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

Percentage Accurate: 98.0% → 99.1%

Time: 5.2s

Alternatives: 8

Speedup: 1.2×

• 32.4% 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% 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 98.8%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
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: 54.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.85 \cdot 10^{-35}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq -9.5 \cdot 10^{-183}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t \cdot z \leq 8.6 \cdot 10^{-275}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t \cdot z \leq 8.2 \cdot 10^{+37}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* t z) -1.85e-35)
   (* t z)
   (if (<= (* t z) -9.5e-183)
     (* x y)
     (if (<= (* t z) 8.6e-275)
       (* a b)
       (if (<= (* t z) 8.2e+37) (* x y) (* t z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t * z) <= -1.85e-35) {
		tmp = t * z;
	} else if ((t * z) <= -9.5e-183) {
		tmp = x * y;
	} else if ((t * z) <= 8.6e-275) {
		tmp = a * b;
	} else if ((t * z) <= 8.2e+37) {
		tmp = x * y;
	} 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) <= (-1.85d-35)) then
        tmp = t * z
    else if ((t * z) <= (-9.5d-183)) then
        tmp = x * y
    else if ((t * z) <= 8.6d-275) then
        tmp = a * b
    else if ((t * z) <= 8.2d+37) then
        tmp = x * y
    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) <= -1.85e-35) {
		tmp = t * z;
	} else if ((t * z) <= -9.5e-183) {
		tmp = x * y;
	} else if ((t * z) <= 8.6e-275) {
		tmp = a * b;
	} else if ((t * z) <= 8.2e+37) {
		tmp = x * y;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t * z) <= -1.85e-35:
		tmp = t * z
	elif (t * z) <= -9.5e-183:
		tmp = x * y
	elif (t * z) <= 8.6e-275:
		tmp = a * b
	elif (t * z) <= 8.2e+37:
		tmp = x * y
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t * z) <= -1.85e-35)
		tmp = Float64(t * z);
	elseif (Float64(t * z) <= -9.5e-183)
		tmp = Float64(x * y);
	elseif (Float64(t * z) <= 8.6e-275)
		tmp = Float64(a * b);
	elseif (Float64(t * z) <= 8.2e+37)
		tmp = Float64(x * y);
	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) <= -1.85e-35)
		tmp = t * z;
	elseif ((t * z) <= -9.5e-183)
		tmp = x * y;
	elseif ((t * z) <= 8.6e-275)
		tmp = a * b;
	elseif ((t * z) <= 8.2e+37)
		tmp = x * y;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(t * z), $MachinePrecision], -1.85e-35], N[(t * z), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], -9.5e-183], N[(x * y), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 8.6e-275], N[(a * b), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 8.2e+37], N[(x * y), $MachinePrecision], N[(t * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.8499999999999999e-35 or 8.1999999999999996e37 < (*.f64 z t)

   1. Initial program 97.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -1.8499999999999999e-35 < (*.f64 z t) < -9.5000000000000008e-183 or 8.59999999999999953e-275 < (*.f64 z t) < 8.1999999999999996e37

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

   if -9.5000000000000008e-183 < (*.f64 z t) < 8.59999999999999953e-275

   1. Initial program 100.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.85 \cdot 10^{-35}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq -9.5 \cdot 10^{-183}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t \cdot z \leq 8.6 \cdot 10^{-275}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;t \cdot z \leq 8.2 \cdot 10^{+37}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -4e+78)
   (fma y x (* t z))
   (if (<= (* x y) 4e+26) (fma z t (* a b)) (fma b a (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -4e+78) {
		tmp = fma(y, x, (t * z));
	} else if ((x * y) <= 4e+26) {
		tmp = fma(z, t, (a * b));
	} else {
		tmp = fma(b, a, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -4e+78)
		tmp = fma(y, x, Float64(t * z));
	elseif (Float64(x * y) <= 4e+26)
		tmp = fma(z, t, Float64(a * b));
	else
		tmp = fma(b, a, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -4e+78], N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+26], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.00000000000000003e78

   1. Initial program 96.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if -4.00000000000000003e78 < (*.f64 x y) < 4.00000000000000019e26

