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

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

Alternatives: 7

Speedup: 1.0×

• 20.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot \left(y + x\right)\\ \mathbf{if}\;1 - z \leq -10000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - z \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) (+ y x))))
   (if (<= (- 1.0 z) -10000000000.0) t_0 (if (<= (- 1.0 z) 2.0) (+ y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * (y + x);
	double tmp;
	if ((1.0 - z) <= -10000000000.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 2.0) {
		tmp = y + x;
	} 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 * (y + x)
    if ((1.0d0 - z) <= (-10000000000.0d0)) then
        tmp = t_0
    else if ((1.0d0 - z) <= 2.0d0) then
        tmp = y + x
    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 * (y + x);
	double tmp;
	if ((1.0 - z) <= -10000000000.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 2.0) {
		tmp = y + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * (y + x)
	tmp = 0
	if (1.0 - z) <= -10000000000.0:
		tmp = t_0
	elif (1.0 - z) <= 2.0:
		tmp = y + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * Float64(y + x))
	tmp = 0.0
	if (Float64(1.0 - z) <= -10000000000.0)
		tmp = t_0;
	elseif (Float64(1.0 - z) <= 2.0)
		tmp = Float64(y + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * (y + x);
	tmp = 0.0;
	if ((1.0 - z) <= -10000000000.0)
		tmp = t_0;
	elseif ((1.0 - z) <= 2.0)
		tmp = y + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - z), $MachinePrecision], -10000000000.0], t$95$0, If[LessEqual[N[(1.0 - z), $MachinePrecision], 2.0], N[(y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -1e10 or 2 < (-.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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -1e10 < (-.f64 #s(literal 1 binary64) z) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   5. Applied rewrites 3.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -10000000000:\\ \;\;\;\;\left(-z\right) \cdot \left(y + x\right)\\ \mathbf{elif}\;1 - z \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) \cdot x\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+193}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 z) x)))
   (if (<= z -1.2e-6)
     t_0
     (if (<= z 1.25e-5) (+ y x) (if (<= z 5.1e+193) t_0 (* (- z) y))))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - z) * x;
	double tmp;
	if (z <= -1.2e-6) {
		tmp = t_0;
	} else if (z <= 1.25e-5) {
		tmp = y + x;
	} else if (z <= 5.1e+193) {
		tmp = t_0;
	} else {
		tmp = -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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - z) * x
    if (z <= (-1.2d-6)) then
        tmp = t_0
    else if (z <= 1.25d-5) then
        tmp = y + x
    else if (z <= 5.1d+193) then
        tmp = t_0
    else
        tmp = -z * y
    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 <= -1.2e-6) {
		tmp = t_0;
	} else if (z <= 1.25e-5) {
		tmp = y + x;
	} else if (z <= 5.1e+193) {
		tmp = t_0;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 - z) * x
	tmp = 0
	if z <= -1.2e-6:
		tmp = t_0
	elif z <= 1.25e-5:
		tmp = y + x
	elif z <= 5.1e+193:
		tmp = t_0
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - z) * x)
	tmp = 0.0
	if (z <= -1.2e-6)
		tmp = t_0;
	elseif (z <= 1.25e-5)
		tmp = Float64(y + x);
	elseif (z <= 5.1e+193)
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - z) * x;
	tmp = 0.0;
	if (z <= -1.2e-6)
		tmp = t_0;
	elseif (z <= 1.25e-5)
		tmp = y + x;
	elseif (z <= 5.1e+193)
		tmp = t_0;
	else
		tmp = -z * y;
	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, -1.2e-6], t$95$0, If[LessEqual[z, 1.25e-5], N[(y + x), $MachinePrecision], If[LessEqual[z, 5.1e+193], t$95$0, N[((-z) * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1999999999999999e-6 or 1.25000000000000006e-5 < z < 5.09999999999999999e193

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      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 55.1

