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

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 8

Speedup: 1.0×

• 20.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] * (-z) + 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. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-neg.f64 100.0

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) \cdot x\\ \mathbf{if}\;z \leq -300:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-15}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+250}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 z) x)))
   (if (<= z -300.0)
     t_0
     (if (<= z 1.52e-15)
       (+ x y)
       (if (<= z 9.5e+159) t_0 (if (<= z 1.25e+250) (* (- z) y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - z) * x;
	double tmp;
	if (z <= -300.0) {
		tmp = t_0;
	} else if (z <= 1.52e-15) {
		tmp = x + y;
	} else if (z <= 9.5e+159) {
		tmp = t_0;
	} else if (z <= 1.25e+250) {
		tmp = -z * y;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - z) * x
    if (z <= (-300.0d0)) then
        tmp = t_0
    else if (z <= 1.52d-15) then
        tmp = x + y
    else if (z <= 9.5d+159) then
        tmp = t_0
    else if (z <= 1.25d+250) then
        tmp = -z * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - z) * x;
	double tmp;
	if (z <= -300.0) {
		tmp = t_0;
	} else if (z <= 1.52e-15) {
		tmp = x + y;
	} else if (z <= 9.5e+159) {
		tmp = t_0;
	} else if (z <= 1.25e+250) {
		tmp = -z * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 - z) * x
	tmp = 0
	if z <= -300.0:
		tmp = t_0
	elif z <= 1.52e-15:
		tmp = x + y
	elif z <= 9.5e+159:
		tmp = t_0
	elif z <= 1.25e+250:
		tmp = -z * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - z) * x)
	tmp = 0.0
	if (z <= -300.0)
		tmp = t_0;
	elseif (z <= 1.52e-15)
		tmp = Float64(x + y);
	elseif (z <= 9.5e+159)
		tmp = t_0;
	elseif (z <= 1.25e+250)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - z) * x;
	tmp = 0.0;
	if (z <= -300.0)
		tmp = t_0;
	elseif (z <= 1.52e-15)
		tmp = x + y;
	elseif (z <= 9.5e+159)
		tmp = t_0;
	elseif (z <= 1.25e+250)
		tmp = -z * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -300.0], t$95$0, If[LessEqual[z, 1.52e-15], N[(x + y), $MachinePrecision], If[LessEqual[z, 9.5e+159], t$95$0, If[LessEqual[z, 1.25e+250], N[((-z) * y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -300 or 1.52000000000000005e-15 < z < 9.5000000000000003e159 or 1.2500000000000001e250 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 55.4

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

   5. Applied rewrites 55.4%

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

   if -300 < z < 1.52000000000000005e-15

   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. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if 9.5000000000000003e159 < z < 1.2500000000000001e250

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 47.5%

      \[\leadsto \left(-z\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -300:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-15}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+250}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;z \leq -300:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+250}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) x)))
   (if (<= z -300.0)
     t_0
     (if (<= z 1.0)
       (+ x y)
       (if (<= z 9.5e+159) t_0 (if (<= z 1.25e+250) (* (- z) y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (z <= -300.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 9.5e+159) {
		tmp = t_0;
	} else if (z <= 1.25e+250) {
		tmp = -z * y;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -z * x
    if (z <= (-300.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = x + y
    else if (z <= 9.5d+159) then
        tmp = t_0
    else if (z <= 1.25d+250) then
        tmp = -z * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (z <= -300.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if (z <= 9.5e+159) {
		tmp = t_0;
	} else if (z <= 1.25e+250) {
		tmp = -z * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * x
	tmp = 0
	if z <= -300.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = x + y
	elif z <= 9.5e+159:
		tmp = t_0
	elif z <= 1.25e+250:
		tmp = -z * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * x)
	tmp = 0.0
	if (z <= -300.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	elseif (z <= 9.5e+159)
		tmp = t_0;
	elseif (z <= 1.25e+250)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * x;
	tmp = 0.0;
	if (z <= -300.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x + y;
	elseif (z <= 9.5e+159)
		tmp = t_0;
	elseif (z <= 1.25e+250)
		tmp = -z * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * x), $MachinePrecision]}, If[LessEqual[z, -300.0], t$95$0, If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 9.5e+159], t$95$0, If[LessEqual[z, 1.25e+250], N[((-z) * y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -300 or 1 < z < 9.5000000000000003e159 or 1.2500000000000001e250 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto \left(-z\right) \cdot x \]

   if -300 < z < 1

   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. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 9.5000000000000003e159 < z < 1.2500000000000001e250

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.5

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot z\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 47.5%

      \[\leadsto \left(-z\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -300:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+250}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;z \leq -300:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) x))) (if (<= z -300.0) t_0 (if (<= z 1.0) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (z <= -300.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -z * x
    if (z <= (-300.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (z <= -300.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * x
	tmp = 0
	if z <= -300.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * x)
	tmp = 0.0
	if (z <= -300.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * x;
	tmp = 0.0;
	if (z <= -300.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * x), $MachinePrecision]}, If[LessEqual[z, -300.0], t$95$0, If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -300 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot z\right) \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto \left(-z\right) \cdot x \]

   if -300 < z < 1

   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. lower-+.f64 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -300:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(-z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-294) (fma (- z) x x) (* (- 1.0 z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-294) {
		tmp = fma(-z, x, x);
	} else {
		tmp = (1.0 - z) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-294)
		tmp = fma(Float64(-z), x, x);
	else
		tmp = Float64(Float64(1.0 - z) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-294], N[((-z) * x + x), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.00000000000000002e-294

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 54.8%

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

   if -1.00000000000000002e-294 < (+.f64 x y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 49.1

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

   5. Applied rewrites 49.1%

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

Alternative 6: 51.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -1e-294) (* (- 1.0 z) x) (* (- 1.0 z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-294) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = (1.0 - z) * 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) <= (-1d-294)) then
        tmp = (1.0d0 - z) * x
    else
        tmp = (1.0d0 - z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -1e-294) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = (1.0 - z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -1e-294:
		tmp = (1.0 - z) * x
	else:
		tmp = (1.0 - z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -1e-294)
		tmp = Float64(Float64(1.0 - z) * x);
	else
		tmp = Float64(Float64(1.0 - z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= -1e-294)
		tmp = (1.0 - z) * x;
	else
		tmp = (1.0 - z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-294], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.00000000000000002e-294

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 54.8

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

   5. Applied rewrites 54.8%

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

   if -1.00000000000000002e-294 < (+.f64 x y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 49.1

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

   5. Applied rewrites 49.1%

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

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 - z) * (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 = (1.0d0 - z) * (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(1.0 - z), $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. Final simplification 100.0%

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

Alternative 8: 51.1% 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. lower-+.f64 50.3

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

5. Applied rewrites 50.3%

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

6. Final simplification 50.3%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))