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

Percentage Accurate: 97.7% → 98.8%

Time: 6.4s

Alternatives: 7

Speedup: 1.2×

• 33.5% 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: 98.8% 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 97.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. 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 egg-rr 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: 54.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -15200:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-289}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 10^{+69}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -15200.0)
   (* a b)
   (if (<= (* a b) 4e-289) (* y x) (if (<= (* a b) 1e+69) (* z t) (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -15200.0) {
		tmp = a * b;
	} else if ((a * b) <= 4e-289) {
		tmp = y * x;
	} else if ((a * b) <= 1e+69) {
		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) <= (-15200.0d0)) then
        tmp = a * b
    else if ((a * b) <= 4d-289) then
        tmp = y * x
    else if ((a * b) <= 1d+69) 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) <= -15200.0) {
		tmp = a * b;
	} else if ((a * b) <= 4e-289) {
		tmp = y * x;
	} else if ((a * b) <= 1e+69) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -15200.0:
		tmp = a * b
	elif (a * b) <= 4e-289:
		tmp = y * x
	elif (a * b) <= 1e+69:
		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) <= -15200.0)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 4e-289)
		tmp = Float64(y * x);
	elseif (Float64(a * b) <= 1e+69)
		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) <= -15200.0)
		tmp = a * b;
	elseif ((a * b) <= 4e-289)
		tmp = y * x;
	elseif ((a * b) <= 1e+69)
		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], -15200.0], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4e-289], N[(y * x), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+69], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -15200 or 1.0000000000000001e69 < (*.f64 a b)

   1. Initial program 95.4%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 -15200 < (*.f64 a b) < 4e-289

   1. Initial program 97.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 59.5

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

   5. Simplified 59.5%

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

   if 4e-289 < (*.f64 a b) < 1.0000000000000001e69

   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 50.0

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

   5. Simplified 50.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -15200:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 4 \cdot 10^{-289}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 10^{+69}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a b (* z t))))
   (if (<= (* z t) -4e+111)
     t_1
     (if (<= (* z t) 2e+29) (fma y x (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(a, b, (z * t));
	double tmp;
	if ((z * t) <= -4e+111) {
		tmp = t_1;
	} else if ((z * t) <= 2e+29) {
		tmp = fma(y, x, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, b, Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -4e+111)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+29)
		tmp = fma(y, x, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -4e+111], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+29], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.99999999999999983e111 or 1.99999999999999983e29 < (*.f64 z t)

   1. Initial program 95.9%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 90.2

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

   5. Simplified 90.2%

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

   if -3.99999999999999983e111 < (*.f64 z t) < 1.99999999999999983e29

   1. Initial program 98.1%

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

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   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 91.0

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

   7. Simplified 91.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a b (* z t))))
   (if (<= (* z t) -4e+111)
     t_1
     (if (<= (* z t) 2e+29) (fma a b (* 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, (z * t));
	double tmp;
	if ((z * t) <= -4e+111) {
		tmp = t_1;
	} else if ((z * t) <= 2e+29) {
		tmp = fma(a, b, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(a, b, Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -4e+111)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+29)
		tmp = fma(a, b, 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[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -4e+111], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+29], N[(a * b + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.99999999999999983e111 or 1.99999999999999983e29 < (*.f64 z t)

   1. Initial program 95.9%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 90.2

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

   5. Simplified 90.2%

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

   if -3.99999999999999983e111 < (*.f64 z t) < 1.99999999999999983e29

   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 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 90.4

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

   5. Simplified 90.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -7.4 \cdot 10^{+110}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 2.9 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* y x) -7.4e+110)
   (* y x)
   (if (<= (* y x) 2.9e+115) (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) <= -7.4e+110) {
		tmp = y * x;
	} else if ((y * x) <= 2.9e+115) {
		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) <= -7.4e+110)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= 2.9e+115)
		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], -7.4e+110], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 2.9e+115], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -7.40000000000000024e110 or 2.90000000000000005e115 < (*.f64 x y)

   1. Initial program 93.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 77.3

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

   5. Simplified 77.3%

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

   if -7.40000000000000024e110 < (*.f64 x y) < 2.90000000000000005e115

   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 85.9

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

   5. Simplified 85.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -7.4 \cdot 10^{+110}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 2.9 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -7 \cdot 10^{+39}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 10^{+69}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -7e+39) (* a b) (if (<= (* a b) 1e+69) (* z t) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -7e+39) {
		tmp = a * b;
	} else if ((a * b) <= 1e+69) {
		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) <= (-7d+39)) then
        tmp = a * b
    else if ((a * b) <= 1d+69) 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) <= -7e+39) {
		tmp = a * b;
	} else if ((a * b) <= 1e+69) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -7e+39:
		tmp = a * b
	elif (a * b) <= 1e+69:
		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) <= -7e+39)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1e+69)
		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) <= -7e+39)
		tmp = a * b;
	elseif ((a * b) <= 1e+69)
		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], -7e+39], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+69], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -7.0000000000000003e39 or 1.0000000000000001e69 < (*.f64 a b)

   1. Initial program 95.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 74.3

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

   5. Simplified 74.3%

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

   if -7.0000000000000003e39 < (*.f64 a b) < 1.0000000000000001e69

   1. Initial program 98.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 45.6

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

   5. Simplified 45.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -7 \cdot 10^{+39}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 10^{+69}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.9% 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 97.3%

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

3. Taylor expanded in a around inf

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

   1. lower-*.f64 37.1

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

5. Simplified 37.1%

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

Reproduce

herbie shell --seed 2024207 
(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)))