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

Percentage Accurate: 92.3% → 97.9%

Time: 11.3s

Alternatives: 7

Speedup: 1.1×

• 25.4% 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: 92.3% 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 91.4%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 96.8

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

4. Applied egg-rr 96.8%

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

Alternative 2: 40.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x (/ (* y (- z x)) t)) 2e+306) x (* t (/ x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + ((y * (z - x)) / t)) <= 2e+306) {
		tmp = x;
	} else {
		tmp = t * (x / 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 ((x + ((y * (z - x)) / t)) <= 2d+306) then
        tmp = x
    else
        tmp = t * (x / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + ((y * (z - x)) / t)) <= 2e+306) {
		tmp = x;
	} else {
		tmp = t * (x / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x + ((y * (z - x)) / t)) <= 2e+306:
		tmp = x
	else:
		tmp = t * (x / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(z - x)) / t)) <= 2e+306)
		tmp = x;
	else
		tmp = Float64(t * Float64(x / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x + ((y * (z - x)) / t)) <= 2e+306)
		tmp = x;
	else
		tmp = t * (x / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 2e+306], x, N[(t * N[(x / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 2.00000000000000003e306

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 42.9%

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

   if 2.00000000000000003e306 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 78.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 98.0

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in t around 0

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

      1. distribute-lft-out-- N/A

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

      2. *-commutative N/A

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. /-lowering-/.f64 N/A

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

      8. associate-+r+ N/A

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

      9. mul-1-neg N/A

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

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

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-out N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. mul-1-neg N/A

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

      17. unsub-neg N/A

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

      18. --lowering--.f64 N/A

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

      19. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, t - y, \color{blue}{y \cdot z + 0}\right)}{t} \]

      20. accelerator-lowering-fma.f64 78.6

          \[\leadsto \frac{\mathsf{fma}\left(x, t - y, \color{blue}{\mathsf{fma}\left(y, z, 0\right)}\right)}{t} \]

   7. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, t - y, \mathsf{fma}\left(y, z, 0\right)\right)}{t}} \]

   8. Taylor expanded in t around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{t \cdot x + 0}}{t} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot t} + 0}{t} \]

      3. accelerator-lowering-fma.f64 2.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, t, 0\right)}}{t} \]

   10. Simplified 2.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, t, 0\right)}}{t} \]
   11. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. /-lowering-/.f64 23.9

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

   12. Applied egg-rr 23.9%

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

Alternative 3: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -34:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -34.0)
   (fma (/ y t) z x)
   (if (<= z 2.9e-62) (* x (- 1.0 (/ y t))) (+ x (* (/ y t) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -34.0) {
		tmp = fma((y / t), z, x);
	} else if (z <= 2.9e-62) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = x + ((y / t) * z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -34.0)
		tmp = fma(Float64(y / t), z, x);
	elseif (z <= 2.9e-62)
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(x + Float64(Float64(y / t) * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -34.0], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 2.9e-62], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -34

   1. Initial program 89.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 89.1%

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

   if -34 < z < 2.89999999999999986e-62

   1. Initial program 92.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{0 - x}, y, y \cdot z\right)}{t} \]

      7. --lowering--.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{0 - x}, y, y \cdot z\right)}{t} \]

      8. *-lowering-*.f64 92.4

         \[\leadsto x + \frac{\mathsf{fma}\left(0 - x, y, \color{blue}{y \cdot z}\right)}{t} \]

   4. Applied egg-rr 92.4%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(0 - x, y, y \cdot z\right)}}{t} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot z + \left(0 - x\right) \cdot y}}{t} \]

      2. sub0-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. sub-neg N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. /-lowering-/.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 95.8

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. *-inverses N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. *-lowering-*.f64 N/A

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

      6. div-sub N/A

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

      7. *-inverses N/A

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

      8. --lowering--.f64 N/A

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

      9. /-lowering-/.f64 88.2

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

   9. Simplified 88.2%

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

   if 2.89999999999999986e-62 < z

   1. Initial program 92.0%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 85.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 90.4

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -34:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;z \leq -490:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= z -490.0) t_1 (if (<= z 9e-62) (* x (- 1.0 (/ y t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (z <= -490.0) {
		tmp = t_1;
	} else if (z <= 9e-62) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (z <= -490.0)
		tmp = t_1;
	elseif (z <= 9e-62)
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -490.0], t$95$1, If[LessEqual[z, 9e-62], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -490 or 9.00000000000000036e-62 < z

   1. Initial program 90.7%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 97.6

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 89.7%

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

   if -490 < z < 9.00000000000000036e-62

   1. Initial program 92.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{0 - x}, y, y \cdot z\right)}{t} \]

      7. --lowering--.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{0 - x}, y, y \cdot z\right)}{t} \]

      8. *-lowering-*.f64 92.4

         \[\leadsto x + \frac{\mathsf{fma}\left(0 - x, y, \color{blue}{y \cdot z}\right)}{t} \]

   4. Applied egg-rr 92.4%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(0 - x, y, y \cdot z\right)}}{t} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot z + \left(0 - x\right) \cdot y}}{t} \]

      2. sub0-neg N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. sub-neg N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. /-lowering-/.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 95.8

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. *-inverses N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. *-lowering-*.f64 N/A

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

      6. div-sub N/A

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

      7. *-inverses N/A

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

      8. --lowering--.f64 N/A

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

      9. /-lowering-/.f64 88.2

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

   9. Simplified 88.2%

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

Alternative 5: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3e+55) x (if (<= t 1.15e-9) (* (/ y t) z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e+55) {
		tmp = x;
	} else if (t <= 1.15e-9) {
		tmp = (y / t) * z;
	} else {
		tmp = x;
	}
	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 (t <= (-3d+55)) then
        tmp = x
    else if (t <= 1.15d-9) then
        tmp = (y / t) * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e+55) {
		tmp = x;
	} else if (t <= 1.15e-9) {
		tmp = (y / t) * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3e+55:
		tmp = x
	elif t <= 1.15e-9:
		tmp = (y / t) * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3e+55)
		tmp = x;
	elseif (t <= 1.15e-9)
		tmp = Float64(Float64(y / t) * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3e+55)
		tmp = x;
	elseif (t <= 1.15e-9)
		tmp = (y / t) * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3e+55], x, If[LessEqual[t, 1.15e-9], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -3.00000000000000017e55 or 1.15e-9 < t

   1. Initial program 82.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 70.2%

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

   if -3.00000000000000017e55 < t < 1.15e-9

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{t}} + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, 0\right)} \]

      4. /-lowering-/.f64 50.9

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{z}{t}}, 0\right) \]

   5. Simplified 50.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 57.6

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

   7. Applied egg-rr 57.6%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+55}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 96.8

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

4. Applied egg-rr 96.8%

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

5. Taylor expanded in z around inf

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

7. Simplified 77.9%

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

Alternative 7: 38.1% accurate, 23.0× 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 91.4%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 37.1%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 90.9% 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 2024195 
(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)))