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

Percentage Accurate: 100.0% → 100.0%

Time: 10.0s

Alternatives: 9

Speedup: 1.0×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(1 - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (- 1.0 z)))

C

double code(double x, double y, double z) {
	return (x + y) * (1.0 - z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) * (1.0d0 - z)
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) * (1.0 - z);
}

Python

def code(x, y, z):
	return (x + y) * (1.0 - z)

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(1.0 - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) * (1.0 - z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(1 - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (- 1.0 z)))

C

double code(double x, double y, double z) {
	return (x + y) * (1.0 - z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) * (1.0d0 - z)
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) * (1.0 - z);
}

Python

def code(x, y, z):
	return (x + y) * (1.0 - z)

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(1.0 - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) * (1.0 - z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0 - z, x + y, x + y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- 0.0 z) (+ x y) (+ x y)))

C

double code(double x, double y, double z) {
	return fma((0.0 - z), (x + y), (x + y));
}

Julia

function code(x, y, z)
	return fma(Float64(0.0 - z), Float64(x + y), Float64(x + y))
end

Wolfram

code[x_, y_, z_] := N[(N[(0.0 - z), $MachinePrecision] * N[(x + y), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(1 - z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

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

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

   6. neg-sub0 N/A

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

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

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

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

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

   9. +-lowering-+.f64 100.0

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - z, x + y, x + y\right)} \]
5. Add Preprocessing

Alternative 2: 51.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-269}:\\ \;\;\;\;\mathsf{fma}\left(0 - z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0 - z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-269) (fma (- 0.0 z) x x) (fma (- 0.0 z) y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-269) {
		tmp = fma((0.0 - z), x, x);
	} else {
		tmp = fma((0.0 - z), y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-269)
		tmp = fma(Float64(0.0 - z), x, x);
	else
		tmp = fma(Float64(0.0 - z), y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-269], N[(N[(0.0 - z), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(0.0 - z), $MachinePrecision] * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.9999999999999996e-270

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

      9. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 70.9%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 53.3%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 53.3

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

   12. Applied egg-rr 53.3%

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

   if -9.9999999999999996e-270 < (+.f64 x y)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

      9. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 68.1%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 42.2%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 42.2

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

   12. Applied egg-rr 42.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-269}:\\ \;\;\;\;\mathsf{fma}\left(0 - z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0 - z, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-269}:\\ \;\;\;\;\mathsf{fma}\left(0 - z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-269) (fma (- 0.0 z) x x) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-269) {
		tmp = fma((0.0 - z), x, x);
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-269)
		tmp = fma(Float64(0.0 - z), x, x);
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-269], N[(N[(0.0 - z), $MachinePrecision] * x + x), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.9999999999999996e-270

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

      9. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 70.9%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 53.3%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 53.3

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

   12. Applied egg-rr 53.3%

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

   if -9.9999999999999996e-270 < (+.f64 x y)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} \cdot \left(1 - z\right) \]
   4. Step-by-step derivation

   5. Simplified 42.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-269}:\\ \;\;\;\;\mathsf{fma}\left(0 - z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-269}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-269) (* x (- 1.0 z)) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-269) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x + y) <= (-1d-269)) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-269) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -1e-269:
		tmp = x * (1.0 - z)
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-269)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= -1e-269)
		tmp = x * (1.0 - z);
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-269], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -9.9999999999999996e-270

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \cdot \left(1 - z\right) \]
   4. Step-by-step derivation

   5. Simplified 53.3%

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

   if -9.9999999999999996e-270 < (+.f64 x y)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y} \cdot \left(1 - z\right) \]
   4. Step-by-step derivation

   5. Simplified 42.2%

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

Alternative 5: 51.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 5 \cdot 10^{-290}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) 5e-290) (* x (- 1.0 z)) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= 5e-290) {
		tmp = x * (1.0 - z);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x + y) <= 5d-290) then
        tmp = x * (1.0d0 - z)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= 5e-290) {
		tmp = x * (1.0 - z);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= 5e-290:
		tmp = x * (1.0 - z)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= 5e-290)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= 5e-290)
		tmp = x * (1.0 - z);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], 5e-290], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 5.0000000000000001e-290

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \cdot \left(1 - z\right) \]
   4. Step-by-step derivation

   5. Simplified 53.0%

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

   if 5.0000000000000001e-290 < (+.f64 x y)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 51.0

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

   5. Simplified 51.0%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 5 \cdot 10^{-290}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 26.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x + y\right) \cdot \left(1 - z\right) \leq -1 \cdot 10^{-269}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (+ x y) (- 1.0 z)) -1e-269) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + y) * (1.0 - z)) <= -1e-269) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x + y) * (1.0d0 - z)) <= (-1d-269)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x + y) * (1.0 - z)) <= -1e-269) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x + y) * (1.0 - z)) <= -1e-269:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + y) * Float64(1.0 - z)) <= -1e-269)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x + y) * (1.0 - z)) <= -1e-269)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], -1e-269], x, y]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) z)) < -9.9999999999999996e-270

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 57.5

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

   5. Simplified 57.5%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 27.9%

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

   if -9.9999999999999996e-270 < (*.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) z))

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 49.3

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

   5. Simplified 49.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 19.3%

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

Alternative 7: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \left(1 - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (- 1.0 z)))

C

double code(double x, double y, double z) {
	return (x + y) * (1.0 - z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) * (1.0d0 - z)
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) * (1.0 - z);
}

Python

def code(x, y, z):
	return (x + y) * (1.0 - z)

Julia

function code(x, y, z)
	return Float64(Float64(x + y) * Float64(1.0 - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) * (1.0 - z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(1 - z\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 8: 51.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x + y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x y))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x + y

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(1 - z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 53.5

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

5. Simplified 53.5%

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

6. Final simplification 53.5%

   \[\leadsto x + y \]
7. Add Preprocessing

Alternative 9: 26.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[\left(x + y\right) \cdot \left(1 - z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 53.5

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

5. Simplified 53.5%

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

6. Taylor expanded in y around 0

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

8. Simplified 30.2%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))