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

Percentage Accurate: 97.9% → 98.9%

Time: 5.5s

Alternatives: 7

Speedup: 1.2×

• 32.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.6

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma t z (* b a))))
   (if (<= (* z t) -1e+131)
     t_1
     (if (<= (* z t) 2e-42)
       (fma y x (* b a))
       (if (<= (* z t) 4e+44) (fma y x (* t z)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, z, (b * a));
	double tmp;
	if ((z * t) <= -1e+131) {
		tmp = t_1;
	} else if ((z * t) <= 2e-42) {
		tmp = fma(y, x, (b * a));
	} else if ((z * t) <= 4e+44) {
		tmp = fma(y, x, (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(t, z, Float64(b * a))
	tmp = 0.0
	if (Float64(z * t) <= -1e+131)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-42)
		tmp = fma(y, x, Float64(b * a));
	elseif (Float64(z * t) <= 4e+44)
		tmp = fma(y, x, Float64(t * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+131], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-42], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+44], N[(y * x + N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999991e130 or 4.0000000000000004e44 < (*.f64 z t)

   1. Initial program 98.8%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

   if -9.9999999999999991e130 < (*.f64 z t) < 2.00000000000000008e-42

   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 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 92.1

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

   5. Applied rewrites 92.1%

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

   if 2.00000000000000008e-42 < (*.f64 z t) < 4.0000000000000004e44

   1. Initial program 99.9%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 8.5%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   10. 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 92.2

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

   11. Applied rewrites 92.2%

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

   13. Applied rewrites 92.3%

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

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(y, x, b \cdot a\right)\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma t z (* b a))))
   (if (<= (* z t) -1e+131)
     t_1
     (if (<= (* z t) 2e-42)
       (fma y x (* b a))
       (if (<= (* z t) 4e+44) (fma t z (* y x)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, z, (b * a));
	double tmp;
	if ((z * t) <= -1e+131) {
		tmp = t_1;
	} else if ((z * t) <= 2e-42) {
		tmp = fma(y, x, (b * a));
	} else if ((z * t) <= 4e+44) {
		tmp = fma(t, z, (y * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(t, z, Float64(b * a))
	tmp = 0.0
	if (Float64(z * t) <= -1e+131)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-42)
		tmp = fma(y, x, Float64(b * a));
	elseif (Float64(z * t) <= 4e+44)
		tmp = fma(t, z, Float64(y * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * z + N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+131], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-42], N[(y * x + N[(b * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+44], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999991e130 or 4.0000000000000004e44 < (*.f64 z t)

   1. Initial program 98.8%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 94.4

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

   5. Applied rewrites 94.4%

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

   if -9.9999999999999991e130 < (*.f64 z t) < 2.00000000000000008e-42

   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 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 92.1

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

   5. Applied rewrites 92.1%

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

   if 2.00000000000000008e-42 < (*.f64 z t) < 4.0000000000000004e44

   1. Initial program 99.9%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 8.5%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   10. 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 92.2

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

   11. Applied rewrites 92.2%

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

Alternative 4: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+139} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+52}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -5e+139) (not (<= (* a b) 2e+52)))
   (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) <= -5e+139) || !((a * b) <= 2e+52)) {
		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) <= -5e+139) || !(Float64(a * b) <= 2e+52))
		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], -5e+139], N[Not[LessEqual[N[(a * b), $MachinePrecision], 2e+52]], $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) < -5.0000000000000003e139 or 2e52 < (*.f64 a b)

   1. Initial program 98.7%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. +-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 92.6

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

   5. Applied rewrites 92.6%

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

   if -5.0000000000000003e139 < (*.f64 a b) < 2e52

   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 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 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 11.6%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{t \cdot z + x \cdot y} \]
   10. 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 90.1

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

   11. Applied rewrites 90.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+139} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+52}\right):\\ \;\;\;\;\mathsf{fma}\left(t, z, b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+139} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+94}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -5e+139) (not (<= (* a b) 2e+94))) (* 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+139) || !((a * b) <= 2e+94)) {
		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+139)) .or. (.not. ((a * b) <= 2d+94))) 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+139) || !((a * b) <= 2e+94)) {
		tmp = b * a;
	} else {
		tmp = t * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -5e+139) or not ((a * b) <= 2e+94):
		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+139) || !(Float64(a * b) <= 2e+94))
		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+139) || ~(((a * b) <= 2e+94)))
		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+139], N[Not[LessEqual[N[(a * b), $MachinePrecision], 2e+94]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(t * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -5.0000000000000003e139 or 2e94 < (*.f64 a b)

   1. Initial program 98.5%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.5%

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

   if -5.0000000000000003e139 < (*.f64 a b) < 2e94

   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 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 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 12.4%

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

   9. Taylor expanded in z around inf

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

       1. lower-*.f64 42.5

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

   11. Applied rewrites 42.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -5 \cdot 10^{+139} \lor \neg \left(a \cdot b \leq 2 \cdot 10^{+94}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% 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 99.6%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
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 63.5

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

5. Applied rewrites 63.5%

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

Alternative 7: 35.2% 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 99.6%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
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 68.1

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 31.8%

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

Reproduce

herbie shell --seed 2024326 
(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)))