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

Percentage Accurate: 97.7% → 98.8%

Time: 7.1s

Alternatives: 8

Speedup: 1.2×

• 32.9% 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: 98.8% 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

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. 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 egg-rr 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: 53.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.2 \cdot 10^{+44}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -2.8 \cdot 10^{-229}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 3.8 \cdot 10^{+82}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -2.2e+44)
   (* x y)
   (if (<= (* x y) -2.8e-229)
     (* a b)
     (if (<= (* x y) 3.8e+82) (* z t) (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2.2e+44) {
		tmp = x * y;
	} else if ((x * y) <= -2.8e-229) {
		tmp = a * b;
	} else if ((x * y) <= 3.8e+82) {
		tmp = z * t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((x * y) <= (-2.2d+44)) then
        tmp = x * y
    else if ((x * y) <= (-2.8d-229)) then
        tmp = a * b
    else if ((x * y) <= 3.8d+82) then
        tmp = z * t
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -2.2e+44) {
		tmp = x * y;
	} else if ((x * y) <= -2.8e-229) {
		tmp = a * b;
	} else if ((x * y) <= 3.8e+82) {
		tmp = z * t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -2.2e+44:
		tmp = x * y
	elif (x * y) <= -2.8e-229:
		tmp = a * b
	elif (x * y) <= 3.8e+82:
		tmp = z * t
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -2.2e+44)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= -2.8e-229)
		tmp = Float64(a * b);
	elseif (Float64(x * y) <= 3.8e+82)
		tmp = Float64(z * t);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -2.2e+44)
		tmp = x * y;
	elseif ((x * y) <= -2.8e-229)
		tmp = a * b;
	elseif ((x * y) <= 3.8e+82)
		tmp = z * t;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -2.2e+44], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -2.8e-229], N[(a * b), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 3.8e+82], N[(z * t), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.19999999999999996e44 or 3.80000000000000033e82 < (*.f64 x y)

   1. Initial program 94.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 69.7

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

   5. Simplified 69.7%

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

   if -2.19999999999999996e44 < (*.f64 x y) < -2.7999999999999999e-229

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 55.3

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

   5. Simplified 55.3%

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

   if -2.7999999999999999e-229 < (*.f64 x y) < 3.80000000000000033e82

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 59.5

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

   5. Simplified 59.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.2 \cdot 10^{+44}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq -2.8 \cdot 10^{-229}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;x \cdot y \leq 3.8 \cdot 10^{+82}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1e+38)
   (fma y x (* a b))
   (if (<= (* a b) 2e+50) (fma t z (* x y)) (fma a b (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1e+38) {
		tmp = fma(y, x, (a * b));
	} else if ((a * b) <= 2e+50) {
		tmp = fma(t, z, (x * y));
	} else {
		tmp = fma(a, b, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -1e+38)
		tmp = fma(y, x, Float64(a * b));
	elseif (Float64(a * b) <= 2e+50)
		tmp = fma(t, z, Float64(x * y));
	else
		tmp = fma(a, b, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -1e+38], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+50], N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -9.99999999999999977e37

   1. Initial program 90.7%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 84.3

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

   5. Simplified 84.3%

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

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 89.0

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

   7. Applied egg-rr 89.0%

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

   if -9.99999999999999977e37 < (*.f64 a b) < 2.0000000000000002e50

   1. Initial program 99.3%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 91.7

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

   5. Simplified 91.7%

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

   if 2.0000000000000002e50 < (*.f64 a b)

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 89.9

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

   5. Simplified 89.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -2.2 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma t z (* x y))))
   (if (<= (* x y) -2.2e+44)
     t_1
     (if (<= (* x y) 40000000000.0) (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(t, z, (x * y));
	double tmp;
	if ((x * y) <= -2.2e+44) {
		tmp = t_1;
	} else if ((x * y) <= 40000000000.0) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(t, z, Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -2.2e+44)
		tmp = t_1;
	elseif (Float64(x * y) <= 40000000000.0)
		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[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2.2e+44], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 40000000000.0], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.19999999999999996e44 or 4e10 < (*.f64 x y)

   1. Initial program 94.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 85.0

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

   5. Simplified 85.0%

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

   if -2.19999999999999996e44 < (*.f64 x y) < 4e10

   1. Initial program 99.2%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 94.6

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

   5. Simplified 94.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.2 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.4 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 3.8 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -3.4e+178)
   (* x y)
   (if (<= (* x y) 3.8e+82) (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) <= -3.4e+178) {
		tmp = x * y;
	} else if ((x * y) <= 3.8e+82) {
		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) <= -3.4e+178)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 3.8e+82)
		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], -3.4e+178], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 3.8e+82], 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) < -3.4000000000000003e178

   1. Initial program 90.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 88.3

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

   5. Simplified 88.3%

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

   if -3.4000000000000003e178 < (*.f64 x y) < 3.80000000000000033e82

   1. Initial program 98.8%

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

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

      2. lower-*.f64 89.9

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

   5. Simplified 89.9%

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

   if 3.80000000000000033e82 < (*.f64 x y)

   1. Initial program 93.6%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 83.6

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

   5. Simplified 83.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.4 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 3.8 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.4 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 6 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -3.4e+178)
   (* x y)
   (if (<= (* x y) 6e+195) (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) <= -3.4e+178) {
		tmp = x * y;
	} else if ((x * y) <= 6e+195) {
		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) <= -3.4e+178)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 6e+195)
		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], -3.4e+178], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 6e+195], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.4000000000000003e178 or 6.0000000000000001e195 < (*.f64 x y)

   1. Initial program 91.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 84.7

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

   5. Simplified 84.7%

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

   if -3.4000000000000003e178 < (*.f64 x y) < 6.0000000000000001e195

   1. Initial program 98.9%

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

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

      2. lower-*.f64 87.9

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

   5. Simplified 87.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.4 \cdot 10^{+178}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 6 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.4 \cdot 10^{+28}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1.4e+28) (* a b) (if (<= (* a b) 1.9e+99) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1.4e+28) {
		tmp = a * b;
	} else if ((a * b) <= 1.9e+99) {
		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) <= (-1.4d+28)) then
        tmp = a * b
    else if ((a * b) <= 1.9d+99) 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) <= -1.4e+28) {
		tmp = a * b;
	} else if ((a * b) <= 1.9e+99) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -1.4e+28:
		tmp = a * b
	elif (a * b) <= 1.9e+99:
		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) <= -1.4e+28)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.9e+99)
		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) <= -1.4e+28)
		tmp = a * b;
	elseif ((a * b) <= 1.9e+99)
		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], -1.4e+28], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.9e+99], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.4000000000000001e28 or 1.9e99 < (*.f64 a b)

   1. Initial program 92.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 70.6

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

   5. Simplified 70.6%

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

   if -1.4000000000000001e28 < (*.f64 a b) < 1.9e99

   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 54.2

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

   5. Simplified 54.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.4 \cdot 10^{+28}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.2% 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 34.0

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

5. Simplified 34.0%

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

Reproduce

herbie shell --seed 2024208 
(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)))