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

Percentage Accurate: 97.6% → 98.7%

Time: 7.9s

Alternatives: 7

Speedup: 0.4×

• 32.2% 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: 97.6% 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: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{a \cdot b}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (+ (* x y) (* z t)))))
   (if (<= t_1 INFINITY) t_1 (* y (+ x (/ (* a b) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + ((x * y) + (z * t));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = y * (x + ((a * b) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + ((x * y) + (z * t));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = y * (x + ((a * b) / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + ((x * y) + (z * t))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = y * (x + ((a * b) / y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(Float64(x * y) + Float64(z * t)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(x + Float64(Float64(a * b) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + ((x * y) + (z * t));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = y * (x + ((a * b) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(y * N[(x + N[(N[(a * b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b))

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

         \[\leadsto \left(t \cdot z\right) \cdot 1 + y \cdot \left(x + \frac{a \cdot b}{y}\right) \]

      8. *-rgt-identity N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot z\right), \color{blue}{\left(y \cdot \left(x + \frac{a \cdot b}{y}\right)\right)}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \left(\color{blue}{y} \cdot \left(x + \frac{a \cdot b}{y}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(y, \color{blue}{\left(x + \frac{a \cdot b}{y}\right)}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{a \cdot b}{y}\right)}\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(a \cdot b\right), \color{blue}{y}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 33.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), y\right)\right)\right)\right) \]

   5. Simplified 33.3%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x + \frac{a \cdot b}{y}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{a \cdot b}{y}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\left(a \cdot b\right), \color{blue}{y}\right)\right)\right) \]

      4. *-lowering-*.f64 66.7%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), y\right)\right)\right) \]

   8. Simplified 66.7%

      \[\leadsto \color{blue}{y \cdot \left(x + \frac{a \cdot b}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b + \left(x \cdot y + z \cdot t\right) \leq \infty:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{a \cdot b}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + z \cdot t\\ t_2 := x \cdot y + a \cdot b\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+193}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-19}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* z t))) (t_2 (+ (* x y) (* a b))))
   (if (<= (* a b) -2e+193)
     t_2
     (if (<= (* a b) -5e+79)
       t_1
       (if (<= (* a b) 2e-19)
         (+ (* x y) (* z t))
         (if (<= (* a b) 1e+117) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (z * t);
	double t_2 = (x * y) + (a * b);
	double tmp;
	if ((a * b) <= -2e+193) {
		tmp = t_2;
	} else if ((a * b) <= -5e+79) {
		tmp = t_1;
	} else if ((a * b) <= 2e-19) {
		tmp = (x * y) + (z * t);
	} else if ((a * b) <= 1e+117) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * b) + (z * t)
    t_2 = (x * y) + (a * b)
    if ((a * b) <= (-2d+193)) then
        tmp = t_2
    else if ((a * b) <= (-5d+79)) then
        tmp = t_1
    else if ((a * b) <= 2d-19) then
        tmp = (x * y) + (z * t)
    else if ((a * b) <= 1d+117) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (z * t);
	double t_2 = (x * y) + (a * b);
	double tmp;
	if ((a * b) <= -2e+193) {
		tmp = t_2;
	} else if ((a * b) <= -5e+79) {
		tmp = t_1;
	} else if ((a * b) <= 2e-19) {
		tmp = (x * y) + (z * t);
	} else if ((a * b) <= 1e+117) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + (z * t)
	t_2 = (x * y) + (a * b)
	tmp = 0
	if (a * b) <= -2e+193:
		tmp = t_2
	elif (a * b) <= -5e+79:
		tmp = t_1
	elif (a * b) <= 2e-19:
		tmp = (x * y) + (z * t)
	elif (a * b) <= 1e+117:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(z * t))
	t_2 = Float64(Float64(x * y) + Float64(a * b))
	tmp = 0.0
	if (Float64(a * b) <= -2e+193)
		tmp = t_2;
	elseif (Float64(a * b) <= -5e+79)
		tmp = t_1;
	elseif (Float64(a * b) <= 2e-19)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	elseif (Float64(a * b) <= 1e+117)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + (z * t);
	t_2 = (x * y) + (a * b);
	tmp = 0.0;
	if ((a * b) <= -2e+193)
		tmp = t_2;
	elseif ((a * b) <= -5e+79)
		tmp = t_1;
	elseif ((a * b) <= 2e-19)
		tmp = (x * y) + (z * t);
	elseif ((a * b) <= 1e+117)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e+193], t$95$2, If[LessEqual[N[(a * b), $MachinePrecision], -5e+79], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2e-19], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+117], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.00000000000000013e193 or 1.00000000000000005e117 < (*.f64 a b)

