Linear.Quaternion:$c/ from linear-1.19.1.3, A

Percentage Accurate: 98.3% → 99.3%

Time: 9.2s

Alternatives: 11

Speedup: 1.0×

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. +-commutative 98.8%

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

   2. fma-def 98.8%

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

   3. associate-+l+ 98.8%

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

   4. fma-def 99.2%

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

   5. distribute-lft-out 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x y (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	return fma(x, y, (z * (z * 3.0)));
}

Julia

function code(x, y, z)
	return fma(x, y, Float64(z * Float64(z * 3.0)))
end

Wolfram

code[x_, y_, z_] := N[(x * y + N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. associate-+l+ 98.8%

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

   2. associate-+l+ 98.8%

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

   3. fma-def 99.2%

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

   4. count-2 99.2%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{2 \cdot \left(z \cdot z\right)} + z \cdot z\right) \]

   5. distribute-lft1-in 99.2%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\left(2 + 1\right) \cdot \left(z \cdot z\right)}\right) \]

   6. metadata-eval 99.2%

      \[\leadsto \mathsf{fma}\left(x, y, \left(2 + \color{blue}{-1 \cdot -1}\right) \cdot \left(z \cdot z\right)\right) \]

   7. *-commutative 99.2%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\left(z \cdot z\right) \cdot \left(2 + -1 \cdot -1\right)}\right) \]

   8. associate-*l* 99.2%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(z \cdot \left(2 + -1 \cdot -1\right)\right)}\right) \]

   9. metadata-eval 99.2%

      \[\leadsto \mathsf{fma}\left(x, y, z \cdot \left(z \cdot \left(2 + \color{blue}{1}\right)\right)\right) \]

   10. metadata-eval 99.2%

       \[\leadsto \mathsf{fma}\left(x, y, z \cdot \left(z \cdot \color{blue}{3}\right)\right) \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)} \]

4. Final simplification 99.2%

   \[\leadsto \mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right) \]

Alternative 3: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-138} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-110}\right):\\ \;\;\;\;z \cdot z + \left(x \cdot y + z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -5e-138) (not (<= (* x y) 2e-110)))
   (+ (* z z) (+ (* x y) (* z z)))
   (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -5e-138) || !((x * y) <= 2e-110)) {
		tmp = (z * z) + ((x * y) + (z * z));
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x * y) <= (-5d-138)) .or. (.not. ((x * y) <= 2d-110))) then
        tmp = (z * z) + ((x * y) + (z * z))
    else
        tmp = 3.0d0 * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -5e-138) || !((x * y) <= 2e-110)) {
		tmp = (z * z) + ((x * y) + (z * z));
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * y) <= -5e-138) or not ((x * y) <= 2e-110):
		tmp = (z * z) + ((x * y) + (z * z))
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -5e-138) || !(Float64(x * y) <= 2e-110))
		tmp = Float64(Float64(z * z) + Float64(Float64(x * y) + Float64(z * z)));
	else
		tmp = Float64(3.0 * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * y) <= -5e-138) || ~(((x * y) <= 2e-110)))
		tmp = (z * z) + ((x * y) + (z * z));
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-138], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e-110]], $MachinePrecision]], N[(N[(z * z), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999989e-138 or 2.0000000000000001e-110 < (*.f64 x y)

