97.9%

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 6: 66.1% 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 94.5%

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

   1. +-commutative 94.5%

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

   2. *-commutative 94.5%

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

   3. associate-/l* 97.1%

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

   4. fma-define 97.1%

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

3. Simplified 97.1%

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

5. Taylor expanded in z around 0 65.0%

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

   1. mul-1-neg 65.0%

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

   2. *-rgt-identity 65.0%

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

   3. associate-/l* 69.0%

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

   4. distribute-rgt-neg-in 69.0%

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

   5. mul-1-neg 69.0%

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

   6. distribute-lft-in 69.0%

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

   7. mul-1-neg 69.0%

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

   8. unsub-neg 69.0%

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

7. Simplified 69.0%

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

8. Final simplification 69.0%

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

Alternative 7: 38.9% 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 94.5%

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

   1. +-commutative 94.5%

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

   2. *-commutative 94.5%

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

   3. associate-/l* 97.1%

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

   4. fma-define 97.1%

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

3. Simplified 97.1%

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

5. Taylor expanded in y around 0 40.4%

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

6. Final simplification 40.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 91.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

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

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