   1. Initial program 94.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 94.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 94.0%

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

   if -2.00000000000000013e193 < (*.f64 a b) < -5e79 or 2e-19 < (*.f64 a b) < 1.00000000000000005e117

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 87.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 87.0%

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

   if -5e79 < (*.f64 a b) < 2e-19

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \left(\color{blue}{x} \cdot y\right)\right) \]

      3. *-lowering-*.f64 89.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   5. Simplified 89.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+193}:\\ \;\;\;\;x \cdot y + a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{+79}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-19}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 10^{+117}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.8 \cdot 10^{+181}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -6.6 \cdot 10^{-285}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 0.00185:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -4.8e+181)
   (* a b)
   (if (<= (* a b) -6.6e-285)
     (* z t)
     (if (<= (* a b) 0.00185) (* x y) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -4.8e+181) {
		tmp = a * b;
	} else if ((a * b) <= -6.6e-285) {
		tmp = z * t;
	} else if ((a * b) <= 0.00185) {
		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) <= (-4.8d+181)) then
        tmp = a * b
    else if ((a * b) <= (-6.6d-285)) then
        tmp = z * t
    else if ((a * b) <= 0.00185d0) 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) <= -4.8e+181) {
		tmp = a * b;
	} else if ((a * b) <= -6.6e-285) {
		tmp = z * t;
	} else if ((a * b) <= 0.00185) {
		tmp = x * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -4.8e+181:
		tmp = a * b
	elif (a * b) <= -6.6e-285:
		tmp = z * t
	elif (a * b) <= 0.00185:
		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) <= -4.8e+181)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -6.6e-285)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 0.00185)
		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) <= -4.8e+181)
		tmp = a * b;
	elseif ((a * b) <= -6.6e-285)
		tmp = z * t;
	elseif ((a * b) <= 0.00185)
		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], -4.8e+181], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -6.6e-285], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 0.00185], N[(x * y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.80000000000000004e181 or 0.0018500000000000001 < (*.f64 a b)

   1. Initial program 95.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 74.4%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

   5. Simplified 74.4%

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

   if -4.80000000000000004e181 < (*.f64 a b) < -6.5999999999999997e-285

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 46.2%

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{z}\right) \]

   5. Simplified 46.2%

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

   if -6.5999999999999997e-285 < (*.f64 a b) < 0.0018500000000000001

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 58.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.8 \cdot 10^{+181}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -6.6 \cdot 10^{-285}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 0.00185:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + z \cdot t\\ \mathbf{if}\;a \cdot b \leq -4.8 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{-13}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* z t))))
   (if (<= (* a b) -4.8e+79)
     t_1
     (if (<= (* a b) 1.75e-13) (+ (* x y) (* z t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (z * t);
	double tmp;
	if ((a * b) <= -4.8e+79) {
		tmp = t_1;
	} else if ((a * b) <= 1.75e-13) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (a * b) + (z * t)
    if ((a * b) <= (-4.8d+79)) then
        tmp = t_1
    else if ((a * b) <= 1.75d-13) then
        tmp = (x * y) + (z * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + (z * t);
	double tmp;
	if ((a * b) <= -4.8e+79) {
		tmp = t_1;
	} else if ((a * b) <= 1.75e-13) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + (z * t)
	tmp = 0
	if (a * b) <= -4.8e+79:
		tmp = t_1
	elif (a * b) <= 1.75e-13:
		tmp = (x * y) + (z * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(z * t))
	tmp = 0.0
	if (Float64(a * b) <= -4.8e+79)
		tmp = t_1;
	elseif (Float64(a * b) <= 1.75e-13)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + (z * t);
	tmp = 0.0;
	if ((a * b) <= -4.8e+79)
		tmp = t_1;
	elseif ((a * b) <= 1.75e-13)
		tmp = (x * y) + (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -4.8e+79], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1.75e-13], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.79999999999999971e79 or 1.7500000000000001e-13 < (*.f64 a b)