   1. Initial program 98.4%

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

   3. Applied egg-rr 88.2%

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

   if -4.99999999999999989e-138 < (*.f64 x y) < 2.0000000000000001e-110

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

      2. associate-+r+ 99.9%

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

      3. distribute-lft1-in 99.9%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

      4. metadata-eval 99.9%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

      5. *-commutative 99.9%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

      6. fma-def 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

      7. unpow2 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

   5. Taylor expanded in z around inf 88.4%

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 88.4%

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   7. Simplified 88.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-138} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-110}\right):\\ \;\;\;\;z \cdot z + \left(x \cdot y + z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 4: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.3 \cdot 10^{-50} \lor \neg \left(z \cdot z \leq 2020000000000\right) \land z \cdot z \leq 2.35 \cdot 10^{+160}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 2.3e-50)
         (and (not (<= (* z z) 2020000000000.0)) (<= (* z z) 2.35e+160)))
   (* x y)
   (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 2.3e-50) || (!((z * z) <= 2020000000000.0) && ((z * z) <= 2.35e+160))) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z * 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 (((z * z) <= 2.3d-50) .or. (.not. ((z * z) <= 2020000000000.0d0)) .and. ((z * z) <= 2.35d+160)) then
        tmp = x * y
    else
        tmp = 3.0d0 * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 2.3e-50) || (!((z * z) <= 2020000000000.0) && ((z * z) <= 2.35e+160))) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 2.3e-50) or (not ((z * z) <= 2020000000000.0) and ((z * z) <= 2.35e+160)):
		tmp = x * y
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 2.3e-50) || (!(Float64(z * z) <= 2020000000000.0) && (Float64(z * z) <= 2.35e+160)))
		tmp = Float64(x * y);
	else
		tmp = Float64(3.0 * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z * z) <= 2.3e-50) || (~(((z * z) <= 2020000000000.0)) && ((z * z) <= 2.35e+160)))
		tmp = x * y;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 2.3e-50], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 2020000000000.0]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 2.35e+160]]], N[(x * y), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.3000000000000002e-50 or 2.02e12 < (*.f64 z z) < 2.34999999999999985e160

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

      2. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

      3. distribute-lft1-in 100.0%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

      5. *-commutative 100.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

      6. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

      7. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

   5. Taylor expanded in z around 0 85.4%

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

   if 2.3000000000000002e-50 < (*.f64 z z) < 2.02e12 or 2.34999999999999985e160 < (*.f64 z z)

   1. Initial program 97.3%

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

   2. Taylor expanded in x around 0 97.3%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 97.3%

         \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

      2. associate-+r+ 97.3%

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

      3. distribute-lft1-in 97.3%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

      4. metadata-eval 97.3%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

      5. *-commutative 97.3%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

      6. fma-def 97.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

      7. unpow2 97.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

   4. Simplified 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

   5. Taylor expanded in z around inf 88.0%

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 88.0%

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   7. Simplified 88.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.3 \cdot 10^{-50} \lor \neg \left(z \cdot z \leq 2020000000000\right) \land z \cdot z \leq 2.35 \cdot 10^{+160}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 5: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-57}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 1000000000:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+159}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-57)
   (* x y)
   (if (<= (* z z) 1000000000.0)
     (* z (* z 3.0))
     (if (<= (* z z) 1e+159) (* x y) (* 3.0 (* z z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-57) {
		tmp = x * y;
	} else if ((z * z) <= 1000000000.0) {
		tmp = z * (z * 3.0);
	} else if ((z * z) <= 1e+159) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z * 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 ((z * z) <= 2d-57) then
        tmp = x * y
    else if ((z * z) <= 1000000000.0d0) then
        tmp = z * (z * 3.0d0)
    else if ((z * z) <= 1d+159) then
        tmp = x * y
    else
        tmp = 3.0d0 * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-57) {
		tmp = x * y;
	} else if ((z * z) <= 1000000000.0) {
		tmp = z * (z * 3.0);
	} else if ((z * z) <= 1e+159) {
		tmp = x * y;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-57:
		tmp = x * y
	elif (z * z) <= 1000000000.0:
		tmp = z * (z * 3.0)
	elif (z * z) <= 1e+159:
		tmp = x * y
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-57)
		tmp = Float64(x * y);
	elseif (Float64(z * z) <= 1000000000.0)
		tmp = Float64(z * Float64(z * 3.0));
	elseif (Float64(z * z) <= 1e+159)
		tmp = Float64(x * y);
	else
		tmp = Float64(3.0 * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-57)
		tmp = x * y;
	elseif ((z * z) <= 1000000000.0)
		tmp = z * (z * 3.0);
	elseif ((z * z) <= 1e+159)
		tmp = x * y;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-57], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1000000000.0], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+159], N[(x * y), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.99999999999999991e-57 or 1e9 < (*.f64 z z) < 9.9999999999999993e158

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

      2. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

      3. distribute-lft1-in 100.0%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

      5. *-commutative 100.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

      6. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

      7. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

   5. Taylor expanded in z around 0 85.4%

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

   if 1.99999999999999991e-57 < (*.f64 z z) < 1e9

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 74.1%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   3. Step-by-step derivation

      1. distribute-lft1-in 74.1%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval 74.1%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative 74.1%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. unpow2 74.1%

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      5. associate-*r* 74.2%

         \[\leadsto \color{blue}{z \cdot \left(z \cdot 3\right)} \]

   4. Simplified 74.2%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot 3\right)} \]

   if 9.9999999999999993e158 < (*.f64 z z)

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 97.0%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 97.0%

