Linear.V3:$cdot from linear-1.19.1.3, B

Percentage Accurate: 97.8% → 99.0%

Time: 4.4s

Alternatives: 10

Speedup: 1.0×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (+ (* x y) (* z t)) (* a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (a * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * y) + (z * t)) + (a * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (+ (* x y) (* z t)) (* a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * t)) + (a * b);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((x * y) + (z * t)) + (a * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (fma x y (fma z t (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(x, y, fma(z, t, (a * b)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(x, y, fma(z, t, Float64(a * b)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * y + N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. fma-def 98.4%

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

   3. fma-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 98.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

   1. associate-+l+ 97.6%

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

   2. fma-def 98.4%

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

   3. fma-def 98.4%

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

3. Simplified 98.4%

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

   1. fma-udef 97.6%

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

   2. fma-udef 97.6%

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

   3. associate-+l+ 97.6%

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

   4. +-commutative 97.6%

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

   5. associate-+r+ 97.6%

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

   6. fma-def 97.6%

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

5. Applied egg-rr 97.6%

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

6. Final simplification 97.6%

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

Alternative 3: 98.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a * b), $MachinePrecision] + N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

   1. fma-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 4: 47.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+167}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+148}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+103}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-174}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-256}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-48}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-16}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.3e+167)
   (* x y)
   (if (<= x -1.7e+148)
     (* z t)
     (if (<= x -8e+103)
       (* x y)
       (if (<= x -5.2e-174)
         (* z t)
         (if (<= x 1.75e-256)
           (* a b)
           (if (<= x 8.2e-48)
             (* z t)
             (if (<= x 7.4e-16) (* a b) (* x y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.3e+167) {
		tmp = x * y;
	} else if (x <= -1.7e+148) {
		tmp = z * t;
	} else if (x <= -8e+103) {
		tmp = x * y;
	} else if (x <= -5.2e-174) {
		tmp = z * t;
	} else if (x <= 1.75e-256) {
		tmp = a * b;
	} else if (x <= 8.2e-48) {
		tmp = z * t;
	} else if (x <= 7.4e-16) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-2.3d+167)) then
        tmp = x * y
    else if (x <= (-1.7d+148)) then
        tmp = z * t
    else if (x <= (-8d+103)) then
        tmp = x * y
    else if (x <= (-5.2d-174)) then
        tmp = z * t
    else if (x <= 1.75d-256) then
        tmp = a * b
    else if (x <= 8.2d-48) then
        tmp = z * t
    else if (x <= 7.4d-16) then
        tmp = a * b
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.3e+167) {
		tmp = x * y;
	} else if (x <= -1.7e+148) {
		tmp = z * t;
	} else if (x <= -8e+103) {
		tmp = x * y;
	} else if (x <= -5.2e-174) {
		tmp = z * t;
	} else if (x <= 1.75e-256) {
		tmp = a * b;
	} else if (x <= 8.2e-48) {
		tmp = z * t;
	} else if (x <= 7.4e-16) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -2.3e+167:
		tmp = x * y
	elif x <= -1.7e+148:
		tmp = z * t
	elif x <= -8e+103:
		tmp = x * y
	elif x <= -5.2e-174:
		tmp = z * t
	elif x <= 1.75e-256:
		tmp = a * b
	elif x <= 8.2e-48:
		tmp = z * t
	elif x <= 7.4e-16:
		tmp = a * b
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.3e+167)
		tmp = Float64(x * y);
	elseif (x <= -1.7e+148)
		tmp = Float64(z * t);
	elseif (x <= -8e+103)
		tmp = Float64(x * y);
	elseif (x <= -5.2e-174)
		tmp = Float64(z * t);
	elseif (x <= 1.75e-256)
		tmp = Float64(a * b);
	elseif (x <= 8.2e-48)
		tmp = Float64(z * t);
	elseif (x <= 7.4e-16)
		tmp = Float64(a * b);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -2.3e+167)
		tmp = x * y;
	elseif (x <= -1.7e+148)
		tmp = z * t;
	elseif (x <= -8e+103)
		tmp = x * y;
	elseif (x <= -5.2e-174)
		tmp = z * t;
	elseif (x <= 1.75e-256)
		tmp = a * b;
	elseif (x <= 8.2e-48)
		tmp = z * t;
	elseif (x <= 7.4e-16)
		tmp = a * b;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.3e+167], N[(x * y), $MachinePrecision], If[LessEqual[x, -1.7e+148], N[(z * t), $MachinePrecision], If[LessEqual[x, -8e+103], N[(x * y), $MachinePrecision], If[LessEqual[x, -5.2e-174], N[(z * t), $MachinePrecision], If[LessEqual[x, 1.75e-256], N[(a * b), $MachinePrecision], If[LessEqual[x, 8.2e-48], N[(z * t), $MachinePrecision], If[LessEqual[x, 7.4e-16], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.29999999999999988e167 or -1.7000000000000001e148 < x < -8e103 or 7.3999999999999999e-16 < x

