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

Percentage Accurate: 98.0% → 99.0%

Time: 9.1s

Alternatives: 8

Speedup: 1.2×

• 33.2% 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: 98.0% 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 98.8%

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

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

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

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

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

   5. *-lowering-*.f64 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 2: 54.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+127}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -4 \cdot 10^{+45}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 10^{-181}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+122}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2e+127)
   (* a b)
   (if (<= (* a b) -4e+45)
     (* z t)
     (if (<= (* a b) 1e-181)
       (* x y)
       (if (<= (* a b) 5e+122) (* z t) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2e+127) {
		tmp = a * b;
	} else if ((a * b) <= -4e+45) {
		tmp = z * t;
	} else if ((a * b) <= 1e-181) {
		tmp = x * y;
	} else if ((a * b) <= 5e+122) {
		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) <= (-2d+127)) then
        tmp = a * b
    else if ((a * b) <= (-4d+45)) then
        tmp = z * t
    else if ((a * b) <= 1d-181) then
        tmp = x * y
    else if ((a * b) <= 5d+122) 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) <= -2e+127) {
		tmp = a * b;
	} else if ((a * b) <= -4e+45) {
		tmp = z * t;
	} else if ((a * b) <= 1e-181) {
		tmp = x * y;
	} else if ((a * b) <= 5e+122) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2e+127:
		tmp = a * b
	elif (a * b) <= -4e+45:
		tmp = z * t
	elif (a * b) <= 1e-181:
		tmp = x * y
	elif (a * b) <= 5e+122:
		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) <= -2e+127)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -4e+45)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 1e-181)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 5e+122)
		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) <= -2e+127)
		tmp = a * b;
	elseif ((a * b) <= -4e+45)
		tmp = z * t;
	elseif ((a * b) <= 1e-181)
		tmp = x * y;
	elseif ((a * b) <= 5e+122)
		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], -2e+127], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -4e+45], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e-181], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+122], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.99999999999999991e127 or 4.99999999999999989e122 < (*.f64 a b)

   1. Initial program 96.8%

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

   3. Taylor expanded in a around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 78.0

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

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]
   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 78.0

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

   7. Applied egg-rr 78.0%

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

   if -1.99999999999999991e127 < (*.f64 a b) < -3.9999999999999997e45 or 1.00000000000000005e-181 < (*.f64 a b) < 4.99999999999999989e122

   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. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 52.6

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

   5. Simplified 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, 0\right)} \]
   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 52.6

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

   7. Applied egg-rr 52.6%

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

   if -3.9999999999999997e45 < (*.f64 a b) < 1.00000000000000005e-181

   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 57.6

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

   5. Simplified 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   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 57.6

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

   7. Applied egg-rr 57.6%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+127}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -4 \cdot 10^{+45}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 10^{-181}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+122}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;z \cdot t \leq 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -0.005)
   (fma z t (* a b))
   (if (<= (* z t) 1e+107) (fma y x (* a b)) (fma z t (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -0.005) {
		tmp = fma(z, t, (a * b));
	} else if ((z * t) <= 1e+107) {
		tmp = fma(y, x, (a * b));
	} else {
		tmp = fma(z, t, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -0.005)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(z * t) <= 1e+107)
		tmp = fma(y, x, Float64(a * b));
	else
		tmp = fma(z, t, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -0.005], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+107], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -0.0050000000000000001

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 88.6

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

   7. Simplified 88.6%

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

   if -0.0050000000000000001 < (*.f64 z t) < 9.9999999999999997e106

   1. Initial program 98.7%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around 0

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

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

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

      2. *-lowering-*.f64 90.2

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

   7. Simplified 90.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 90.2

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

   9. Applied egg-rr 90.2%

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

   if 9.9999999999999997e106 < (*.f64 z t)

   1. Initial program 97.5%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 97.5

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

   7. Simplified 97.5%

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

Alternative 4: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{if}\;z \cdot t \leq -0.005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z t (* a b))))
   (if (<= (* z t) -0.005) t_1 (if (<= (* z t) 1e-28) (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(z, t, (a * b));
	double tmp;
	if ((z * t) <= -0.005) {
		tmp = t_1;
	} else if ((z * t) <= 1e-28) {
		tmp = fma(y, x, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, t, Float64(a * b))
	tmp = 0.0
	if (Float64(z * t) <= -0.005)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e-28)
		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[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -0.005], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e-28], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -0.0050000000000000001 or 9.99999999999999971e-29 < (*.f64 z t)

