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

Percentage Accurate: 98.1% → 99.0%

Time: 8.1s

Alternatives: 7

Speedup: 1.2×

• 33.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% 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

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

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 98.0%

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

   1. associate-+l+ N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 53.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+31}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-175}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 50000000:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+71}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1e+31)
   (* a b)
   (if (<= (* a b) -2e-175)
     (* y x)
     (if (<= (* a b) 50000000.0)
       (* z t)
       (if (<= (* a b) 2e+71) (* y x) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1e+31) {
		tmp = a * b;
	} else if ((a * b) <= -2e-175) {
		tmp = y * x;
	} else if ((a * b) <= 50000000.0) {
		tmp = z * t;
	} else if ((a * b) <= 2e+71) {
		tmp = y * x;
	} 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) <= (-1d+31)) then
        tmp = a * b
    else if ((a * b) <= (-2d-175)) then
        tmp = y * x
    else if ((a * b) <= 50000000.0d0) then
        tmp = z * t
    else if ((a * b) <= 2d+71) then
        tmp = y * x
    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) <= -1e+31) {
		tmp = a * b;
	} else if ((a * b) <= -2e-175) {
		tmp = y * x;
	} else if ((a * b) <= 50000000.0) {
		tmp = z * t;
	} else if ((a * b) <= 2e+71) {
		tmp = y * x;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -1e+31:
		tmp = a * b
	elif (a * b) <= -2e-175:
		tmp = y * x
	elif (a * b) <= 50000000.0:
		tmp = z * t
	elif (a * b) <= 2e+71:
		tmp = y * x
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -1e+31)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -2e-175)
		tmp = Float64(y * x);
	elseif (Float64(a * b) <= 50000000.0)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 2e+71)
		tmp = Float64(y * x);
	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) <= -1e+31)
		tmp = a * b;
	elseif ((a * b) <= -2e-175)
		tmp = y * x;
	elseif ((a * b) <= 50000000.0)
		tmp = z * t;
	elseif ((a * b) <= 2e+71)
		tmp = y * x;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -1e+31], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2e-175], N[(y * x), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 50000000.0], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+71], N[(y * x), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -9.9999999999999996e30 or 2.0000000000000001e71 < (*.f64 a b)

   1. Initial program 95.7%

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

   3. Taylor expanded in a around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{a \cdot b + 0} \]

      2. accelerator-lowering-fma.f64 70.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]

   5. Simplified 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 70.5

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

   7. Applied egg-rr 70.5%

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

   if -9.9999999999999996e30 < (*.f64 a b) < -2e-175 or 5e7 < (*.f64 a b) < 2.0000000000000001e71

   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 inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 58.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 58.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 58.0

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

   7. Applied egg-rr 58.0%

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

   if -2e-175 < (*.f64 a b) < 5e7

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{t \cdot z + 0} \]

      2. accelerator-lowering-fma.f64 56.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, 0\right)} \]

   5. Simplified 56.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 56.0

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

   7. Applied egg-rr 56.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+31}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-175}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;a \cdot b \leq 50000000:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+71}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{elif}\;a \cdot b \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1e+31)
   (fma a b (* z t))
   (if (<= (* a b) 50000000.0) (fma y x (* z t)) (fma y x (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1e+31) {
		tmp = fma(a, b, (z * t));
	} else if ((a * b) <= 50000000.0) {
		tmp = fma(y, x, (z * t));
	} else {
		tmp = fma(y, x, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -1e+31)
		tmp = fma(a, b, Float64(z * t));
	elseif (Float64(a * b) <= 50000000.0)
		tmp = fma(y, x, Float64(z * t));
	else
		tmp = fma(y, x, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -1e+31], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 50000000.0], N[(y * x + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -9.9999999999999996e30

   1. Initial program 92.7%

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

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

      2. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{t \cdot z + 0}\right) \]

      3. accelerator-lowering-fma.f64 85.6

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{\mathsf{fma}\left(t, z, 0\right)}\right) \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, \mathsf{fma}\left(t, z, 0\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 85.6

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

   7. Applied egg-rr 85.6%

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

   if -9.9999999999999996e30 < (*.f64 a b) < 5e7

   1. Initial program 99.3%

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

      1. associate-+l+ N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 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. *-lowering-*.f64 91.3