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

   5. Applied rewrites 55.1%

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

   if -1.1999999999999999e-6 < z < 1.25000000000000006e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   5. Applied rewrites 3.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 99.8

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

   10. Applied rewrites 99.8%

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

   if 5.09999999999999999e193 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      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 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+193}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;z \leq -12.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+193}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) x)))
   (if (<= z -12.5)
     t_0
     (if (<= z 1.0) (+ y x) (if (<= z 5.1e+193) t_0 (* (- z) y))))))

C

double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (z <= -12.5) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = y + x;
	} else if (z <= 5.1e+193) {
		tmp = t_0;
	} else {
		tmp = -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) :: t_0
    real(8) :: tmp
    t_0 = -z * x
    if (z <= (-12.5d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = y + x
    else if (z <= 5.1d+193) then
        tmp = t_0
    else
        tmp = -z * y
    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 <= -12.5) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = y + x;
	} else if (z <= 5.1e+193) {
		tmp = t_0;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * x
	tmp = 0
	if z <= -12.5:
		tmp = t_0
	elif z <= 1.0:
		tmp = y + x
	elif z <= 5.1e+193:
		tmp = t_0
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * x)
	tmp = 0.0
	if (z <= -12.5)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(y + x);
	elseif (z <= 5.1e+193)
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * x;
	tmp = 0.0;
	if (z <= -12.5)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = y + x;
	elseif (z <= 5.1e+193)
		tmp = t_0;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * x), $MachinePrecision]}, If[LessEqual[z, -12.5], t$95$0, If[LessEqual[z, 1.0], N[(y + x), $MachinePrecision], If[LessEqual[z, 5.1e+193], t$95$0, N[((-z) * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -12.5 or 1 < z < 5.09999999999999999e193

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      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 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.4%

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

   if -12.5 < 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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   5. Applied rewrites 3.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 99.8

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

   10. Applied rewrites 99.8%

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

   if 5.09999999999999999e193 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      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 49.4

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -12.5:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+193}:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;z \leq -12.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) x))) (if (<= z -12.5) t_0 (if (<= z 1.0) (+ y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (z <= -12.5) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = y + x;
	} 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 <= (-12.5d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = y + x
    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 <= -12.5) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = y + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * x
	tmp = 0
	if z <= -12.5:
		tmp = t_0
	elif z <= 1.0:
		tmp = y + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * x)
	tmp = 0.0
	if (z <= -12.5)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(y + x);
	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 <= -12.5)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = y + x;
	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, -12.5], t$95$0, If[LessEqual[z, 1.0], N[(y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -12.5 or 1 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      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 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 55.8%

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

   if -12.5 < 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 inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   5. Applied rewrites 3.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 3.2

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in z around 0

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

      1. lower-+.f64 99.8

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

   10. Applied rewrites 99.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -12.5:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ y x) -1e-283) (* (- 1.0 z) x) (* (- 1.0 z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y + x) <= -1e-283) {
		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 ((y + x) <= (-1d-283)) 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 ((y + x) <= -1e-283) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = (1.0 - z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y + x) <= -1e-283:
		tmp = (1.0 - z) * x
	else:
		tmp = (1.0 - z) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y + x) <= -1e-283)
		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 ((y + x) <= -1e-283)
		tmp = (1.0 - z) * x;
	else
		tmp = (1.0 - z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y + x), $MachinePrecision], -1e-283], 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) < -9.99999999999999947e-284

   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

      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 54.5

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

   5. Applied rewrites 54.5%

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

   if -9.99999999999999947e-284 < (+.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

      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 50.7

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

   5. Applied rewrites 50.7%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return y + 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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 45.9

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

5. Applied rewrites 45.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 44.0

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

7. Applied rewrites 44.0%

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

8. Taylor expanded in z around 0

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

   1. lower-+.f64 56.8

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

10. Applied rewrites 56.8%

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

11. Final simplification 56.8%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))