         \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

      2. associate-+r+ 97.0%

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

      3. distribute-lft1-in 97.0%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

      4. metadata-eval 97.0%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

      5. *-commutative 97.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

      6. fma-def 97.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

      7. unpow2 97.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

   4. Simplified 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

   5. Taylor expanded in z around inf 90.0%

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 90.0%

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   7. Simplified 90.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-57}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 1000000000:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+159}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 6: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-57}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 1000000000:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+159}:\\ \;\;\;\;x \cdot y + 0.75\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-57)
   (* x y)
   (if (<= (* z z) 1000000000.0)
     (* z (* z 3.0))
     (if (<= (* z z) 1e+159) (+ (* x y) 0.75) (* 3.0 (* z z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-57) {
		tmp = x * y;
	} else if ((z * z) <= 1000000000.0) {
		tmp = z * (z * 3.0);
	} else if ((z * z) <= 1e+159) {
		tmp = (x * y) + 0.75;
	} else {
		tmp = 3.0 * (z * 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 ((z * z) <= 2d-57) then
        tmp = x * y
    else if ((z * z) <= 1000000000.0d0) then
        tmp = z * (z * 3.0d0)
    else if ((z * z) <= 1d+159) then
        tmp = (x * y) + 0.75d0
    else
        tmp = 3.0d0 * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-57) {
		tmp = x * y;
	} else if ((z * z) <= 1000000000.0) {
		tmp = z * (z * 3.0);
	} else if ((z * z) <= 1e+159) {
		tmp = (x * y) + 0.75;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-57:
		tmp = x * y
	elif (z * z) <= 1000000000.0:
		tmp = z * (z * 3.0)
	elif (z * z) <= 1e+159:
		tmp = (x * y) + 0.75
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-57)
		tmp = Float64(x * y);
	elseif (Float64(z * z) <= 1000000000.0)
		tmp = Float64(z * Float64(z * 3.0));
	elseif (Float64(z * z) <= 1e+159)
		tmp = Float64(Float64(x * y) + 0.75);
	else
		tmp = Float64(3.0 * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-57)
		tmp = x * y;
	elseif ((z * z) <= 1000000000.0)
		tmp = z * (z * 3.0);
	elseif ((z * z) <= 1e+159)
		tmp = (x * y) + 0.75;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-57], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1000000000.0], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e+159], N[(N[(x * y), $MachinePrecision] + 0.75), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z z) < 1.99999999999999991e-57

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

      2. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

      3. distribute-lft1-in 100.0%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

      4. metadata-eval 100.0%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

      5. *-commutative 100.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

      6. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

      7. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

   5. Taylor expanded in z around 0 90.0%

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

   if 1.99999999999999991e-57 < (*.f64 z z) < 1e9

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 74.1%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   3. Step-by-step derivation

      1. distribute-lft1-in 74.1%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval 74.1%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative 74.1%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. unpow2 74.1%

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      5. associate-*r* 74.2%

         \[\leadsto \color{blue}{z \cdot \left(z \cdot 3\right)} \]

   4. Simplified 74.2%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot 3\right)} \]

   if 1e9 < (*.f64 z z) < 9.9999999999999993e158

   1. Initial program 99.8%

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

      1. associate-+l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. fma-def 99.9%

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

      4. count-2 99.9%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{2 \cdot \left(z \cdot z\right)} + z \cdot z\right) \]

      5. distribute-lft1-in 99.9%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\left(2 + 1\right) \cdot \left(z \cdot z\right)}\right) \]

      6. metadata-eval 99.9%

         \[\leadsto \mathsf{fma}\left(x, y, \left(2 + \color{blue}{-1 \cdot -1}\right) \cdot \left(z \cdot z\right)\right) \]

      7. *-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\left(z \cdot z\right) \cdot \left(2 + -1 \cdot -1\right)}\right) \]

      8. associate-*l* 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{z \cdot \left(z \cdot \left(2 + -1 \cdot -1\right)\right)}\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(x, y, z \cdot \left(z \cdot \left(2 + \color{blue}{1}\right)\right)\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(x, y, z \cdot \left(z \cdot \color{blue}{3}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)} \]

   5. Applied egg-rr 64.7%

      \[\leadsto \color{blue}{x \cdot y + 0.75} \]

   if 9.9999999999999993e158 < (*.f64 z z)

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 97.0%

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 97.0%

         \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

      2. associate-+r+ 97.0%

         \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

      3. distribute-lft1-in 97.0%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

      4. metadata-eval 97.0%

         \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

      5. *-commutative 97.0%

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

      6. fma-def 97.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

      7. unpow2 97.0%

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

   4. Simplified 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

   5. Taylor expanded in z around inf 90.0%

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 90.0%

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   7. Simplified 90.0%

      \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-57}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 1000000000:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{elif}\;z \cdot z \leq 10^{+159}:\\ \;\;\;\;x \cdot y + 0.75\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

