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

Percentage Accurate: 98.1% → 99.4%

Time: 4.2s

Alternatives: 6

Speedup: 1.7×

• 35.2% 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: 98.1% 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: 99.4% 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 99.0%

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

   1. +-commutative 99.0%

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

   2. fma-def 99.1%

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

   3. associate-+l+ 99.1%

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

   4. fma-def 99.9%

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

   5. distribute-lft-out 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.4% 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 99.0%

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

   1. associate-+l+ 99.0%

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

   2. associate-+l+ 99.0%

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

   3. fma-def 99.8%

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

   4. count-2 99.8%

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

   5. distribute-rgt1-in 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*l* 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-101} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+159}\right) \land z \cdot z \leq 10^{+182}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 1e-101)
         (and (not (<= (* z z) 2e+159)) (<= (* z z) 1e+182)))
   (* x y)
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 1e-101) || (!((z * z) <= 2e+159) && ((z * z) <= 1e+182))) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	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) :: tmp
    if (((z * z) <= 1d-101) .or. (.not. ((z * z) <= 2d+159)) .and. ((z * z) <= 1d+182)) then
        tmp = x * y
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 1e-101) || (!((z * z) <= 2e+159) && ((z * z) <= 1e+182))) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 1e-101) or (not ((z * z) <= 2e+159) and ((z * z) <= 1e+182)):
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 1e-101) || (!(Float64(z * z) <= 2e+159) && (Float64(z * z) <= 1e+182)))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z * z) <= 1e-101) || (~(((z * z) <= 2e+159)) && ((z * z) <= 1e+182)))
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 1e-101], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 2e+159]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 1e+182]]], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000005e-101 or 1.9999999999999999e159 < (*.f64 z z) < 1.0000000000000001e182

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate-+l+ 99.9%

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

      3. fma-def 99.9%

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

      4. count-2 99.9%

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

      5. distribute-rgt1-in 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l* 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in z around 0 89.6%

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

   if 1.00000000000000005e-101 < (*.f64 z z) < 1.9999999999999999e159 or 1.0000000000000001e182 < (*.f64 z z)

   1. Initial program 98.4%

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

   2. Taylor expanded in x around 0 81.7%

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

      1. unpow2 81.7%

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

      2. unpow2 81.7%

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

      3. distribute-rgt1-in 81.7%

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

      4. metadata-eval 81.7%

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

      5. *-commutative 81.7%

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

      6. associate-*r* 81.8%

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

   4. Simplified 81.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-101} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+159}\right) \land z \cdot z \leq 10^{+182}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 4: 98.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-+l+ 99.0%

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

   2. associate-+l+ 99.0%

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

   3. fma-def 99.8%

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

   4. count-2 99.8%

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

   5. distribute-rgt1-in 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*l* 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.1%

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

   2. +-commutative 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

   \[\leadsto z \cdot \left(z \cdot 3\right) + x \cdot y \]

Alternative 5: 75.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 6.5 \cdot 10^{+247}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 6.5e+247) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 6.5e+247) {
		tmp = x * y;
	} else {
		tmp = 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) <= 6.5d+247) then
        tmp = x * y
    else
        tmp = z * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 6.5e+247) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 6.5e+247:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 6.5e+247)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 6.5e+247)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 6.5e+247], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 6.50000000000000023e247

   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+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. fma-def 99.8%

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

      4. count-2 99.8%

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

      5. distribute-rgt1-in 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around 0 65.0%

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

   if 6.50000000000000023e247 < (*.f64 z z)

   1. Initial program 97.1%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. unpow2 97.4%

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

      2. *-commutative 97.4%

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

      3. associate-*l* 97.4%

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

      4. *-commutative 97.4%

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

      5. count-2 97.4%

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

   4. Simplified 97.4%

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

      1. flip-+ 0.0%

         \[\leadsto z \cdot \color{blue}{\frac{z \cdot z - z \cdot z}{z - z}} + z \cdot z \]

