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

Percentage Accurate: 92.8% → 97.5%

Time: 8.8s

Alternatives: 7

Speedup: 1.1×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.8% 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: 97.5% 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 92.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 98.0

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

4. Applied egg-rr 98.0%

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

Alternative 2: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, -x, x\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- x) x)))
   (if (<= x -5e-57) t_1 (if (<= x 0.0033) (fma z (/ y t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), -x, x);
	double tmp;
	if (x <= -5e-57) {
		tmp = t_1;
	} else if (x <= 0.0033) {
		tmp = fma(z, (y / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), Float64(-x), x)
	tmp = 0.0
	if (x <= -5e-57)
		tmp = t_1;
	elseif (x <= 0.0033)
		tmp = fma(z, Float64(y / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * (-x) + x), $MachinePrecision]}, If[LessEqual[x, -5e-57], t$95$1, If[LessEqual[x, 0.0033], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-57 or 0.0033 < x

   1. Initial program 91.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.8

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

   7. Simplified 89.8%

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

   if -5.0000000000000002e-57 < x < 0.0033

   1. Initial program 92.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 85.7

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

   5. Simplified 85.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 91.4

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

   7. Applied egg-rr 91.4%

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

Alternative 3: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{if}\;t \leq -2.52 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ y t) x)))
   (if (<= t -2.52e-26) t_1 (if (<= t 7.6e-20) (/ (* y (- z x)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (y / t), x);
	double tmp;
	if (t <= -2.52e-26) {
		tmp = t_1;
	} else if (t <= 7.6e-20) {
		tmp = (y * (z - x)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(y / t), x)
	tmp = 0.0
	if (t <= -2.52e-26)
		tmp = t_1;
	elseif (t <= 7.6e-20)
		tmp = Float64(Float64(y * Float64(z - x)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2.52e-26], t$95$1, If[LessEqual[t, 7.6e-20], N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.52000000000000004e-26 or 7.5999999999999995e-20 < t

   1. Initial program 86.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 90.8

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

   7. Applied egg-rr 90.8%

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

   if -2.52000000000000004e-26 < t < 7.5999999999999995e-20

   1. Initial program 98.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 95.8

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 87.4

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

   7. Simplified 87.4%

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

Alternative 4: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - x}{t}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ (- z x) t))))
   (if (<= y -3.2e+67) t_1 (if (<= y 6.4e+153) (fma z (/ y t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * ((z - x) / t);
	double tmp;
	if (y <= -3.2e+67) {
		tmp = t_1;
	} else if (y <= 6.4e+153) {
		tmp = fma(z, (y / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(Float64(z - x) / t))
	tmp = 0.0
	if (y <= -3.2e+67)
		tmp = t_1;
	elseif (y <= 6.4e+153)
		tmp = fma(z, Float64(y / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+67], t$95$1, If[LessEqual[y, 6.4e+153], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999983e67 or 6.4000000000000003e153 < y

   1. Initial program 83.5%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 93.8

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

   5. Simplified 93.8%

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

   if -3.19999999999999983e67 < y < 6.4000000000000003e153

   1. Initial program 96.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 82.4

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

   5. Simplified 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 84.6

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

   7. Applied egg-rr 84.6%

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

Alternative 5: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{-241}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-193}:\\ \;\;\;\;-\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ y t) x)))
   (if (<= t -4.7e-241) t_1 (if (<= t 6.5e-193) (- (* (/ y t) x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (y / t), x);
	double tmp;
	if (t <= -4.7e-241) {
		tmp = t_1;
	} else if (t <= 6.5e-193) {
		tmp = -((y / t) * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(y / t), x)
	tmp = 0.0
	if (t <= -4.7e-241)
		tmp = t_1;
	elseif (t <= 6.5e-193)
		tmp = Float64(-Float64(Float64(y / t) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -4.7e-241], t$95$1, If[LessEqual[t, 6.5e-193], (-N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.6999999999999999e-241 or 6.5000000000000004e-193 < t

   1. Initial program 91.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 76.3

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

   5. Simplified 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 82.2

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

   7. Applied egg-rr 82.2%

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

   if -4.6999999999999999e-241 < t < 6.5000000000000004e-193

   1. Initial program 96.9%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 71.6

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

   5. Simplified 71.6%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 70.1

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

   8. Simplified 70.1%

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

      1. distribute-rgt-neg-out N/A

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

      2. distribute-lft-neg-in N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 73.1

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

   10. Applied egg-rr 73.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-193}:\\ \;\;\;\;-\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 71.0

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

5. Simplified 71.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lift-/.f64 N/A

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

   9. lower-fma.f64 77.2

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

7. Applied egg-rr 77.2%

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

Alternative 7: 40.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = (y / t) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 33.0

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

5. Simplified 33.0%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lift-/.f64 N/A

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

   4. lower-*.f64 38.0

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

7. Applied egg-rr 38.0%

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

8. Final simplification 38.0%

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

Developer Target 1: 90.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024207 
(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)))