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

Percentage Accurate: 97.6% → 98.8%

Time: 5.9s

Alternatives: 7

Speedup: 1.2×

• 33.8% 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: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.2

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 54.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+26}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 10^{+111}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2e+26)
   (* x y)
   (if (<= (* x y) 0.0) (* a b) (if (<= (* x y) 1e+111) (* t z) (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e+26) {
		tmp = x * y;
	} else if ((x * y) <= 0.0) {
		tmp = a * b;
	} else if ((x * y) <= 1e+111) {
		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) <= (-2d+26)) then
        tmp = x * y
    else if ((x * y) <= 0.0d0) then
        tmp = a * b
    else if ((x * y) <= 1d+111) 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) <= -2e+26) {
		tmp = x * y;
	} else if ((x * y) <= 0.0) {
		tmp = a * b;
	} else if ((x * y) <= 1e+111) {
		tmp = t * z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -2e+26:
		tmp = x * y
	elif (x * y) <= 0.0:
		tmp = a * b
	elif (x * y) <= 1e+111:
		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) <= -2e+26)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 0.0)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 1e+111)
		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) <= -2e+26)
		tmp = x * y;
	elseif ((x * y) <= 0.0)
		tmp = a * b;
	elseif ((x * y) <= 1e+111)
		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], -2e+26], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 0.0], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+111], N[(t * z), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.0000000000000001e26 or 9.99999999999999957e110 < (*.f64 x y)

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 37.1

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 34.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 70.9

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

   11. Applied rewrites 70.9%

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

   if -2.0000000000000001e26 < (*.f64 x y) < 0.0

   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 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.5%

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

   if 0.0 < (*.f64 x y) < 9.99999999999999957e110

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto t \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 52.6%

       \[\leadsto t \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+26}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 0:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 10^{+111}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b a (* t z))))
   (if (<= (* t z) -1e+144)
     t_1
     (if (<= (* t z) 2e+35) (fma b a (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, (t * z));
	double tmp;
	if ((t * z) <= -1e+144) {
		tmp = t_1;
	} else if ((t * z) <= 2e+35) {
		tmp = fma(b, a, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, a, Float64(t * z))
	tmp = 0.0
	if (Float64(t * z) <= -1e+144)
		tmp = t_1;
	elseif (Float64(t * z) <= 2e+35)
		tmp = fma(b, a, Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t * z), $MachinePrecision], -1e+144], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 2e+35], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000002e144 or 1.9999999999999999e35 < (*.f64 z t)

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if -1.00000000000000002e144 < (*.f64 z t) < 1.9999999999999999e35

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+267}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2e+267)
   (* x y)
   (if (<= (* x y) 2e+130) (fma b a (* t z)) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e+267) {
		tmp = x * y;
	} else if ((x * y) <= 2e+130) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2e+267)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 2e+130)
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+267], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+130], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.9999999999999999e267 or 2.0000000000000001e130 < (*.f64 x y)

   1. Initial program 84.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 21.1

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 26.4%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 90.6

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

   11. Applied rewrites 90.6%

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

   if -1.9999999999999999e267 < (*.f64 x y) < 2.0000000000000001e130

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, a, t \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+267}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -4 \cdot 10^{+148}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{-17}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* t z) -4e+148) (* t z) (if (<= (* t z) 5e-17) (* a b) (* t z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t * z) <= -4e+148) {
		tmp = t * z;
	} else if ((t * z) <= 5e-17) {
		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 ((t * z) <= (-4d+148)) then
        tmp = t * z
    else if ((t * z) <= 5d-17) 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 ((t * z) <= -4e+148) {
		tmp = t * z;
	} else if ((t * z) <= 5e-17) {
		tmp = a * b;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t * z) <= -4e+148:
		tmp = t * z
	elif (t * z) <= 5e-17:
		tmp = a * b
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t * z) <= -4e+148)
		tmp = Float64(t * z);
	elseif (Float64(t * z) <= 5e-17)
		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 ((t * z) <= -4e+148)
		tmp = t * z;
	elseif ((t * z) <= 5e-17)
		tmp = a * b;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(t * z), $MachinePrecision], -4e+148], N[(t * z), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 5e-17], N[(a * b), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.0000000000000002e148 or 4.9999999999999999e-17 < (*.f64 z t)

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in z around inf

      \[\leadsto t \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 71.4%

       \[\leadsto t \cdot z \]

   if -4.0000000000000002e148 < (*.f64 z t) < 4.9999999999999999e-17

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.2%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -4 \cdot 10^{+148}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 5 \cdot 10^{-17}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 98.0

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 7: 35.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return 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 = t * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 70.0

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

5. Applied rewrites 70.0%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 59.3%

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

9. Taylor expanded in z around inf

   \[\leadsto t \cdot z \]
10. Step-by-step derivation

11. Applied rewrites 38.2%

    \[\leadsto t \cdot z \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(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)))