   1. Initial program 99.3%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 93.2

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

   7. Applied rewrites 93.2%

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

   if 4.00000000000000019e26 < (*.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in t 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -4e+78)
   (fma y x (* t z))
   (if (<= (* x y) 4e+26) (fma b a (* t z)) (fma b a (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -4e+78) {
		tmp = fma(y, x, (t * z));
	} else if ((x * y) <= 4e+26) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = fma(b, a, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -4e+78)
		tmp = fma(y, x, Float64(t * z));
	elseif (Float64(x * y) <= 4e+26)
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = fma(b, a, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -4e+78], N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+26], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.00000000000000003e78

   1. Initial program 96.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if -4.00000000000000003e78 < (*.f64 x y) < 4.00000000000000019e26

   1. Initial program 99.3%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      4. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 4.00000000000000019e26 < (*.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in t 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 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{if}\;t \cdot z \leq -1.85 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \cdot z \leq 8.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b a (* t z))))
   (if (<= (* t z) -1.85e-35)
     t_1
     (if (<= (* t z) 8.2e+37) (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) <= -1.85e-35) {
		tmp = t_1;
	} else if ((t * z) <= 8.2e+37) {
		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) <= -1.85e-35)
		tmp = t_1;
	elseif (Float64(t * z) <= 8.2e+37)
		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], -1.85e-35], t$95$1, If[LessEqual[N[(t * z), $MachinePrecision], 8.2e+37], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.8499999999999999e-35 or 8.1999999999999996e37 < (*.f64 z t)

   1. Initial program 97.6%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      4. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -1.8499999999999999e-35 < (*.f64 z t) < 8.1999999999999996e37

   1. Initial program 100.0%

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

   3. Taylor expanded in t 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 92.0

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

   5. Applied rewrites 92.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.85 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{elif}\;t \cdot z \leq 8.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -7 \cdot 10^{+197}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 2.2 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* t z) -7e+197)
   (* t z)
   (if (<= (* t z) 2.2e+193) (fma b a (* x y)) (* t z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t * z) <= -7e+197) {
		tmp = t * z;
	} else if ((t * z) <= 2.2e+193) {
		tmp = fma(b, a, (x * y));
	} else {
		tmp = t * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(t * z) <= -7e+197)
		tmp = Float64(t * z);
	elseif (Float64(t * z) <= 2.2e+193)
		tmp = fma(b, a, Float64(x * y));
	else
		tmp = Float64(t * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(t * z), $MachinePrecision], -7e+197], N[(t * z), $MachinePrecision], If[LessEqual[N[(t * z), $MachinePrecision], 2.2e+193], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(t * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -6.99999999999999999e197 or 2.19999999999999986e193 < (*.f64 z t)

   1. Initial program 95.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -6.99999999999999999e197 < (*.f64 z t) < 2.19999999999999986e193

   1. Initial program 100.0%

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

   3. Taylor expanded in t 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 80.9

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

   5. Applied rewrites 80.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -7 \cdot 10^{+197}:\\ \;\;\;\;t \cdot z\\ \mathbf{elif}\;t \cdot z \leq 2.2 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+93}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2000000000:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2e+93)
   (* x y)
   (if (<= (* x y) 2000000000.0) (* a b) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e+93) {
		tmp = x * y;
	} else if ((x * y) <= 2000000000.0) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((x * y) <= (-2d+93)) then
        tmp = x * y
    else if ((x * y) <= 2000000000.0d0) then
        tmp = a * b
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2e+93) {
		tmp = x * y;
	} else if ((x * y) <= 2000000000.0) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -2e+93:
		tmp = x * y
	elif (x * y) <= 2000000000.0:
		tmp = a * b
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2e+93)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 2000000000.0)
		tmp = Float64(a * b);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -2e+93)
		tmp = x * y;
	elseif ((x * y) <= 2000000000.0)
		tmp = a * b;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+93], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2000000000.0], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000009e93 or 2e9 < (*.f64 x y)

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -2.00000000000000009e93 < (*.f64 x y) < 2e9

   1. Initial program 98.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 47.9

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

   5. Applied rewrites 47.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+93}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2000000000:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 33.7

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

5. Applied rewrites 33.7%

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

6. Final simplification 33.7%

   \[\leadsto a \cdot b \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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)))