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

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 8

Speedup: 1.0×

• 20.5% 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

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

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. Final simplification 100.0%

   \[\leadsto \left(1 - z\right) \cdot \left(x + y\right) \]

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -20 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 z) -20.0) (not (<= (- 1.0 z) 2.0)))
   (* z (- (- y) x))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) <= -20.0) || !((1.0 - z) <= 2.0)) {
		tmp = z * (-y - x);
	} 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 (((1.0d0 - z) <= (-20.0d0)) .or. (.not. ((1.0d0 - z) <= 2.0d0))) then
        tmp = z * (-y - x)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) <= -20.0) || !((1.0 - z) <= 2.0)) {
		tmp = z * (-y - x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((1.0 - z) <= -20.0) or not ((1.0 - z) <= 2.0):
		tmp = z * (-y - x)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(1.0 - z) <= -20.0) || !(Float64(1.0 - z) <= 2.0))
		tmp = Float64(z * Float64(Float64(-y) - x));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((1.0 - z) <= -20.0) || ~(((1.0 - z) <= 2.0)))
		tmp = z * (-y - x);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(1.0 - z), $MachinePrecision], -20.0], N[Not[LessEqual[N[(1.0 - z), $MachinePrecision], 2.0]], $MachinePrecision]], N[(z * N[((-y) - x), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 z) < -20 or 2 < (-.f64 1 z)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]

   2. Taylor expanded in z around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. +-commutative 99.1%

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

      3. distribute-rgt-neg-out 99.1%

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

      4. +-commutative 99.1%

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

   4. Simplified 99.1%

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

   if -20 < (-.f64 1 z) < 2

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]

   2. Taylor expanded in z around 0 97.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -20 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-y\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - z \leq 0.9999999999995 \lor \neg \left(1 - z \leq 10^{+18}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 z) 0.9999999999995) (not (<= (- 1.0 z) 1e+18)))
   (* x (- 1.0 z))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) <= 0.9999999999995) || !((1.0 - z) <= 1e+18)) {
		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 (((1.0d0 - z) <= 0.9999999999995d0) .or. (.not. ((1.0d0 - z) <= 1d+18))) 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 (((1.0 - z) <= 0.9999999999995) || !((1.0 - z) <= 1e+18)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((1.0 - z) <= 0.9999999999995) or not ((1.0 - z) <= 1e+18):
		tmp = x * (1.0 - z)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(1.0 - z) <= 0.9999999999995) || !(Float64(1.0 - z) <= 1e+18))
		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 (((1.0 - z) <= 0.9999999999995) || ~(((1.0 - z) <= 1e+18)))
		tmp = x * (1.0 - z);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(1.0 - z), $MachinePrecision], 0.9999999999995], N[Not[LessEqual[N[(1.0 - z), $MachinePrecision], 1e+18]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 z) < 0.99999999999949996 or 1e18 < (-.f64 1 z)

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]

   2. Taylor expanded in x around inf 54.4%

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

   if 0.99999999999949996 < (-.f64 1 z) < 1e18

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]

   2. Taylor expanded in z around 0 99.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq 0.9999999999995 \lor \neg \left(1 - z \leq 10^{+18}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+17} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.8e+17) (not (<= z 1.0))) (* x (- z)) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.8e+17) || !(z <= 1.0)) {
		tmp = x * -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 ((z <= (-9.8d+17)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * -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 ((z <= -9.8e+17) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.8e+17) or not (z <= 1.0):
		tmp = x * -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.8e+17) || !(z <= 1.0))
		tmp = Float64(x * Float64(-z));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9.8e+17) || ~((z <= 1.0)))
		tmp = x * -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.8e+17], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.8e17 or 1 < z

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-commutative 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. flip-+ 69.0%

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

      6. div-inv 69.0%

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

      7. fma-def 69.0%

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

   3. Applied egg-rr 69.0%

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

   4. Taylor expanded in x around inf 43.0%

      \[\leadsto \mathsf{fma}\left(x \cdot x - y \cdot y, \frac{1}{x - y}, \color{blue}{-1 \cdot \left(z \cdot x\right)}\right) \]
   5. Step-by-step derivation

      1. associate-*r* 43.0%

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

      2. mul-1-neg 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in z around inf 53.5%

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

      1. associate-*r* 53.5%

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

      2. mul-1-neg 53.5%

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

   9. Simplified 53.5%

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

   if -9.8e17 < z < 1

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]

   2. Taylor expanded in z around 0 96.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+17} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.8e-107) (* x (- 1.0 z)) (- y (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8e-107) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y - (y * 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 <= (-5.8d-107)) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8e-107) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.8e-107:
		tmp = x * (1.0 - z)
	else:
		tmp = y - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.8e-107)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.8e-107)
		tmp = x * (1.0 - z);
	else
		tmp = y - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.8e-107], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.7999999999999996e-107

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]

   2. Taylor expanded in x around inf 67.8%

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

   if -5.7999999999999996e-107 < x

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]

   2. Taylor expanded in x around 0 61.0%

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

      1. sub-neg 61.0%

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

      2. distribute-lft-in 61.0%

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

      3. distribute-rgt-neg-out 61.0%

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

      4. unsub-neg 61.0%

         \[\leadsto \color{blue}{y \cdot 1 - y \cdot z} \]

      5. *-rgt-identity 61.0%

         \[\leadsto \color{blue}{y} - y \cdot z \]

   4. Simplified 61.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot z\\ \end{array} \]

Alternative 6: 31.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 6e-57) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e-57) {
		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 (y <= 6d-57) 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 (y <= 6e-57) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e-57:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e-57)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e-57)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e-57], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000001e-57

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]

   2. Taylor expanded in x around inf 57.8%

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

   3. Taylor expanded in z around 0 31.9%

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

   if 6.00000000000000001e-57 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-commutative 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. flip-+ 60.6%

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

      6. div-inv 60.5%

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

      7. fma-def 60.5%

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

   3. Applied egg-rr 60.5%

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

   4. Taylor expanded in x around inf 45.7%

      \[\leadsto \mathsf{fma}\left(x \cdot x - y \cdot y, \frac{1}{x - y}, \color{blue}{-1 \cdot \left(z \cdot x\right)}\right) \]
   5. Step-by-step derivation

      1. associate-*r* 45.7%

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

      2. mul-1-neg 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in x around 0 35.2%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-57}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 50.2% accurate, 2.3× 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. Taylor expanded in z around 0 52.4%

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

3. Final simplification 52.4%

   \[\leadsto x + y \]

Alternative 8: 26.0% accurate, 7.0× 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. Taylor expanded in x around inf 52.6%

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

3. Taylor expanded in z around 0 27.0%

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

4. Final simplification 27.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023242 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))