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

Percentage Accurate: 97.7% → 98.9%

Time: 6.5s

Alternatives: 9

Speedup: 1.2×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
2. 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. lift-*.f64 N/A

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

   8. lower-fma.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 53.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2900000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 9 \cdot 10^{-294}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{+93}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2900000.0)
   (* a b)
   (if (<= (* a b) 9e-294)
     (* y x)
     (if (<= (* a b) 1.75e+93) (* z t) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2900000.0) {
		tmp = a * b;
	} else if ((a * b) <= 9e-294) {
		tmp = y * x;
	} else if ((a * b) <= 1.75e+93) {
		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) <= (-2900000.0d0)) then
        tmp = a * b
    else if ((a * b) <= 9d-294) then
        tmp = y * x
    else if ((a * b) <= 1.75d+93) 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) <= -2900000.0) {
		tmp = a * b;
	} else if ((a * b) <= 9e-294) {
		tmp = y * x;
	} else if ((a * b) <= 1.75e+93) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2900000.0:
		tmp = a * b
	elif (a * b) <= 9e-294:
		tmp = y * x
	elif (a * b) <= 1.75e+93:
		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) <= -2900000.0)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 9e-294)
		tmp = Float64(y * x);
	elseif (Float64(a * b) <= 1.75e+93)
		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) <= -2900000.0)
		tmp = a * b;
	elseif ((a * b) <= 9e-294)
		tmp = y * x;
	elseif ((a * b) <= 1.75e+93)
		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], -2900000.0], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 9e-294], N[(y * x), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.75e+93], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.9e6 or 1.74999999999999999e93 < (*.f64 a b)

   1. Initial program 98.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -2.9e6 < (*.f64 a b) < 8.99999999999999963e-294

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

   if 8.99999999999999963e-294 < (*.f64 a b) < 1.74999999999999999e93

   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. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2900000:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 9 \cdot 10^{-294}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{+93}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2e-32)
   (fma y x (* a b))
   (if (<= (* a b) 5e+86) (fma y x (* z t)) (fma a b (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2e-32) {
		tmp = fma(y, x, (a * b));
	} else if ((a * b) <= 5e+86) {
		tmp = fma(y, x, (z * t));
	} else {
		tmp = fma(a, b, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2e-32)
		tmp = fma(y, x, Float64(a * b));
	elseif (Float64(a * b) <= 5e+86)
		tmp = fma(y, x, Float64(z * t));
	else
		tmp = fma(a, b, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e-32], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+86], N[(y * x + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.00000000000000011e-32

   1. Initial program 100.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lift-*.f64 N/A

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

      8. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 85.8

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

   7. Applied rewrites 85.8%

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

   if -2.00000000000000011e-32 < (*.f64 a b) < 4.9999999999999998e86

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lift-*.f64 N/A

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

      8. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

   if 4.9999999999999998e86 < (*.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 z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2e-32)
   (fma y x (* a b))
   (if (<= (* a b) 5e+86) (fma t z (* y x)) (fma a b (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2e-32) {
		tmp = fma(y, x, (a * b));
	} else if ((a * b) <= 5e+86) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = fma(a, b, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2e-32)
		tmp = fma(y, x, Float64(a * b));
	elseif (Float64(a * b) <= 5e+86)
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = fma(a, b, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e-32], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+86], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.00000000000000011e-32

   1. Initial program 100.0%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lift-*.f64 N/A

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

      8. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 85.8

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

   7. Applied rewrites 85.8%

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

   if -2.00000000000000011e-32 < (*.f64 a b) < 4.9999999999999998e86

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if 4.9999999999999998e86 < (*.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 z around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a b (* y x))))
   (if (<= (* a b) -2e-32) t_1 (if (<= (* a b) 5e+86) (fma t z (* y x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, b, (y * x));
	double tmp;
	if ((a * b) <= -2e-32) {
		tmp = t_1;
	} else if ((a * b) <= 5e+86) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, b, Float64(y * x))
	tmp = 0.0
	if (Float64(a * b) <= -2e-32)
		tmp = t_1;
	elseif (Float64(a * b) <= 5e+86)
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -2e-32], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 5e+86], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.00000000000000011e-32 or 4.9999999999999998e86 < (*.f64 a b)

   1. Initial program 98.4%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lower-fma.f64 N/A

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

      2. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -2.00000000000000011e-32 < (*.f64 a b) < 4.9999999999999998e86

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{if}\;y \cdot x \leq -4.1 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 4.4 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a b (* y x))))
   (if (<= (* y x) -4.1e+62)
     t_1
     (if (<= (* y x) 4.4e-52) (fma a b (* z t)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, b, (y * x));
	double tmp;
	if ((y * x) <= -4.1e+62) {
		tmp = t_1;
	} else if ((y * x) <= 4.4e-52) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, b, Float64(y * x))
	tmp = 0.0
	if (Float64(y * x) <= -4.1e+62)
		tmp = t_1;
	elseif (Float64(y * x) <= 4.4e-52)
		tmp = fma(a, b, Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -4.1e+62], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 4.4e-52], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.09999999999999984e62 or 4.40000000000000018e-52 < (*.f64 x y)

   1. Initial program 97.6%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. lower-fma.f64 N/A

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

      2. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -4.09999999999999984e62 < (*.f64 x y) < 4.40000000000000018e-52

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4.1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{elif}\;y \cdot x \leq 4.4 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -3.9 \cdot 10^{+169}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 7.6 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* y x) -3.9e+169)
   (* y x)
   (if (<= (* y x) 7.6e+161) (fma a b (* z t)) (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y * x) <= -3.9e+169) {
		tmp = y * x;
	} else if ((y * x) <= 7.6e+161) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(y * x) <= -3.9e+169)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= 7.6e+161)
		tmp = fma(a, b, Float64(z * t));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(y * x), $MachinePrecision], -3.9e+169], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 7.6e+161], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.89999999999999983e169 or 7.6000000000000005e161 < (*.f64 x y)

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if -3.89999999999999983e169 < (*.f64 x y) < 7.6000000000000005e161

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -3.9 \cdot 10^{+169}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 7.6 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.2 \cdot 10^{-53}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{+93}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -5.2e-53) (* a b) (if (<= (* a b) 1.75e+93) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -5.2e-53) {
		tmp = a * b;
	} else if ((a * b) <= 1.75e+93) {
		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) <= (-5.2d-53)) then
        tmp = a * b
    else if ((a * b) <= 1.75d+93) 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) <= -5.2e-53) {
		tmp = a * b;
	} else if ((a * b) <= 1.75e+93) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -5.2e-53:
		tmp = a * b
	elif (a * b) <= 1.75e+93:
		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) <= -5.2e-53)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.75e+93)
		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) <= -5.2e-53)
		tmp = a * b;
	elseif ((a * b) <= 1.75e+93)
		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], -5.2e-53], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.75e+93], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.19999999999999993e-53 or 1.74999999999999999e93 < (*.f64 a b)

   1. Initial program 98.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if -5.19999999999999993e-53 < (*.f64 a b) < 1.74999999999999999e93

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.2 \cdot 10^{-53}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{+93}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.6% 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 a around inf

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

   1. lower-*.f64 37.8

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

5. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2024233 
(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)))