   1. Initial program 95.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 83.1%

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

   if -4.79999999999999971e79 < (*.f64 a b) < 1.7500000000000001e-13

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(t \cdot z\right), \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \left(\color{blue}{x} \cdot y\right)\right) \]

      3. *-lowering-*.f64 89.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   5. Simplified 89.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.8 \cdot 10^{+79}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{-13}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.6 \cdot 10^{+164}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 8.6 \cdot 10^{+68}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2.6e+164)
   (* x y)
   (if (<= (* x y) 8.6e+68) (+ (* 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 * y) <= -2.6e+164) {
		tmp = x * y;
	} else if ((x * y) <= 8.6e+68) {
		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 * y) <= (-2.6d+164)) then
        tmp = x * y
    else if ((x * y) <= 8.6d+68) 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 * y) <= -2.6e+164) {
		tmp = x * y;
	} else if ((x * y) <= 8.6e+68) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -2.6e+164:
		tmp = x * y
	elif (x * y) <= 8.6e+68:
		tmp = (a * b) + (z * t)
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2.6e+164)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 8.6e+68)
		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 * y) <= -2.6e+164)
		tmp = x * y;
	elseif ((x * y) <= 8.6e+68)
		tmp = (a * b) + (z * t);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2.6e+164], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 8.6e+68], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.5999999999999999e164 or 8.6000000000000002e68 < (*.f64 x y)

   1. Initial program 94.6%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 74.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 74.6%

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

   if -2.5999999999999999e164 < (*.f64 x y) < 8.6000000000000002e68

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(t \cdot z\right)}, \mathsf{*.f64}\left(a, b\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(t, z\right), \mathsf{*.f64}\left(\color{blue}{a}, b\right)\right) \]

   5. Simplified 85.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.6 \cdot 10^{+164}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 8.6 \cdot 10^{+68}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.8 \cdot 10^{+181}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 20:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -4.8e+181) (* a b) (if (<= (* a b) 20.0) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -4.8e+181) {
		tmp = a * b;
	} else if ((a * b) <= 20.0) {
		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) <= (-4.8d+181)) then
        tmp = a * b
    else if ((a * b) <= 20.0d0) 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) <= -4.8e+181) {
		tmp = a * b;
	} else if ((a * b) <= 20.0) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -4.8e+181:
		tmp = a * b
	elif (a * b) <= 20.0:
		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) <= -4.8e+181)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 20.0)
		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) <= -4.8e+181)
		tmp = a * b;
	elseif ((a * b) <= 20.0)
		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], -4.8e+181], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 20.0], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.80000000000000004e181 or 20 < (*.f64 a b)

   1. Initial program 95.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot b} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

   5. Simplified 75.0%

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

   if -4.80000000000000004e181 < (*.f64 a b) < 20

   1. Initial program 99.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 43.2%

         \[\leadsto \mathsf{*.f64}\left(t, \color{blue}{z}\right) \]

   5. Simplified 43.2%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.8 \cdot 10^{+181}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 20:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.8% 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. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot b} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 39.4%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{b}\right) \]

5. Simplified 39.4%

   \[\leadsto \color{blue}{a \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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)))