2. Final simplification 98.8%

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

Alternative 8: 52.2% accurate, 5.0× speedup

Math

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

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 98.8%

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

2. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
3. Step-by-step derivation

   1. +-commutative 98.8%

      \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

   2. associate-+r+ 98.8%

      \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

   3. distribute-lft1-in 98.8%

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

   4. metadata-eval 98.8%

      \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

   5. *-commutative 98.8%

      \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

   6. fma-def 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

   7. unpow2 98.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

4. Simplified 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

5. Taylor expanded in z around 0 55.8%

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

6. Final simplification 55.8%

   \[\leadsto x \cdot y \]

Alternative 9: 2.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -4.0)

C

double code(double x, double y, double z) {
	return -4.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -4.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -4.0;
}

Python

def code(x, y, z):
	return -4.0

Julia

function code(x, y, z)
	return -4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -4.0;
end

Wolfram

code[x_, y_, z_] := -4.0

Derivation

1. Initial program 98.8%

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

2. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
3. Step-by-step derivation

   1. +-commutative 98.8%

      \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

   2. associate-+r+ 98.8%

      \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

   3. distribute-lft1-in 98.8%

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

   4. metadata-eval 98.8%

      \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

   5. *-commutative 98.8%

      \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

   6. fma-def 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

   7. unpow2 98.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

4. Simplified 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

6. Applied egg-rr 1.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, -2\right) + \mathsf{fma}\left(-2, x \cdot y, \mathsf{fma}\left(x, y, -2\right)\right)} \]
7. Step-by-step derivation

   1. fma-udef 1.9%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(-2 \cdot \left(x \cdot y\right) + \mathsf{fma}\left(x, y, -2\right)\right)} \]

   2. *-commutative 1.9%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \left(\color{blue}{\left(x \cdot y\right) \cdot -2} + \mathsf{fma}\left(x, y, -2\right)\right) \]

   3. associate-*r* 1.9%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \left(\color{blue}{x \cdot \left(y \cdot -2\right)} + \mathsf{fma}\left(x, y, -2\right)\right) \]

   4. fma-def 1.9%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \left(x \cdot \left(y \cdot -2\right) + \color{blue}{\left(x \cdot y + -2\right)}\right) \]

   5. associate-+r+ 1.9%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(\left(x \cdot \left(y \cdot -2\right) + x \cdot y\right) + -2\right)} \]

   6. associate-+r+ 1.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, y, -2\right) + \left(x \cdot \left(y \cdot -2\right) + x \cdot y\right)\right) + -2} \]

   7. associate-*r* 1.9%

      \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \left(\color{blue}{\left(x \cdot y\right) \cdot -2} + x \cdot y\right)\right) + -2 \]

   8. *-commutative 1.9%

      \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \left(\color{blue}{-2 \cdot \left(x \cdot y\right)} + x \cdot y\right)\right) + -2 \]

   9. distribute-lft1-in 1.9%

      \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(-2 + 1\right) \cdot \left(x \cdot y\right)}\right) + -2 \]

   10. metadata-eval 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \color{blue}{-1} \cdot \left(x \cdot y\right)\right) + -2 \]

   11. neg-mul-1 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(-x \cdot y\right)}\right) + -2 \]

   12. neg-sub0 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(0 - x \cdot y\right)}\right) + -2 \]

   13. metadata-eval 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \left(\color{blue}{\left(-2 - -2\right)} - x \cdot y\right)\right) + -2 \]

   14. associate--r+ 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(-2 - \left(-2 + x \cdot y\right)\right)}\right) + -2 \]

   15. +-commutative 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \left(-2 - \color{blue}{\left(x \cdot y + -2\right)}\right)\right) + -2 \]

   16. fma-def 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \left(-2 - \color{blue}{\mathsf{fma}\left(x, y, -2\right)}\right)\right) + -2 \]

   17. sub-neg 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(-2 + \left(-\mathsf{fma}\left(x, y, -2\right)\right)\right)}\right) + -2 \]

   18. +-commutative 1.9%

       \[\leadsto \left(\mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(\left(-\mathsf{fma}\left(x, y, -2\right)\right) + -2\right)}\right) + -2 \]

   19. associate-+r+ 1.9%

       \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, y, -2\right) + \left(-\mathsf{fma}\left(x, y, -2\right)\right)\right) + -2\right)} + -2 \]