   1. Initial program 99.2%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 85.6

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

   7. Simplified 85.6%

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

   if -0.0050000000000000001 < (*.f64 z t) < 9.99999999999999971e-29

   1. Initial program 98.5%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. *-lowering-*.f64 99.2

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in z around 0

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

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

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

      2. *-lowering-*.f64 93.5

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

   7. Simplified 93.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 93.5

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

   9. Applied egg-rr 93.5%

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

Alternative 5: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -6.5 \cdot 10^{+78}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(y, x, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -6.5e+78)
   (* z t)
   (if (<= (* z t) 5e+142) (fma y x (* a b)) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -6.5e+78) {
		tmp = z * t;
	} else if ((z * t) <= 5e+142) {
		tmp = fma(y, x, (a * b));
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -6.5e+78)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 5e+142)
		tmp = fma(y, x, Float64(a * b));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -6.5e+78], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+142], N[(y * x + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -6.50000000000000036e78 or 5.0000000000000001e142 < (*.f64 z t)

   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. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 77.5

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

   5. Simplified 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, 0\right)} \]
   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 77.5

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

   7. Applied egg-rr 77.5%

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

   if -6.50000000000000036e78 < (*.f64 z t) < 5.0000000000000001e142

   1. Initial program 98.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around 0

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

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

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

      2. *-lowering-*.f64 86.6

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

   7. Simplified 86.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 86.6

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

   9. Applied egg-rr 86.6%

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

Alternative 6: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -6.5 \cdot 10^{+78}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -6.5e+78)
   (* z t)
   (if (<= (* z t) 5e+142) (fma a b (* x y)) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -6.5e+78) {
		tmp = z * t;
	} else if ((z * t) <= 5e+142) {
		tmp = fma(a, b, (x * y));
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -6.5e+78)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 5e+142)
		tmp = fma(a, b, Float64(x * y));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -6.5e+78], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+142], N[(a * b + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -6.50000000000000036e78 or 5.0000000000000001e142 < (*.f64 z t)

   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. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 77.5

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

   5. Simplified 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, 0\right)} \]
   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 77.5

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

   7. Applied egg-rr 77.5%

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

   if -6.50000000000000036e78 < (*.f64 z t) < 5.0000000000000001e142

   1. Initial program 98.9%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

      5. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around 0

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

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

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

      2. *-lowering-*.f64 86.6

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

   7. Simplified 86.6%

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

Alternative 7: 54.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+144}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 10^{+117}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1e+144) (* a b) (if (<= (* a b) 1e+117) (* x y) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1e+144) {
		tmp = a * b;
	} else if ((a * b) <= 1e+117) {
		tmp = x * y;
	} 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+144)) then
        tmp = a * b
    else if ((a * b) <= 1d+117) then
        tmp = x * y
    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+144) {
		tmp = a * b;
	} else if ((a * b) <= 1e+117) {
		tmp = x * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -1e+144:
		tmp = a * b
	elif (a * b) <= 1e+117:
		tmp = x * y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -1e+144)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1e+117)
		tmp = Float64(x * y);
	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+144)
		tmp = a * b;
	elseif ((a * b) <= 1e+117)
		tmp = x * y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -1e+144], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+117], N[(x * y), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.00000000000000002e144 or 1.00000000000000005e117 < (*.f64 a b)

   1. Initial program 96.7%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 77.8

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

   5. Simplified 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]
   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 77.8

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

   7. Applied egg-rr 77.8%

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

   if -1.00000000000000002e144 < (*.f64 a b) < 1.00000000000000005e117

   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 49.5

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

   5. Simplified 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   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 49.5

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

   7. Applied egg-rr 49.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+144}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 10^{+117}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 98.8%

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

3. Taylor expanded in a around inf

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

   1. +-rgt-identity N/A

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

   2. accelerator-lowering-fma.f64 36.8

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

5. Simplified 36.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, 0\right)} \]
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 36.8

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

7. Applied egg-rr 36.8%

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

8. Final simplification 36.8%

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

Reproduce

herbie shell --seed 2024196 
(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)))