   1. Initial program 94.8%

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

      1. associate-+l+ 94.8%

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

      2. fma-def 95.9%

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

      3. fma-def 95.9%

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

   3. Simplified 95.9%

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

      1. fma-udef 94.8%

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

      2. fma-udef 94.8%

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

      3. associate-+l+ 94.8%

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

      4. +-commutative 94.8%

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

      5. associate-+r+ 94.8%

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

      6. fma-def 94.8%

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

   5. Applied egg-rr 94.8%

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

   6. Taylor expanded in x around inf 63.5%

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

   if -2.29999999999999988e167 < x < -1.7000000000000001e148 or -8e103 < x < -5.2000000000000004e-174 or 1.75000000000000007e-256 < x < 8.20000000000000028e-48

   1. Initial program 99.1%

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

      1. associate-+l+ 99.1%

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

      2. fma-def 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 99.1%

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

      2. fma-udef 99.1%

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

      3. associate-+l+ 99.1%

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

      4. +-commutative 99.1%

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

      5. associate-+r+ 99.1%

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

      6. fma-def 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in z around inf 53.6%

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

   if -5.2000000000000004e-174 < x < 1.75000000000000007e-256 or 8.20000000000000028e-48 < x < 7.3999999999999999e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf 54.9%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+167}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+148}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+103}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-174}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-256}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-48}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-16}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+167} \lor \neg \left(x \leq -1.95 \cdot 10^{+149} \lor \neg \left(x \leq -3.6 \cdot 10^{+112}\right) \land x \leq 8.2 \cdot 10^{-48}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= x -4.4e+167)
         (not
          (or (<= x -1.95e+149) (and (not (<= x -3.6e+112)) (<= x 8.2e-48)))))
   (+ (* a b) (* x y))
   (+ (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x <= -4.4e+167) || !((x <= -1.95e+149) || (!(x <= -3.6e+112) && (x <= 8.2e-48)))) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((x <= (-4.4d+167)) .or. (.not. (x <= (-1.95d+149)) .or. (.not. (x <= (-3.6d+112))) .and. (x <= 8.2d-48))) then
        tmp = (a * b) + (x * y)
    else
        tmp = (a * b) + (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x <= -4.4e+167) || !((x <= -1.95e+149) || (!(x <= -3.6e+112) && (x <= 8.2e-48)))) {
		tmp = (a * b) + (x * y);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x <= -4.4e+167) or not ((x <= -1.95e+149) or (not (x <= -3.6e+112) and (x <= 8.2e-48))):
		tmp = (a * b) + (x * y)
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((x <= -4.4e+167) || !((x <= -1.95e+149) || (!(x <= -3.6e+112) && (x <= 8.2e-48))))
		tmp = Float64(Float64(a * b) + Float64(x * y));
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x <= -4.4e+167) || ~(((x <= -1.95e+149) || (~((x <= -3.6e+112)) && (x <= 8.2e-48)))))
		tmp = (a * b) + (x * y);
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[x, -4.4e+167], N[Not[Or[LessEqual[x, -1.95e+149], And[N[Not[LessEqual[x, -3.6e+112]], $MachinePrecision], LessEqual[x, 8.2e-48]]]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.40000000000000007e167 or -1.95e149 < x < -3.6e112 or 8.20000000000000028e-48 < x

   1. Initial program 95.2%

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

   2. Taylor expanded in x around inf 86.0%

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

   if -4.40000000000000007e167 < x < -1.95e149 or -3.6e112 < x < 8.20000000000000028e-48

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0 84.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+167} \lor \neg \left(x \leq -1.95 \cdot 10^{+149} \lor \neg \left(x \leq -3.6 \cdot 10^{+112}\right) \land x \leq 8.2 \cdot 10^{-48}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 6: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.8 \cdot 10^{+123}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2900000000000:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2.8e+123)
   (+ (* a b) (* x y))
   (if (<= (* a b) 2900000000000.0) (+ (* x y) (* z t)) (+ (* a b) (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2.8e+123) {
		tmp = (a * b) + (x * y);
	} else if ((a * b) <= 2900000000000.0) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * b) <= (-2.8d+123)) then
        tmp = (a * b) + (x * y)
    else if ((a * b) <= 2900000000000.0d0) then
        tmp = (x * y) + (z * t)
    else
        tmp = (a * b) + (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2.8e+123) {
		tmp = (a * b) + (x * y);
	} else if ((a * b) <= 2900000000000.0) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2.8e+123:
		tmp = (a * b) + (x * y)
	elif (a * b) <= 2900000000000.0:
		tmp = (x * y) + (z * t)
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2.8e+123)
		tmp = Float64(Float64(a * b) + Float64(x * y));
	elseif (Float64(a * b) <= 2900000000000.0)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -2.8e+123)
		tmp = (a * b) + (x * y);
	elseif ((a * b) <= 2900000000000.0)
		tmp = (x * y) + (z * t);
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2.8e+123], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2900000000000.0], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.80000000000000011e123