   20. sub-neg 1.9%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, y, -2\right) - \mathsf{fma}\left(x, y, -2\right)\right)} + -2\right) + -2 \]

   21. +-inverses 2.2%

       \[\leadsto \left(\color{blue}{0} + -2\right) + -2 \]

   22. metadata-eval 2.2%

       \[\leadsto \color{blue}{-2} + -2 \]

8. Simplified 2.2%

   \[\leadsto \color{blue}{-4} \]

9. Final simplification 2.2%

   \[\leadsto -4 \]

Alternative 10: 2.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -0.177978515625 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -0.177978515625)

C

double code(double x, double y, double z) {
	return -0.177978515625;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -0.177978515625d0
end function

Java

public static double code(double x, double y, double z) {
	return -0.177978515625;
}

Python

def code(x, y, z):
	return -0.177978515625

Julia

function code(x, y, z)
	return -0.177978515625
end

MATLAB

function tmp = code(x, y, z)
	tmp = -0.177978515625;
end

Wolfram

code[x_, y_, z_] := -0.177978515625

Derivation

1. Initial program 98.8%

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

2. Taylor expanded in x around 0 52.5%

   \[\leadsto \color{blue}{2 \cdot {z}^{2}} + z \cdot z \]
3. Step-by-step derivation

   1. unpow2 52.5%

      \[\leadsto 2 \cdot \color{blue}{\left(z \cdot z\right)} + z \cdot z \]

   2. associate-*r* 52.5%

      \[\leadsto \color{blue}{\left(2 \cdot z\right) \cdot z} + z \cdot z \]

   3. count-2 52.5%

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

   4. *-commutative 52.5%

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

4. Simplified 52.5%

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

6. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{-0.177978515625} \]

7. Final simplification 2.2%

   \[\leadsto -0.177978515625 \]

Alternative 11: 5.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 98.8%

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

2. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{2 \cdot {z}^{2} + \left(x \cdot y + {z}^{2}\right)} \]
3. Step-by-step derivation

   1. +-commutative 98.8%

      \[\leadsto 2 \cdot {z}^{2} + \color{blue}{\left({z}^{2} + x \cdot y\right)} \]

   2. associate-+r+ 98.8%

      \[\leadsto \color{blue}{\left(2 \cdot {z}^{2} + {z}^{2}\right) + x \cdot y} \]

   3. distribute-lft1-in 98.8%

      \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} + x \cdot y \]

   4. metadata-eval 98.8%

      \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

   5. *-commutative 98.8%

      \[\leadsto \color{blue}{{z}^{2} \cdot 3} + x \cdot y \]

   6. fma-def 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

   7. unpow2 98.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot z}, 3, x \cdot y\right) \]

4. Simplified 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot z, 3, x \cdot y\right)} \]

6. Applied egg-rr 4.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, -2\right) + \mathsf{fma}\left(\mathsf{fma}\left(x, y, -2\right), -2, \mathsf{fma}\left(x, y, -2\right)\right)} \]
7. Step-by-step derivation

   1. fma-udef 4.8%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(\mathsf{fma}\left(x, y, -2\right) \cdot -2 + \mathsf{fma}\left(x, y, -2\right)\right)} \]

   2. *-commutative 4.8%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \left(\color{blue}{-2 \cdot \mathsf{fma}\left(x, y, -2\right)} + \mathsf{fma}\left(x, y, -2\right)\right) \]

   3. distribute-lft1-in 4.9%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(-2 + 1\right) \cdot \mathsf{fma}\left(x, y, -2\right)} \]

   4. metadata-eval 4.9%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \color{blue}{-1} \cdot \mathsf{fma}\left(x, y, -2\right) \]

   5. neg-mul-1 4.9%

      \[\leadsto \mathsf{fma}\left(x, y, -2\right) + \color{blue}{\left(-\mathsf{fma}\left(x, y, -2\right)\right)} \]

   6. sub-neg 4.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, -2\right) - \mathsf{fma}\left(x, y, -2\right)} \]

   7. +-inverses 5.1%

      \[\leadsto \color{blue}{0} \]

8. Simplified 5.1%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 5.1%

   \[\leadsto 0 \]

Developer target: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot z\right) \cdot z + y \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (* 3.0 z) z) (* y x)))

C

double code(double x, double y, double z) {
	return ((3.0 * z) * 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 = ((3.0d0 * z) * z) + (y * x)
end function

Java

public static double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * x);
}

Python

def code(x, y, z):
	return ((3.0 * z) * z) + (y * x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(3.0 * z) * z) + Float64(y * x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((3.0 * z) * z) + (y * x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))