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

Percentage Accurate: 93.4% → 95.4%

Time: 9.7s

Alternatives: 8

Speedup: 1.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.4% 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: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4e-153) (fma (/ y t) (- z x) x) (+ x (/ y (/ t (- z x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4e-153) {
		tmp = fma((y / t), (z - x), x);
	} else {
		tmp = x + (y / (t / (z - x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4e-153)
		tmp = fma(Float64(y / t), Float64(z - x), x);
	else
		tmp = Float64(x + Float64(y / Float64(t / Float64(z - x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4e-153], N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y / N[(t / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000016e-153

   1. Initial program 90.4%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   if -4.00000000000000016e-153 < x

   1. Initial program 91.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. --lowering--.f64 98.5

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

   4. Applied egg-rr 98.5%

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

Alternative 2: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- 1.0 (/ y t)))))
   (if (<= x -8.5e+26) t_1 (if (<= x 3.5e-11) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (1.0 - (y / t));
	double tmp;
	if (x <= -8.5e+26) {
		tmp = t_1;
	} else if (x <= 3.5e-11) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(1.0 - Float64(y / t)))
	tmp = 0.0
	if (x <= -8.5e+26)
		tmp = t_1;
	elseif (x <= 3.5e-11)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e+26], t$95$1, If[LessEqual[x, 3.5e-11], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5e26 or 3.50000000000000019e-11 < x

   1. Initial program 90.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 99.9

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

   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

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

      1. *-rgt-identity N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. distribute-lft-in N/A

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

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

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

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

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

      11. /-lowering-/.f64 91.6

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

   7. Simplified 91.6%

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

   if -8.5e26 < x < 3.50000000000000019e-11

   1. Initial program 91.4%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 81.0%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. /-lowering-/.f64 86.7

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

   7. Applied egg-rr 86.7%

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

Alternative 3: 55.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.55e+15) (* y (/ z t)) (if (<= y 1.6e-75) x (* (/ y t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.55e+15) {
		tmp = y * (z / t);
	} else if (y <= 1.6e-75) {
		tmp = x;
	} else {
		tmp = (y / t) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.55d+15)) then
        tmp = y * (z / t)
    else if (y <= 1.6d-75) then
        tmp = x
    else
        tmp = (y / t) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.55e+15) {
		tmp = y * (z / t);
	} else if (y <= 1.6e-75) {
		tmp = x;
	} else {
		tmp = (y / t) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.55e+15:
		tmp = y * (z / t)
	elif y <= 1.6e-75:
		tmp = x
	else:
		tmp = (y / t) * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.55e+15)
		tmp = Float64(y * Float64(z / t));
	elseif (y <= 1.6e-75)
		tmp = x;
	else
		tmp = Float64(Float64(y / t) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.55e+15)
		tmp = y * (z / t);
	elseif (y <= 1.6e-75)
		tmp = x;
	else
		tmp = (y / t) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.55e+15], N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-75], x, N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55e15

   1. Initial program 86.1%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 97.0

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

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. remove-double-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(-1 \cdot y\right) \cdot z}}{t} \]

      6. sub-neg N/A

         \[\leadsto \frac{\color{blue}{0 + \left(\mathsf{neg}\left(\left(-1 \cdot y\right) \cdot z\right)\right)}}{t} \]

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 50.4

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

   7. Simplified 50.4%

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

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 58.7

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

   9. Applied egg-rr 58.7%

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

   if -1.55e15 < y < 1.59999999999999988e-75

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 65.1%

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

   if 1.59999999999999988e-75 < y

   1. Initial program 86.1%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 95.6

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

   4. Applied egg-rr 95.6%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. remove-double-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(-1 \cdot y\right) \cdot z}}{t} \]

      6. sub-neg N/A

         \[\leadsto \frac{\color{blue}{0 + \left(\mathsf{neg}\left(\left(-1 \cdot y\right) \cdot z\right)\right)}}{t} \]

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 40.9

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

   7. Simplified 40.9%

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

      1. +-rgt-identity N/A

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

      2. associate-*l/ N/A

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

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

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

      4. /-lowering-/.f64 49.0

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

   9. Applied egg-rr 49.0%

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

Alternative 4: 51.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-99}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.8e-21) x (if (<= x 1.1e-99) (* y (/ z t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.8e-21) {
		tmp = x;
	} else if (x <= 1.1e-99) {
		tmp = y * (z / t);
	} 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 (x <= (-5.8d-21)) then
        tmp = x
    else if (x <= 1.1d-99) then
        tmp = y * (z / t)
    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 (x <= -5.8e-21) {
		tmp = x;
	} else if (x <= 1.1e-99) {
		tmp = y * (z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.8e-21:
		tmp = x
	elif x <= 1.1e-99:
		tmp = y * (z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.8e-21)
		tmp = x;
	elseif (x <= 1.1e-99)
		tmp = Float64(y * Float64(z / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -5.8e-21)
		tmp = x;
	elseif (x <= 1.1e-99)
		tmp = y * (z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.8e-21], x, If[LessEqual[x, 1.1e-99], N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e-21 or 1.10000000000000002e-99 < x

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 52.1%

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

   if -5.8e-21 < x < 1.10000000000000002e-99

   1. Initial program 90.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 92.7

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

   4. Applied egg-rr 92.7%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. remove-double-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - \left(-1 \cdot y\right) \cdot z}}{t} \]

      6. sub-neg N/A

         \[\leadsto \frac{\color{blue}{0 + \left(\mathsf{neg}\left(\left(-1 \cdot y\right) \cdot z\right)\right)}}{t} \]

      7. mul-1-neg N/A

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

      8. distribute-lft-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 63.4

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

   7. Simplified 63.4%

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

      1. +-rgt-identity N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 68.8

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

   9. Applied egg-rr 68.8%

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

Alternative 5: 95.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - x}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.52e-135) (fma (/ y t) (- z x) x) (fma (/ (- z x) t) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.52e-135) {
		tmp = fma((y / t), (z - x), x);
	} else {
		tmp = fma(((z - x) / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.52e-135)
		tmp = fma(Float64(y / t), Float64(z - x), x);
	else
		tmp = fma(Float64(Float64(z - x) / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.52e-135], N[(N[(y / t), $MachinePrecision] * N[(z - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5200000000000001e-135

   1. Initial program 89.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   if -1.5200000000000001e-135 < x

   1. Initial program 91.8%

      \[x + \frac{y \cdot \left(z - x\right)}{t} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. --lowering--.f64 98.1

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

   4. Applied egg-rr 98.1%

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

Alternative 6: 97.6% 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 91.1%

   \[x + \frac{y \cdot \left(z - x\right)}{t} \]
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 97.0

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

4. Applied egg-rr 97.0%

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

Alternative 7: 77.6% 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 91.1%

   \[x + \frac{y \cdot \left(z - x\right)}{t} \]
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 97.0

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

4. Applied egg-rr 97.0%

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

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 77.4%

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

Alternative 8: 37.6% 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 91.1%

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

3. Taylor expanded in y around 0

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

5. Simplified 39.4%

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

Developer Target 1: 90.4% 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 2024196 
(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)))