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

Percentage Accurate: 98.2% → 99.3%

Time: 7.3s

Alternatives: 6

Speedup: 1.8×

• 34.4% 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: 98.2% 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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 2: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-41} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-262}\right):\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -1e-41) (not (<= (* x y) 2e-262)))
   (fma z z (* y x))
   (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -1e-41) || !((x * y) <= 2e-262)) {
		tmp = fma(z, z, (y * x));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -1e-41) || !(Float64(x * y) <= 2e-262))
		tmp = fma(z, z, Float64(y * x));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e-41], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e-262]], $MachinePrecision]], N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000001e-41 or 2.00000000000000002e-262 < (*.f64 x y)

   1. Initial program 98.3%

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

   4. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-rgt-identity 86.7

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

   6. Applied rewrites 86.7%

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

   if -1.00000000000000001e-41 < (*.f64 x y) < 2.00000000000000002e-262

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 87.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-41} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{-262}\right):\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{-35} \lor \neg \left(z \leq 1.3 \cdot 10^{-8} \lor \neg \left(z \leq 6.8 \cdot 10^{+25}\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 1.35e-35) (not (or (<= z 1.3e-8) (not (<= z 6.8e+25)))))
   (* y x)
   (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1.35e-35) || !((z <= 1.3e-8) || !(z <= 6.8e+25))) {
		tmp = y * x;
	} 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 <= 1.35d-35) .or. (.not. (z <= 1.3d-8) .or. (.not. (z <= 6.8d+25)))) then
        tmp = y * x
    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 <= 1.35e-35) || !((z <= 1.3e-8) || !(z <= 6.8e+25))) {
		tmp = y * x;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 1.35e-35) or not ((z <= 1.3e-8) or not (z <= 6.8e+25)):
		tmp = y * x
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 1.35e-35) || !((z <= 1.3e-8) || !(z <= 6.8e+25)))
		tmp = Float64(y * x);
	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 <= 1.35e-35) || ~(((z <= 1.3e-8) || ~((z <= 6.8e+25)))))
		tmp = y * x;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 1.35e-35], N[Not[Or[LessEqual[z, 1.3e-8], N[Not[LessEqual[z, 6.8e+25]], $MachinePrecision]]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.3499999999999999e-35 or 1.3000000000000001e-8 < z < 6.79999999999999967e25

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 32.3%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 62.7

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

   13. Applied rewrites 62.7%

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

   if 1.3499999999999999e-35 < z < 1.3000000000000001e-8 or 6.79999999999999967e25 < z

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{-35} \lor \neg \left(z \leq 1.3 \cdot 10^{-8} \lor \neg \left(z \leq 6.8 \cdot 10^{+25}\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{-35}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.35e-35)
   (* y x)
   (if (<= z 1.3e-8)
     (* (* 3.0 z) z)
     (if (<= z 6.8e+25) (* y x) (* 3.0 (* z z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.35e-35) {
		tmp = y * x;
	} else if (z <= 1.3e-8) {
		tmp = (3.0 * z) * z;
	} else if (z <= 6.8e+25) {
		tmp = y * x;
	} 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 <= 1.35d-35) then
        tmp = y * x
    else if (z <= 1.3d-8) then
        tmp = (3.0d0 * z) * z
    else if (z <= 6.8d+25) then
        tmp = y * x
    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 <= 1.35e-35) {
		tmp = y * x;
	} else if (z <= 1.3e-8) {
		tmp = (3.0 * z) * z;
	} else if (z <= 6.8e+25) {
		tmp = y * x;
	} else {
		tmp = 3.0 * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.35e-35:
		tmp = y * x
	elif z <= 1.3e-8:
		tmp = (3.0 * z) * z
	elif z <= 6.8e+25:
		tmp = y * x
	else:
		tmp = 3.0 * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.35e-35)
		tmp = Float64(y * x);
	elseif (z <= 1.3e-8)
		tmp = Float64(Float64(3.0 * z) * z);
	elseif (z <= 6.8e+25)
		tmp = Float64(y * x);
	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 <= 1.35e-35)
		tmp = y * x;
	elseif (z <= 1.3e-8)
		tmp = (3.0 * z) * z;
	elseif (z <= 6.8e+25)
		tmp = y * x;
	else
		tmp = 3.0 * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.35e-35], N[(y * x), $MachinePrecision], If[LessEqual[z, 1.3e-8], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 6.8e+25], N[(y * x), $MachinePrecision], N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.3499999999999999e-35 or 1.3000000000000001e-8 < z < 6.79999999999999967e25

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 32.3%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 62.7

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

   13. Applied rewrites 62.7%

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

   if 1.3499999999999999e-35 < z < 1.3000000000000001e-8

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 95.9

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

   5. Applied rewrites 95.9%

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

   7. Applied rewrites 96.3%

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

   if 6.79999999999999967e25 < z

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

Alternative 5: 63.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+106}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 4e+106) (* y x) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4e+106) {
		tmp = y * x;
	} 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 <= 4d+106) then
        tmp = y * x
    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 <= 4e+106) {
		tmp = y * x;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 4e+106:
		tmp = y * x
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4e+106)
		tmp = Float64(y * x);
	else
		tmp = Float64(z * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 4e+106)
		tmp = y * x;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4e+106], N[(y * x), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.00000000000000036e106

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 38.2%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 60.2

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

   13. Applied rewrites 60.2%

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

   if 4.00000000000000036e106 < z

   1. Initial program 97.3%

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

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

Alternative 6: 33.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ z \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

4. Applied rewrites 78.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]

5. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 33.2

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

7. Applied rewrites 33.2%

   \[\leadsto \color{blue}{z \cdot z} \]
8. Add Preprocessing

Developer Target 1: 98.2% accurate, 1.6× 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 2024337 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* 3 z) z) (* y x)))

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