      2. +-inverses 0.0%

         \[\leadsto z \cdot \frac{z \cdot z - z \cdot z}{\color{blue}{0}} + z \cdot z \]

      3. +-inverses 0.0%

         \[\leadsto z \cdot \frac{z \cdot z - z \cdot z}{\color{blue}{z \cdot z - z \cdot z}} + z \cdot z \]

      4. associate-*r/ 0.0%

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

      5. +-inverses 0.0%

         \[\leadsto \frac{z \cdot \color{blue}{0}}{z \cdot z - z \cdot z} + z \cdot z \]

      6. +-inverses 0.0%

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

      7. distribute-lft-out-- 0.0%

         \[\leadsto \frac{\color{blue}{z \cdot z - z \cdot z}}{z \cdot z - z \cdot z} + z \cdot z \]

      8. +-inverses 0.0%

         \[\leadsto \frac{z \cdot z - z \cdot z}{\color{blue}{0}} + z \cdot z \]

      9. +-inverses 0.0%

         \[\leadsto \frac{z \cdot z - z \cdot z}{\color{blue}{z - z}} + z \cdot z \]

      10. flip-+ 79.8%

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

      11. add-log-exp 34.4%

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

      12. log1p-expm1-u 33.1%

          \[\leadsto \log \left(e^{z + z}\right) + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(z \cdot z\right)\right)} \]

      13. log1p-udef 33.1%

          \[\leadsto \log \left(e^{z + z}\right) + \color{blue}{\log \left(1 + \mathsf{expm1}\left(z \cdot z\right)\right)} \]

      14. sum-log 33.1%

          \[\leadsto \color{blue}{\log \left(e^{z + z} \cdot \left(1 + \mathsf{expm1}\left(z \cdot z\right)\right)\right)} \]

      15. count-2 33.1%

          \[\leadsto \log \left(e^{\color{blue}{2 \cdot z}} \cdot \left(1 + \mathsf{expm1}\left(z \cdot z\right)\right)\right) \]

      16. exp-prod 33.1%

          \[\leadsto \log \left(\color{blue}{{\left(e^{2}\right)}^{z}} \cdot \left(1 + \mathsf{expm1}\left(z \cdot z\right)\right)\right) \]

   6. Applied egg-rr 33.1%

      \[\leadsto \color{blue}{\log \left({\left(e^{2}\right)}^{z} \cdot \left(1 + \mathsf{expm1}\left(z \cdot z\right)\right)\right)} \]
   7. Step-by-step derivation

      1. log-prod 33.1%

         \[\leadsto \color{blue}{\log \left({\left(e^{2}\right)}^{z}\right) + \log \left(1 + \mathsf{expm1}\left(z \cdot z\right)\right)} \]

      2. +-commutative 33.1%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(z \cdot z\right)\right) + \log \left({\left(e^{2}\right)}^{z}\right)} \]

      3. log1p-def 33.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(z \cdot z\right)\right)} + \log \left({\left(e^{2}\right)}^{z}\right) \]

      4. log1p-expm1 34.4%

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

      5. log-pow 79.8%

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

      6. rem-log-exp 79.8%

         \[\leadsto z \cdot z + z \cdot \color{blue}{2} \]

      7. distribute-lft-in 79.8%

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

   8. Simplified 79.8%

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

   9. Taylor expanded in z around inf 79.8%

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

   11. Simplified 79.8%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 6.5 \cdot 10^{+247}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]

Alternative 6: 53.1% 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 99.0%

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

   1. associate-+l+ 99.0%

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

   2. associate-+l+ 99.0%

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

   3. fma-def 99.8%

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

   4. count-2 99.8%

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

   5. distribute-rgt1-in 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*l* 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.1%

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

   2. +-commutative 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in z around 0 47.5%

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

7. Final simplification 47.5%

   \[\leadsto x \cdot y \]

Developer target: 98.1% 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 2023229 
(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)))