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

Percentage Accurate: 97.7% → 99.0%

Time: 5.6s

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: 97.7% 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: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

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

   8. lower-fma.f64 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 52.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.45 \cdot 10^{-17}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2.45e-17)
   (* x y)
   (if (<= (* x y) 2.1e-92)
     (* z t)
     (if (<= (* x y) 1.05e+160) (* a b) (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2.45e-17) {
		tmp = x * y;
	} else if ((x * y) <= 2.1e-92) {
		tmp = z * t;
	} else if ((x * y) <= 1.05e+160) {
		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) <= (-2.45d-17)) then
        tmp = x * y
    else if ((x * y) <= 2.1d-92) then
        tmp = z * t
    else if ((x * y) <= 1.05d+160) 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) <= -2.45e-17) {
		tmp = x * y;
	} else if ((x * y) <= 2.1e-92) {
		tmp = z * t;
	} else if ((x * y) <= 1.05e+160) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -2.45e-17:
		tmp = x * y
	elif (x * y) <= 2.1e-92:
		tmp = z * t
	elif (x * y) <= 1.05e+160:
		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) <= -2.45e-17)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 2.1e-92)
		tmp = Float64(z * t);
	elseif (Float64(x * y) <= 1.05e+160)
		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) <= -2.45e-17)
		tmp = x * y;
	elseif ((x * y) <= 2.1e-92)
		tmp = z * t;
	elseif ((x * y) <= 1.05e+160)
		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], -2.45e-17], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2.1e-92], N[(z * t), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.05e+160], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.45000000000000006e-17 or 1.04999999999999998e160 < (*.f64 x y)

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if -2.45000000000000006e-17 < (*.f64 x y) < 2.1e-92

   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 53.4

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

   5. Applied rewrites 53.4%

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

   if 2.1e-92 < (*.f64 x y) < 1.04999999999999998e160

   1. Initial program 97.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 49.4

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

   5. Applied rewrites 49.4%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.45 \cdot 10^{-17}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \cdot y \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.22 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5.6 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -1.22e-17)
   (fma t z (* x y))
   (if (<= (* x y) 5.6e+204) (fma a b (* z t)) (fma a b (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -1.22e-17) {
		tmp = fma(t, z, (x * y));
	} else if ((x * y) <= 5.6e+204) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = fma(a, b, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -1.22e-17)
		tmp = fma(t, z, Float64(x * y));
	elseif (Float64(x * y) <= 5.6e+204)
		tmp = fma(a, b, Float64(z * t));
	else
		tmp = fma(a, b, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.22e-17], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5.6e+204], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.22e-17

   1. Initial program 98.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.22e-17 < (*.f64 x y) < 5.60000000000000049e204

   1. Initial program 99.3%

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

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

      2. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 5.60000000000000049e204 < (*.f64 x y)

   1. Initial program 81.3%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 90.6

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

   5. Applied rewrites 90.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.22 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5.6 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -2.45 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5.6 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a b (* x y))))
   (if (<= (* x y) -2.45e-17)
     t_1
     (if (<= (* x y) 5.6e+204) (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, (x * y));
	double tmp;
	if ((x * y) <= -2.45e-17) {
		tmp = t_1;
	} else if ((x * y) <= 5.6e+204) {
		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(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -2.45e-17)
		tmp = t_1;
	elseif (Float64(x * y) <= 5.6e+204)
		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[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.45e-17], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5.6e+204], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.45000000000000006e-17 or 5.60000000000000049e204 < (*.f64 x y)

   1. Initial program 92.9%

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

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

      2. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   if -2.45000000000000006e-17 < (*.f64 x y) < 5.60000000000000049e204

   1. Initial program 99.3%

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

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

      2. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.45 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 5.6 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -8.5 \cdot 10^{+42}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 1.22 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -8.5e+42)
   (* x y)
   (if (<= (* x y) 1.22e+205) (fma a b (* z t)) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -8.5e+42) {
		tmp = x * y;
	} else if ((x * y) <= 1.22e+205) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -8.5e+42)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 1.22e+205)
		tmp = fma(a, b, Float64(z * t));
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -8.5e+42], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.22e+205], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -8.5000000000000003e42 or 1.21999999999999989e205 < (*.f64 x y)

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -8.5000000000000003e42 < (*.f64 x y) < 1.21999999999999989e205

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a \cdot b + t \cdot z} \]
   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 88.8

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

   5. Applied rewrites 88.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -8.5 \cdot 10^{+42}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 1.22 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.8 \cdot 10^{+74}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.06 \cdot 10^{+36}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -5.8e+74) (* a b) (if (<= (* a b) 1.06e+36) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -5.8e+74) {
		tmp = a * b;
	} else if ((a * b) <= 1.06e+36) {
		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) <= (-5.8d+74)) then
        tmp = a * b
    else if ((a * b) <= 1.06d+36) 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) <= -5.8e+74) {
		tmp = a * b;
	} else if ((a * b) <= 1.06e+36) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -5.8e+74:
		tmp = a * b
	elif (a * b) <= 1.06e+36:
		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) <= -5.8e+74)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.06e+36)
		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) <= -5.8e+74)
		tmp = a * b;
	elseif ((a * b) <= 1.06e+36)
		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], -5.8e+74], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.06e+36], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.8000000000000005e74 or 1.06000000000000002e36 < (*.f64 a b)

   1. Initial program 92.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -5.8000000000000005e74 < (*.f64 a b) < 1.06000000000000002e36

   1. Initial program 99.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5.8 \cdot 10^{+74}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.06 \cdot 10^{+36}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% 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 96.9%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 33.3

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

5. Applied rewrites 33.3%

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

Reproduce

herbie shell --seed 2024228 
(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)))