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

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 8

Speedup: 1.0×

• 21.1% 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: 41.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 10^{-291}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{elif}\;x + y \leq 10^{+235}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) 1e-291)
   (* (- 1.0 z) x)
   (if (<= (+ x y) 1e+235) (* (- y) z) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= 1e-291) {
		tmp = (1.0 - z) * x;
	} else if ((x + y) <= 1e+235) {
		tmp = -y * 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) <= 1d-291) then
        tmp = (1.0d0 - z) * x
    else if ((x + y) <= 1d+235) then
        tmp = -y * 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) <= 1e-291) {
		tmp = (1.0 - z) * x;
	} else if ((x + y) <= 1e+235) {
		tmp = -y * z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= 1e-291:
		tmp = (1.0 - z) * x
	elif (x + y) <= 1e+235:
		tmp = -y * z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= 1e-291)
		tmp = Float64(Float64(1.0 - z) * x);
	elseif (Float64(x + y) <= 1e+235)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x + y) <= 1e-291)
		tmp = (1.0 - z) * x;
	elseif ((x + y) <= 1e+235)
		tmp = -y * z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], 1e-291], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e+235], N[((-y) * z), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < 9.99999999999999962e-292

   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 51.6

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

   5. Applied rewrites 51.6%

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

   if 9.99999999999999962e-292 < (+.f64 x y) < 1.0000000000000001e235

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 29.5%

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

   if 1.0000000000000001e235 < (+.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 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 38.6

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

   5. Applied rewrites 38.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 34.9

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

   7. Applied rewrites 34.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 62.5

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

   10. Applied rewrites 62.5%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 10^{-291}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{elif}\;x + y \leq 10^{+235}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -40:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{elif}\;1 - z \leq 10000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- 1.0 z) -40.0)
   (* (- y) z)
   (if (<= (- 1.0 z) 10000000000.0) (+ x y) (* (- x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - z) <= -40.0) {
		tmp = -y * z;
	} else if ((1.0 - z) <= 10000000000.0) {
		tmp = x + y;
	} else {
		tmp = -x * 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 ((1.0d0 - z) <= (-40.0d0)) then
        tmp = -y * z
    else if ((1.0d0 - z) <= 10000000000.0d0) then
        tmp = x + y
    else
        tmp = -x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 - z) <= -40.0) {
		tmp = -y * z;
	} else if ((1.0 - z) <= 10000000000.0) {
		tmp = x + y;
	} else {
		tmp = -x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (1.0 - z) <= -40.0:
		tmp = -y * z
	elif (1.0 - z) <= 10000000000.0:
		tmp = x + y
	else:
		tmp = -x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 - z) <= -40.0)
		tmp = Float64(Float64(-y) * z);
	elseif (Float64(1.0 - z) <= 10000000000.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(-x) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 - z) <= -40.0)
		tmp = -y * z;
	elseif ((1.0 - z) <= 10000000000.0)
		tmp = x + y;
	else
		tmp = -x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 - z), $MachinePrecision], -40.0], N[((-y) * z), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], 10000000000.0], N[(x + y), $MachinePrecision], N[((-x) * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -40

   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. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 50.9%

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

   if -40 < (-.f64 #s(literal 1 binary64) z) < 1e10

   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 4.1

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

   5. Applied rewrites 4.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 4.1

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.8

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

   10. Applied rewrites 96.8%

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

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

   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 51.2

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

   5. Applied rewrites 51.2%

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

   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 51.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -40:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{elif}\;1 - z \leq 10000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) \cdot z\\ \mathbf{if}\;1 - z \leq -40:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;1 - z \leq 10000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x) z)))
   (if (<= (- 1.0 z) -40.0)
     t_0
     (if (<= (- 1.0 z) 10000000000.0) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -x * z;
	double tmp;
	if ((1.0 - z) <= -40.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 10000000000.0) {
		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 = -x * z
    if ((1.0d0 - z) <= (-40.0d0)) then
        tmp = t_0
    else if ((1.0d0 - z) <= 10000000000.0d0) 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 = -x * z;
	double tmp;
	if ((1.0 - z) <= -40.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 10000000000.0) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x * z
	tmp = 0
	if (1.0 - z) <= -40.0:
		tmp = t_0
	elif (1.0 - z) <= 10000000000.0:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) * z)
	tmp = 0.0
	if (Float64(1.0 - z) <= -40.0)
		tmp = t_0;
	elseif (Float64(1.0 - z) <= 10000000000.0)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x * z;
	tmp = 0.0;
	if ((1.0 - z) <= -40.0)
		tmp = t_0;
	elseif ((1.0 - z) <= 10000000000.0)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-x) * z), $MachinePrecision]}, If[LessEqual[N[(1.0 - z), $MachinePrecision], -40.0], t$95$0, If[LessEqual[N[(1.0 - z), $MachinePrecision], 10000000000.0], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) z) < -40 or 1e10 < (-.f64 #s(literal 1 binary64) z)

   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 50.7

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

   5. Applied rewrites 50.7%

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

   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 49.8%

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

   if -40 < (-.f64 #s(literal 1 binary64) z) < 1e10

   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 4.1

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

   5. Applied rewrites 4.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 4.1

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 96.8

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

   10. Applied rewrites 96.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -40:\\ \;\;\;\;\left(-x\right) \cdot z\\ \mathbf{elif}\;1 - z \leq 10000000000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-279}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -5e-279) (* (- 1.0 z) x) (fma (- z) y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -5e-279) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = fma(-z, y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + y) <= -5e-279)
		tmp = Float64(Float64(1.0 - z) * x);
	else
		tmp = fma(Float64(-z), y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-279], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision], N[((-z) * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -4.99999999999999969e-279

   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 51.9

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

   5. Applied rewrites 51.9%

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

   if -4.99999999999999969e-279 < (+.f64 x y)

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(1 - z\right) \]
   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. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 50.3

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

   7. Applied rewrites 50.3%

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

   9. Applied rewrites 50.3%

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

   11. Applied rewrites 50.3%

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

Alternative 6: 51.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-279}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x y) -5e-279) (* (- 1.0 z) x) (* (- 1.0 z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + y) <= -5e-279) {
		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) <= (-5d-279)) 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) <= -5e-279) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = (1.0 - z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x + y) <= -5e-279:
		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) <= -5e-279)
		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) <= -5e-279)
		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], -5e-279], 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) < -4.99999999999999969e-279

   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 51.9

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

   5. Applied rewrites 51.9%

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

   if -4.99999999999999969e-279 < (+.f64 x y)

   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 50.3

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

   5. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\left(1 - z\right) \cdot y} \]
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: 49.7% 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 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 55.2

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

5. Applied rewrites 55.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 54.8

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

7. Applied rewrites 54.8%

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

8. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 45.7

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

10. Applied rewrites 45.7%

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

11. Final simplification 45.7%

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

Reproduce

herbie shell --seed 2024308 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))