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

Percentage Accurate: 97.8% → 98.9%

Time: 4.5s

Alternatives: 6

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: 97.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(z * t + N[(b * a + N[(y * x), $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. 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 99.2

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

   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 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 85.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -500 \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -500.0) (not (<= (* a b) 5e+36)))
   (fma t z (* b a))
   (fma t z (* y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -500.0) || !((a * b) <= 5e+36)) {
		tmp = fma(t, z, (b * a));
	} else {
		tmp = fma(t, z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -500.0) || !(Float64(a * b) <= 5e+36))
		tmp = fma(t, z, Float64(b * a));
	else
		tmp = fma(t, z, Float64(y * x));
	end
	return 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], 5e+36]], $MachinePrecision]], N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -500 or 4.99999999999999977e36 < (*.f64 a b)

   1. Initial program 96.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -500 < (*.f64 a b) < 4.99999999999999977e36

   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 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in a around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. distribute-lft-neg-out N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. mul-1-neg N/A

         \[\leadsto t \cdot z - x \cdot \color{blue}{\left(-1 \cdot y\right)} \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{t \cdot z + \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)} \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot z + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(-1 \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, \left(-1 \cdot x\right) \cdot \left(-1 \cdot y\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-1 \cdot x\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot x\right)\right)}\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{y \cdot \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}\right) \]

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 88.5

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

   8. Applied rewrites 88.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -500 \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+36}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.44 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -3.44e+91)
   (fma y x (* b a))
   (if (<= (* a b) 5e+36) (fma t z (* y x)) (fma t z (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -3.44e+91) {
		tmp = fma(y, x, (b * a));
	} else if ((a * b) <= 5e+36) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = fma(t, z, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -3.44e+91)
		tmp = fma(y, x, Float64(b * a));
	elseif (Float64(a * b) <= 5e+36)
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = fma(t, z, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -3.44e+91], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+36], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -3.4399999999999998e91

   1. Initial program 95.8%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if -3.4399999999999998e91 < (*.f64 a b) < 4.99999999999999977e36

   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. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in a around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. distribute-lft-neg-out N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. mul-1-neg N/A

         \[\leadsto t \cdot z - x \cdot \color{blue}{\left(-1 \cdot y\right)} \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{t \cdot z + \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)} \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot z + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(-1 \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, \left(-1 \cdot x\right) \cdot \left(-1 \cdot y\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{\left(-1 \cdot y\right) \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-1 \cdot x\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{\mathsf{neg}\left(y \cdot \left(-1 \cdot x\right)\right)}\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(t, z, \color{blue}{y \cdot \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}\right) \]

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 87.4

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

   8. Applied rewrites 87.4%

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

   if 4.99999999999999977e36 < (*.f64 a b)

   1. Initial program 96.3%

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

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

Alternative 4: 52.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+101} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{-55}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -5e+101) (not (<= (* a b) 5e-55))) (* b a) (* t z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -5e+101) || !((a * b) <= 5e-55)) {
		tmp = b * a;
	} 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) <= (-5d+101)) .or. (.not. ((a * b) <= 5d-55))) then
        tmp = b * a
    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) <= -5e+101) || !((a * b) <= 5e-55)) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -5e+101) or not ((a * b) <= 5e-55):
		tmp = b * a
	else:
		tmp = t * z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -5e+101) || !(Float64(a * b) <= 5e-55))
		tmp = Float64(b * a);
	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) <= -5e+101) || ~(((a * b) <= 5e-55)))
		tmp = b * a;
	else
		tmp = t * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -5e+101], N[Not[LessEqual[N[(a * b), $MachinePrecision], 5e-55]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(t * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -4.99999999999999989e101 or 5.0000000000000002e-55 < (*.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 z around 0

      \[\leadsto \color{blue}{a \cdot b + x \cdot y} \]
   4. 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. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.0%

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

   if -4.99999999999999989e101 < (*.f64 a b) < 5.0000000000000002e-55

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 45.7

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

   8. Applied rewrites 45.7%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+101} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{-55}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(t * z + N[(b * a), $MachinePrecision]), $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 x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 67.1

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

5. Applied rewrites 67.1%

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

Alternative 6: 35.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ b \cdot a \end{array} \]

FPCore

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

C

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

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 = b * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(b * a), $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 z around 0

   \[\leadsto \color{blue}{a \cdot b + x \cdot y} \]
4. 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. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 71.3

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

5. Applied rewrites 71.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2024320 
(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)))