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

Percentage Accurate: 97.7% → 98.9%

Time: 5.7s

Alternatives: 10

Speedup: 1.0×

• 33.2% 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: 97.7% 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: 98.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

2. Taylor expanded in x around 0 98.8%

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

   1. fma-def 99.2%

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

   2. fma-def 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 98.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. fma-def 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 98.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. +-commutative 98.8%

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

   2. associate-+l+ 98.8%

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

   3. fma-def 99.2%

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

   4. fma-def 99.2%

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

3. Simplified 99.2%

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

   1. fma-udef 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 54.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.8 \cdot 10^{+61}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 4.8 \cdot 10^{-241}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq 1.45 \cdot 10^{-60}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 9.2 \cdot 10^{+98}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -4.8e+61)
   (* x y)
   (if (<= (* x y) 4.8e-241)
     (* t z)
     (if (<= (* x y) 1.45e-60)
       (* a b)
       (if (<= (* x y) 9.2e+98) (* t z) (* x y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -4.8e+61) {
		tmp = x * y;
	} else if ((x * y) <= 4.8e-241) {
		tmp = t * z;
	} else if ((x * y) <= 1.45e-60) {
		tmp = a * b;
	} else if ((x * y) <= 9.2e+98) {
		tmp = t * z;
	} 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 * y) <= (-4.8d+61)) then
        tmp = x * y
    else if ((x * y) <= 4.8d-241) then
        tmp = t * z
    else if ((x * y) <= 1.45d-60) then
        tmp = a * b
    else if ((x * y) <= 9.2d+98) then
        tmp = t * z
    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 * y) <= -4.8e+61) {
		tmp = x * y;
	} else if ((x * y) <= 4.8e-241) {
		tmp = t * z;
	} else if ((x * y) <= 1.45e-60) {
		tmp = a * b;
	} else if ((x * y) <= 9.2e+98) {
		tmp = t * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -4.8e+61:
		tmp = x * y
	elif (x * y) <= 4.8e-241:
		tmp = t * z
	elif (x * y) <= 1.45e-60:
		tmp = a * b
	elif (x * y) <= 9.2e+98:
		tmp = t * z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -4.8e+61)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 4.8e-241)
		tmp = Float64(t * z);
	elseif (Float64(x * y) <= 1.45e-60)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 9.2e+98)
		tmp = Float64(t * z);
	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 * y) <= -4.8e+61)
		tmp = x * y;
	elseif ((x * y) <= 4.8e-241)
		tmp = t * z;
	elseif ((x * y) <= 1.45e-60)
		tmp = a * b;
	elseif ((x * y) <= 9.2e+98)
		tmp = t * z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -4.8e+61], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4.8e-241], N[(t * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.45e-60], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 9.2e+98], N[(t * z), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.7999999999999998e61 or 9.20000000000000053e98 < (*.f64 x y)

   1. Initial program 96.6%

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

   2. Taylor expanded in x around 0 96.6%

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

      1. fma-def 97.8%

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

      2. fma-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 74.9%

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

   if -4.7999999999999998e61 < (*.f64 x y) < 4.8e-241 or 1.45e-60 < (*.f64 x y) < 9.20000000000000053e98

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. fma-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 55.1%

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

   if 4.8e-241 < (*.f64 x y) < 1.45e-60

   1. Initial program 100.0%

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

   2. Taylor expanded in a around inf 60.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4.8 \cdot 10^{+61}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 4.8 \cdot 10^{-241}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;x \cdot y \leq 1.45 \cdot 10^{-60}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 9.2 \cdot 10^{+98}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.7 \cdot 10^{+73} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+105}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -1.7e+73) (not (<= (* x y) 2e+105)))
   (* x y)
   (+ (* a b) (* t z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -1.7e+73) || !((x * y) <= 2e+105)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (t * z);
	}
	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 * y) <= (-1.7d+73)) .or. (.not. ((x * y) <= 2d+105))) then
        tmp = x * y
    else
        tmp = (a * b) + (t * z)
    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 * y) <= -1.7e+73) || !((x * y) <= 2e+105)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * y) <= -1.7e+73) or not ((x * y) <= 2e+105):
		tmp = x * y
	else:
		tmp = (a * b) + (t * z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -1.7e+73) || !(Float64(x * y) <= 2e+105))
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(a * b) + Float64(t * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * y) <= -1.7e+73) || ~(((x * y) <= 2e+105)))
		tmp = x * y;
	else
		tmp = (a * b) + (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1.7e+73], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+105]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.7000000000000001e73 or 1.9999999999999999e105 < (*.f64 x y)

   1. Initial program 96.5%

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

   2. Taylor expanded in x around 0 96.5%

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

      1. fma-def 97.7%

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

      2. fma-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 77.2%

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

   if -1.7000000000000001e73 < (*.f64 x y) < 1.9999999999999999e105

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.7 \cdot 10^{+73} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+105}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + t \cdot z\\ \end{array} \]

