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

Percentage Accurate: 97.9% → 98.9%

Time: 4.9s

Alternatives: 7

Speedup: 1.2×

• 33.1% 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: 97.9% 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: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 98.8

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-97} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+155}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -1e-97) (not (<= (* x y) 5e+155)))
   (fma b a (* x y))
   (fma b a (* t z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -1e-97) || !((x * y) <= 5e+155)) {
		tmp = fma(b, a, (x * y));
	} else {
		tmp = fma(b, a, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -1e-97) || !(Float64(x * y) <= 5e+155))
		tmp = fma(b, a, Float64(x * y));
	else
		tmp = fma(b, a, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e-97], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+155]], $MachinePrecision]], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000004e-97 or 4.9999999999999999e155 < (*.f64 x y)

   1. Initial program 96.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -1.00000000000000004e-97 < (*.f64 x y) < 4.9999999999999999e155

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-97} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+155}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.05 \cdot 10^{+140} \lor \neg \left(t \cdot z \leq 9.8 \cdot 10^{+250}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* t z) -1.05e+140) (not (<= (* t z) 9.8e+250)))
   (* t z)
   (fma b a (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t * z) <= -1.05e+140) || !((t * z) <= 9.8e+250)) {
		tmp = t * z;
	} else {
		tmp = fma(b, a, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(t * z) <= -1.05e+140) || !(Float64(t * z) <= 9.8e+250))
		tmp = Float64(t * z);
	else
		tmp = fma(b, a, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(t * z), $MachinePrecision], -1.05e+140], N[Not[LessEqual[N[(t * z), $MachinePrecision], 9.8e+250]], $MachinePrecision]], N[(t * z), $MachinePrecision], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.0500000000000001e140 or 9.79999999999999986e250 < (*.f64 z t)

   1. Initial program 91.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -1.0500000000000001e140 < (*.f64 z t) < 9.79999999999999986e250

   1. Initial program 99.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot z \leq -1.05 \cdot 10^{+140} \lor \neg \left(t \cdot z \leq 9.8 \cdot 10^{+250}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -0.001)
   (fma y x (* t z))
   (if (<= (* x y) 5e+155) (fma b a (* t z)) (fma b a (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -0.001) {
		tmp = fma(y, x, (t * z));
	} else if ((x * y) <= 5e+155) {
		tmp = fma(b, a, (t * z));
	} else {
		tmp = fma(b, a, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -0.001)
		tmp = fma(y, x, Float64(t * z));
	elseif (Float64(x * y) <= 5e+155)
		tmp = fma(b, a, Float64(t * z));
	else
		tmp = fma(b, a, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -0.001], N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+155], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e-3

   1. Initial program 93.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -1e-3 < (*.f64 x y) < 4.9999999999999999e155

   1. Initial program 98.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if 4.9999999999999999e155 < (*.f64 x y)

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -500 \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -500.0) (not (<= (* a b) 2e+15))) (* a b) (* t z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -500.0) || !((a * b) <= 2e+15)) {
		tmp = a * b;
	} else {
		tmp = t * z;
	}
	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) <= (-500.0d0)) .or. (.not. ((a * b) <= 2d+15))) then
        tmp = a * b
    else
        tmp = t * z
    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) <= -500.0) || !((a * b) <= 2e+15)) {
		tmp = a * b;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -500.0) or not ((a * b) <= 2e+15):
		tmp = a * b
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -500.0) || !(Float64(a * b) <= 2e+15))
		tmp = Float64(a * b);
	else
		tmp = Float64(t * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((a * b) <= -500.0) || ~(((a * b) <= 2e+15)))
		tmp = a * b;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -500.0], N[Not[LessEqual[N[(a * b), $MachinePrecision], 2e+15]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(t * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -500 or 2e15 < (*.f64 a b)

   1. Initial program 96.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if -500 < (*.f64 a b) < 2e15

   1. Initial program 99.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -500 \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+15}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.001 \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+155}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -0.001) (not (<= (* x y) 5e+155))) (* x y) (* a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -0.001) || !((x * y) <= 5e+155)) {
		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 (((x * y) <= (-0.001d0)) .or. (.not. ((x * y) <= 5d+155))) 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 (((x * y) <= -0.001) || !((x * y) <= 5e+155)) {
		tmp = x * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * y) <= -0.001) or not ((x * y) <= 5e+155):
		tmp = x * y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -0.001) || !(Float64(x * y) <= 5e+155))
		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 (((x * y) <= -0.001) || ~(((x * y) <= 5e+155)))
		tmp = x * y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -0.001], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+155]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(a * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e-3 or 4.9999999999999999e155 < (*.f64 x y)

   1. Initial program 96.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if -1e-3 < (*.f64 x y) < 4.9999999999999999e155

   1. Initial program 98.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -0.001 \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+155}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.3% 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 97.7%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 39.8

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

5. Applied rewrites 39.8%

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

6. Final simplification 39.8%

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

Reproduce

herbie shell --seed 2024271 
(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)))