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

Percentage Accurate: 93.0% → 97.9%

Time: 8.1s

Alternatives: 10

Speedup: 1.1×

• 24.6% 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: 93.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{t}, z - x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 97.0

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

4. Applied rewrites 97.0%

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

Alternative 2: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -2.55e-40) t_1 (if (<= t 5.8e-108) (/ (* (- z x) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (t <= -2.55e-40) {
		tmp = t_1;
	} else if (t <= 5.8e-108) {
		tmp = ((z - x) * y) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (t <= -2.55e-40)
		tmp = t_1;
	elseif (t <= 5.8e-108)
		tmp = Float64(Float64(Float64(z - x) * y) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.55e-40], t$95$1, If[LessEqual[t, 5.8e-108], N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.55000000000000019e-40 or 5.8000000000000002e-108 < t

   1. Initial program 92.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -2.55000000000000019e-40 < t < 5.8000000000000002e-108

   1. Initial program 98.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 3: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -2.55e-40) t_1 (if (<= t 1.5e-109) (* (/ (- z x) t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (t <= -2.55e-40) {
		tmp = t_1;
	} else if (t <= 1.5e-109) {
		tmp = ((z - x) / t) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (t <= -2.55e-40)
		tmp = t_1;
	elseif (t <= 1.5e-109)
		tmp = Float64(Float64(Float64(z - x) / t) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.55e-40], t$95$1, If[LessEqual[t, 1.5e-109], N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.55000000000000019e-40 or 1.50000000000000011e-109 < t

   1. Initial program 92.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -2.55000000000000019e-40 < t < 1.50000000000000011e-109

   1. Initial program 98.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 83.6%

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

Alternative 4: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;z \leq 2.65 \cdot 10^{-307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-217}:\\ \;\;\;\;\frac{\left(-x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= z 2.65e-307) t_1 (if (<= z 2e-217) (/ (* (- x) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (z <= 2.65e-307) {
		tmp = t_1;
	} else if (z <= 2e-217) {
		tmp = (-x * y) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (z <= 2.65e-307)
		tmp = t_1;
	elseif (z <= 2e-217)
		tmp = Float64(Float64(Float64(-x) * y) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, 2.65e-307], t$95$1, If[LessEqual[z, 2e-217], N[(N[((-x) * y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.6499999999999999e-307 or 2.00000000000000016e-217 < z

   1. Initial program 94.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if 2.6499999999999999e-307 < z < 2.00000000000000016e-217

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.0%

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

Alternative 5: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;z \leq 2.65 \cdot 10^{-307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-217}:\\ \;\;\;\;\frac{-x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= z 2.65e-307) t_1 (if (<= z 1.05e-217) (* (/ (- x) t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (z <= 2.65e-307) {
		tmp = t_1;
	} else if (z <= 1.05e-217) {
		tmp = (-x / t) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (z <= 2.65e-307)
		tmp = t_1;
	elseif (z <= 1.05e-217)
		tmp = Float64(Float64(Float64(-x) / t) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, 2.65e-307], t$95$1, If[LessEqual[z, 1.05e-217], N[(N[((-x) / t), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.6499999999999999e-307 or 1.05e-217 < z

   1. Initial program 94.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 78.1

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

   7. Applied rewrites 78.1%

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

   if 2.6499999999999999e-307 < z < 1.05e-217

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.6%

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

Alternative 6: 72.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-150}:\\ \;\;\;\;\left(-x\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -1.2e-157) t_1 (if (<= t 9.6e-150) (* (- x) (/ y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (t <= -1.2e-157) {
		tmp = t_1;
	} else if (t <= 9.6e-150) {
		tmp = -x * (y / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (t <= -1.2e-157)
		tmp = t_1;
	elseif (t <= 9.6e-150)
		tmp = Float64(Float64(-x) * Float64(y / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -1.2e-157], t$95$1, If[LessEqual[t, 9.6e-150], N[((-x) * N[(y / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2e-157 or 9.6e-150 < t

   1. Initial program 93.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 84.2

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

   7. Applied rewrites 84.2%

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

   if -1.2e-157 < t < 9.6e-150

   1. Initial program 98.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 56.0%

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

   10. Applied rewrites 61.3%

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

Alternative 7: 73.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z}{t}, y, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (/ z t) y x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 93.5

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

4. Applied rewrites 93.5%

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

5. Taylor expanded in z around inf

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

   1. lower-/.f64 73.1

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

7. Applied rewrites 73.1%

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

Alternative 8: 40.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ z \cdot \frac{y}{t} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 93.5

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

4. Applied rewrites 93.5%

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

5. Taylor expanded in z around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 41.5

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

7. Applied rewrites 41.5%

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

8. Final simplification 41.5%

   \[\leadsto z \cdot \frac{y}{t} \]
9. Add Preprocessing

Alternative 9: 37.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{z \cdot y}{t} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

3. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 39.7

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

5. Applied rewrites 39.7%

   \[\leadsto \color{blue}{\frac{z \cdot y}{t}} \]
6. Add Preprocessing

Alternative 10: 37.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{z}{t} \cdot y \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.0%

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

3. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 39.7

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

5. Applied rewrites 39.7%

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

7. Applied rewrites 39.4%

   \[\leadsto \frac{z}{t} \cdot \color{blue}{y} \]
8. Add Preprocessing

Developer Target 1: 91.1% 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 2024243 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (+ (* x (/ y t)) (* (- z) (/ y t)))))

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