Alternative 6: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.28 \cdot 10^{+26} \lor \neg \left(a \cdot b \leq 3.5 \cdot 10^{+24}\right):\\ \;\;\;\;a \cdot b + t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -1.28e+26) (not (<= (* a b) 3.5e+24)))
   (+ (* a b) (* t z))
   (+ (* x y) (* t z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -1.28e+26) || !((a * b) <= 3.5e+24)) {
		tmp = (a * b) + (t * z);
	} else {
		tmp = (x * y) + (t * z);
	}
	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) <= (-1.28d+26)) .or. (.not. ((a * b) <= 3.5d+24))) then
        tmp = (a * b) + (t * z)
    else
        tmp = (x * y) + (t * z)
    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) <= -1.28e+26) || !((a * b) <= 3.5e+24)) {
		tmp = (a * b) + (t * z);
	} else {
		tmp = (x * y) + (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -1.28e+26) or not ((a * b) <= 3.5e+24):
		tmp = (a * b) + (t * z)
	else:
		tmp = (x * y) + (t * z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -1.28e+26) || !(Float64(a * b) <= 3.5e+24))
		tmp = Float64(Float64(a * b) + Float64(t * z));
	else
		tmp = Float64(Float64(x * y) + Float64(t * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((a * b) <= -1.28e+26) || ~(((a * b) <= 3.5e+24)))
		tmp = (a * b) + (t * z);
	else
		tmp = (x * y) + (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -1.28e+26], N[Not[LessEqual[N[(a * b), $MachinePrecision], 3.5e+24]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.28e26 or 3.5000000000000002e24 < (*.f64 a b)

   1. Initial program 98.1%

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

   2. Taylor expanded in x around 0 85.8%

      \[\leadsto \color{blue}{t \cdot z} + a \cdot b \]

   if -1.28e26 < (*.f64 a b) < 3.5000000000000002e24

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0 99.3%

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

      1. fma-def 99.3%

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

      2. fma-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in a around 0 94.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.28 \cdot 10^{+26} \lor \neg \left(a \cdot b \leq 3.5 \cdot 10^{+24}\right):\\ \;\;\;\;a \cdot b + t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + t \cdot z\\ \end{array} \]

Alternative 7: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.1 \cdot 10^{+106}:\\ \;\;\;\;x \cdot y + a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 8 \cdot 10^{+28}:\\ \;\;\;\;x \cdot y + t \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2.1e+106)
   (+ (* x y) (* a b))
   (if (<= (* a b) 8e+28) (+ (* x y) (* t z)) (+ (* a b) (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2.1e+106) {
		tmp = (x * y) + (a * b);
	} else if ((a * b) <= 8e+28) {
		tmp = (x * y) + (t * z);
	} else {
		tmp = (a * b) + (t * z);
	}
	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) <= (-2.1d+106)) then
        tmp = (x * y) + (a * b)
    else if ((a * b) <= 8d+28) then
        tmp = (x * y) + (t * z)
    else
        tmp = (a * b) + (t * z)
    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) <= -2.1e+106) {
		tmp = (x * y) + (a * b);
	} else if ((a * b) <= 8e+28) {
		tmp = (x * y) + (t * z);
	} else {
		tmp = (a * b) + (t * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2.1e+106:
		tmp = (x * y) + (a * b)
	elif (a * b) <= 8e+28:
		tmp = (x * y) + (t * z)
	else:
		tmp = (a * b) + (t * z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2.1e+106)
		tmp = Float64(Float64(x * y) + Float64(a * b));
	elseif (Float64(a * b) <= 8e+28)
		tmp = Float64(Float64(x * y) + Float64(t * z));
	else
		tmp = Float64(Float64(a * b) + Float64(t * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -2.1e+106)
		tmp = (x * y) + (a * b);
	elseif ((a * b) <= 8e+28)
		tmp = (x * y) + (t * z);
	else
		tmp = (a * b) + (t * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2.1e+106], N[(N[(x * y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 8e+28], N[(N[(x * y), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.10000000000000005e106

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 85.9%

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

   if -2.10000000000000005e106 < (*.f64 a b) < 7.99999999999999967e28

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. fma-def 99.4%

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

      2. fma-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in a around 0 91.2%

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

   if 7.99999999999999967e28 < (*.f64 a b)

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0 95.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.1 \cdot 10^{+106}:\\ \;\;\;\;x \cdot y + a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 8 \cdot 10^{+28}:\\ \;\;\;\;x \cdot y + t \cdot z\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + t \cdot z\\ \end{array} \]

Alternative 8: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.5 \cdot 10^{+133} \lor \neg \left(a \cdot b \leq 4.4 \cdot 10^{+21}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -5.5e+133) (not (<= (* a b) 4.4e+21))) (* a b) (* t z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -5.5e+133) || !((a * b) <= 4.4e+21)) {
		tmp = a * b;
	} else {
		tmp = t * z;
	}
	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) <= (-5.5d+133)) .or. (.not. ((a * b) <= 4.4d+21))) then
        tmp = a * b
    else
        tmp = t * z
    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) <= -5.5e+133) || !((a * b) <= 4.4e+21)) {
		tmp = a * b;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -5.5e+133) or not ((a * b) <= 4.4e+21):
		tmp = a * b
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -5.5e+133) || !(Float64(a * b) <= 4.4e+21))
		tmp = Float64(a * b);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((a * b) <= -5.5e+133) || ~(((a * b) <= 4.4e+21)))
		tmp = a * b;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -5.5e+133], N[Not[LessEqual[N[(a * b), $MachinePrecision], 4.4e+21]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(t * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.5e133 or 4.4e21 < (*.f64 a b)

   1. Initial program 97.6%

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

   2. Taylor expanded in a around inf 75.0%

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

   if -5.5e133 < (*.f64 a b) < 4.4e21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. fma-def 99.4%

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

      2. fma-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in t around inf 46.4%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.5 \cdot 10^{+133} \lor \neg \left(a \cdot b \leq 4.4 \cdot 10^{+21}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]

Alternative 9: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + (t * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

2. Final simplification 98.8%

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

Alternative 10: 36.1% 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 98.8%

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

2. Taylor expanded in a around inf 32.4%

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

3. Final simplification 32.4%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023310 
(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)))