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

Percentage Accurate: 97.7% → 98.9%

Time: 3.7s

Alternatives: 8

Speedup: 1.0×

• 32.6% 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

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

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 98.8%

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

   1. associate-+l+ 98.8%

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

   2. fma-def 99.6%

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

   3. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 98.1% 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 98.8%

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

   1. fma-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 50.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8 \cdot 10^{-133}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-22} \lor \neg \left(a \cdot b \leq 1.9 \cdot 10^{+69}\right) \land a \cdot b \leq 8.8 \cdot 10^{+175}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -8e-133)
   (* a b)
   (if (or (<= (* a b) 4e-22)
           (and (not (<= (* a b) 1.9e+69)) (<= (* a b) 8.8e+175)))
     (* z t)
     (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -8e-133) {
		tmp = a * b;
	} else if (((a * b) <= 4e-22) || (!((a * b) <= 1.9e+69) && ((a * b) <= 8.8e+175))) {
		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) <= (-8d-133)) then
        tmp = a * b
    else if (((a * b) <= 4d-22) .or. (.not. ((a * b) <= 1.9d+69)) .and. ((a * b) <= 8.8d+175)) 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) <= -8e-133) {
		tmp = a * b;
	} else if (((a * b) <= 4e-22) || (!((a * b) <= 1.9e+69) && ((a * b) <= 8.8e+175))) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -8e-133:
		tmp = a * b
	elif ((a * b) <= 4e-22) or (not ((a * b) <= 1.9e+69) and ((a * b) <= 8.8e+175)):
		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) <= -8e-133)
		tmp = Float64(a * b);
	elseif ((Float64(a * b) <= 4e-22) || (!(Float64(a * b) <= 1.9e+69) && (Float64(a * b) <= 8.8e+175)))
		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) <= -8e-133)
		tmp = a * b;
	elseif (((a * b) <= 4e-22) || (~(((a * b) <= 1.9e+69)) && ((a * b) <= 8.8e+175)))
		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], -8e-133], N[(a * b), $MachinePrecision], If[Or[LessEqual[N[(a * b), $MachinePrecision], 4e-22], And[N[Not[LessEqual[N[(a * b), $MachinePrecision], 1.9e+69]], $MachinePrecision], LessEqual[N[(a * b), $MachinePrecision], 8.8e+175]]], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -8.0000000000000005e-133 or 4.0000000000000002e-22 < (*.f64 a b) < 1.90000000000000014e69 or 8.7999999999999997e175 < (*.f64 a b)

   1. Initial program 97.9%

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

   2. Taylor expanded in a around inf 62.4%

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

   if -8.0000000000000005e-133 < (*.f64 a b) < 4.0000000000000002e-22 or 1.90000000000000014e69 < (*.f64 a b) < 8.7999999999999997e175