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 92.8%

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

   if -2.80000000000000011e123 < (*.f64 a b) < 2.9e12

   1. Initial program 97.3%

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

      1. associate-+l+ 97.3%

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

      2. fma-def 98.0%

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

      3. fma-def 98.0%

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

   3. Simplified 98.0%

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

      1. fma-udef 97.3%

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

      2. fma-udef 97.3%

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

      3. associate-+l+ 97.3%

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

      4. +-commutative 97.3%

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

      5. associate-+r+ 97.3%

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

      6. fma-def 97.3%

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

   5. Applied egg-rr 97.3%

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

   6. Taylor expanded in a around 0 87.9%

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

   if 2.9e12 < (*.f64 a b)

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 83.6%

      \[\leadsto \color{blue}{t \cdot z} + a \cdot b \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.8 \cdot 10^{+123}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2900000000000:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 7: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.8 \cdot 10^{+123}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{+40}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2.8e+123)
   (* a b)
   (if (<= (* a b) 1.75e+40) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2.8e+123) {
		tmp = a * b;
	} else if ((a * b) <= 1.75e+40) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * b) <= (-2.8d+123)) then
        tmp = a * b
    else if ((a * b) <= 1.75d+40) then
        tmp = z * t
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2.8e+123) {
		tmp = a * b;
	} else if ((a * b) <= 1.75e+40) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2.8e+123:
		tmp = a * b
	elif (a * b) <= 1.75e+40:
		tmp = z * t
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2.8e+123)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.75e+40)
		tmp = Float64(z * t);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -2.8e+123)
		tmp = a * b;
	elseif ((a * b) <= 1.75e+40)
		tmp = z * t;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2.8e+123], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.75e+40], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.80000000000000011e123 or 1.75e40 < (*.f64 a b)

   1. Initial program 98.0%

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

   2. Taylor expanded in a around inf 70.2%

      \[\leadsto \color{blue}{a \cdot b} \]

   if -2.80000000000000011e123 < (*.f64 a b) < 1.75e40

   1. Initial program 97.4%

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

      1. associate-+l+ 97.4%

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

      2. fma-def 98.1%

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

      3. fma-def 98.1%

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

   3. Simplified 98.1%

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

      1. fma-udef 97.4%

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

      2. fma-udef 97.4%

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

      3. associate-+l+ 97.4%

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

      4. +-commutative 97.4%

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

      5. associate-+r+ 97.4%

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

      6. fma-def 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around inf 48.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.8 \cdot 10^{+123}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{+40}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 8: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+198}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.00035:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.4e+198) (* x y) (if (<= x 0.00035) (+ (* a b) (* z t)) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.4e+198) {
		tmp = x * y;
	} else if (x <= 0.00035) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-2.4d+198)) then
        tmp = x * y
    else if (x <= 0.00035d0) then
        tmp = (a * b) + (z * t)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.4e+198) {
		tmp = x * y;
	} else if (x <= 0.00035) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -2.4e+198:
		tmp = x * y
	elif x <= 0.00035:
		tmp = (a * b) + (z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.4e+198)
		tmp = Float64(x * y);
	elseif (x <= 0.00035)
		tmp = Float64(Float64(a * b) + Float64(z * t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -2.4e+198)
		tmp = x * y;
	elseif (x <= 0.00035)
		tmp = (a * b) + (z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.4e+198], N[(x * y), $MachinePrecision], If[LessEqual[x, 0.00035], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e198 or 3.49999999999999996e-4 < x

   1. Initial program 94.0%

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

      1. associate-+l+ 94.0%

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

      2. fma-def 95.2%

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

      3. fma-def 95.2%

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

   3. Simplified 95.2%

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

      1. fma-udef 94.0%

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

      2. fma-udef 94.0%

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

      3. associate-+l+ 94.0%

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

      4. +-commutative 94.0%

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

      5. associate-+r+ 94.0%

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

      6. fma-def 94.0%

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

   5. Applied egg-rr 94.0%

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

   6. Taylor expanded in x around inf 63.6%

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

   if -2.4000000000000001e198 < x < 3.49999999999999996e-4

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 81.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+198}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 0.00035:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9: 97.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (* a b) (+ (* x y) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (a * b) + ((x * y) + (z * t));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * b) + ((x * y) + (z * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a * b), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

2. Final simplification 97.6%

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

Alternative 10: 35.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ a \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* a b))

C

double code(double x, double y, double z, double t, double a, double b) {
	return a * b;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * b
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return a * b

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(a * b), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

2. Taylor expanded in a around inf 35.0%

   \[\leadsto \color{blue}{a \cdot b} \]

3. Final simplification 35.0%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y z t a b)
  :name "Linear.V3:$cdot from linear-1.19.1.3, B"
  :precision binary64
  (+ (+ (* x y) (* z t)) (* a b)))