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

   7. Simplified 91.3%

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

   if 5e7 < (*.f64 a b)

   1. Initial program 98.4%

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

      1. associate-+l+ N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 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. *-lowering-*.f64 90.6

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

   7. Simplified 90.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{elif}\;a \cdot b \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y x (* a b))))
   (if (<= (* y x) -4e+51) t_1 (if (<= (* y x) 1e+51) (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(y, x, (a * b));
	double tmp;
	if ((y * x) <= -4e+51) {
		tmp = t_1;
	} else if ((y * x) <= 1e+51) {
		tmp = fma(a, b, (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, x, Float64(a * b))
	tmp = 0.0
	if (Float64(y * x) <= -4e+51)
		tmp = t_1;
	elseif (Float64(y * x) <= 1e+51)
		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[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -4e+51], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 1e+51], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4e51 or 1e51 < (*.f64 x y)

   1. Initial program 95.4%

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

      1. associate-+l+ N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 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. *-lowering-*.f64 85.7

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

   7. Simplified 85.7%

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

   if -4e51 < (*.f64 x y) < 1e51

   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. accelerator-lowering-fma.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{t \cdot z + 0}\right) \]

      3. accelerator-lowering-fma.f64 89.0

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{\mathsf{fma}\left(t, z, 0\right)}\right) \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, \mathsf{fma}\left(t, z, 0\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 89.0

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

   7. Applied egg-rr 89.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -4 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{elif}\;y \cdot x \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{+227}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* y x) -2e+227)
   (* y x)
   (if (<= (* y x) 1e+51) (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) <= -2e+227) {
		tmp = y * x;
	} else if ((y * x) <= 1e+51) {
		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) <= -2e+227)
		tmp = Float64(y * x);
	elseif (Float64(y * x) <= 1e+51)
		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], -2e+227], N[(y * x), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1e+51], N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.0000000000000002e227 or 1e51 < (*.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. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 74.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 74.8

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

   7. Applied egg-rr 74.8%

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

   if -2.0000000000000002e227 < (*.f64 x y) < 1e51

   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. accelerator-lowering-fma.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{t \cdot z + 0}\right) \]

      3. accelerator-lowering-fma.f64 83.0

         \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{\mathsf{fma}\left(t, z, 0\right)}\right) \]

   5. Simplified 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, \mathsf{fma}\left(t, z, 0\right)\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 83.0

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

   7. Applied egg-rr 83.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{+227}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \cdot x \leq 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+31}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+71}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1e+31) (* a b) (if (<= (* a b) 2e+71) (* y x) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1e+31) {
		tmp = a * b;
	} else if ((a * b) <= 2e+71) {
		tmp = y * x;
	} 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) <= (-1d+31)) then
        tmp = a * b
    else if ((a * b) <= 2d+71) then
        tmp = y * x
    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) <= -1e+31) {
		tmp = a * b;
	} else if ((a * b) <= 2e+71) {
		tmp = y * x;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -1e+31:
		tmp = a * b
	elif (a * b) <= 2e+71:
		tmp = y * x
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -1e+31)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 2e+71)
		tmp = Float64(y * x);
	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) <= -1e+31)
		tmp = a * b;
	elseif ((a * b) <= 2e+71)
		tmp = y * x;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -1e+31], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+71], N[(y * x), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -9.9999999999999996e30 or 2.0000000000000001e71 < (*.f64 a b)

   1. Initial program 95.7%

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

   3. Taylor expanded in a around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{a \cdot b + 0} \]

      2. accelerator-lowering-fma.f64 70.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]

   5. Simplified 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 70.5

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

   7. Applied egg-rr 70.5%

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

   if -9.9999999999999996e30 < (*.f64 a b) < 2.0000000000000001e71

   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 inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 48.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 48.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 48.1

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

   7. Applied egg-rr 48.1%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+31}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+71}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.9% 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 98.0%

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

3. Taylor expanded in a around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{a \cdot b + 0} \]

   2. accelerator-lowering-fma.f64 33.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]

5. Simplified 33.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 33.3

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

7. Applied egg-rr 33.3%

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

8. Final simplification 33.3%

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

Reproduce

herbie shell --seed 2024195 
(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)))