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 43.2%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8 \cdot 10^{-133}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-22} \lor \neg \left(a \cdot b \leq 1.9 \cdot 10^{+69}\right) \land a \cdot b \leq 8.8 \cdot 10^{+175}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 4: 52.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.4 \cdot 10^{+38}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.15 \cdot 10^{-196}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-315}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 8.8 \cdot 10^{+175}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -5.4e+38)
   (* a b)
   (if (<= (* a b) -1.15e-196)
     (* x y)
     (if (<= (* a b) -1e-315)
       (* z t)
       (if (<= (* a b) 8.8e+175) (* x y) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -5.4e+38) {
		tmp = a * b;
	} else if ((a * b) <= -1.15e-196) {
		tmp = x * y;
	} else if ((a * b) <= -1e-315) {
		tmp = z * t;
	} else if ((a * b) <= 8.8e+175) {
		tmp = x * y;
	} 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) <= (-5.4d+38)) then
        tmp = a * b
    else if ((a * b) <= (-1.15d-196)) then
        tmp = x * y
    else if ((a * b) <= (-1d-315)) then
        tmp = z * t
    else if ((a * b) <= 8.8d+175) then
        tmp = x * y
    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) <= -5.4e+38) {
		tmp = a * b;
	} else if ((a * b) <= -1.15e-196) {
		tmp = x * y;
	} else if ((a * b) <= -1e-315) {
		tmp = z * t;
	} else if ((a * b) <= 8.8e+175) {
		tmp = x * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -5.4e+38:
		tmp = a * b
	elif (a * b) <= -1.15e-196:
		tmp = x * y
	elif (a * b) <= -1e-315:
		tmp = z * t
	elif (a * b) <= 8.8e+175:
		tmp = x * y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -5.4e+38)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -1.15e-196)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= -1e-315)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 8.8e+175)
		tmp = Float64(x * y);
	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) <= -5.4e+38)
		tmp = a * b;
	elseif ((a * b) <= -1.15e-196)
		tmp = x * y;
	elseif ((a * b) <= -1e-315)
		tmp = z * t;
	elseif ((a * b) <= 8.8e+175)
		tmp = x * y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -5.4e+38], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -1.15e-196], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -1e-315], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 8.8e+175], N[(x * y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -5.39999999999999992e38 or 8.7999999999999997e175 < (*.f64 a b)

   1. Initial program 96.7%

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

   2. Taylor expanded in a around inf 78.8%

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

   if -5.39999999999999992e38 < (*.f64 a b) < -1.1500000000000001e-196 or -9.999999985e-316 < (*.f64 a b) < 8.7999999999999997e175

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 55.6%

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

   if -1.1500000000000001e-196 < (*.f64 a b) < -9.999999985e-316

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 72.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.4 \cdot 10^{+38}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.15 \cdot 10^{-196}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-315}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 8.8 \cdot 10^{+175}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 5: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-34} \lor \neg \left(y \leq 1.05 \cdot 10^{+176}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.4e-34) (not (<= y 1.05e+176))) (* x y) (+ (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.4e-34) || !(y <= 1.05e+176)) {
		tmp = 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 ((y <= (-6.4d-34)) .or. (.not. (y <= 1.05d+176))) then
        tmp = 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 ((y <= -6.4e-34) || !(y <= 1.05e+176)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.4e-34) or not (y <= 1.05e+176):
		tmp = x * y
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.4e-34) || !(y <= 1.05e+176))
		tmp = 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 ((y <= -6.4e-34) || ~((y <= 1.05e+176)))
		tmp = x * y;
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.4e-34], N[Not[LessEqual[y, 1.05e+176]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.40000000000000005e-34 or 1.05e176 < y

   1. Initial program 97.1%

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

   2. Taylor expanded in x around inf 65.2%

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

   if -6.40000000000000005e-34 < y < 1.05e176

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-34} \lor \neg \left(y \leq 1.05 \cdot 10^{+176}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 6: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -660000 \lor \neg \left(x \leq 1.95 \cdot 10^{-62}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= x -660000.0) (not (<= x 1.95e-62)))
   (+ (* 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 <= -660000.0) || !(x <= 1.95e-62)) {
		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 <= (-660000.0d0)) .or. (.not. (x <= 1.95d-62))) 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 <= -660000.0) || !(x <= 1.95e-62)) {
		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 <= -660000.0) or not (x <= 1.95e-62):
		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 <= -660000.0) || !(x <= 1.95e-62))
		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 <= -660000.0) || ~((x <= 1.95e-62)))
		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, -660000.0], N[Not[LessEqual[x, 1.95e-62]], $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 < -6.6e5 or 1.9500000000000002e-62 < x

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0 80.9%

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

   if -6.6e5 < x < 1.9500000000000002e-62

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 86.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -660000 \lor \neg \left(x \leq 1.95 \cdot 10^{-62}\right):\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 7: 97.7% 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 98.8%

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

2. Final simplification 98.8%

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

Alternative 8: 35.3% 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 38.5%

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

3. Final simplification 38.5%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023200 
(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)))