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

Percentage Accurate: 97.8% → 98.9%

Time: 5.4s

Alternatives: 8

Speedup: 1.2×

• 33.4% 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 97.6%

   \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
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. +-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 98.4

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

   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 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+76} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+31}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -2e+76) (not (<= (* x y) 2e+31)))
   (fma b a (* y x))
   (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) <= -2e+76) || !((x * y) <= 2e+31)) {
		tmp = fma(b, a, (y * x));
	} 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) <= -2e+76) || !(Float64(x * y) <= 2e+31))
		tmp = fma(b, a, Float64(y * x));
	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], -2e+76], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2e+31]], $MachinePrecision]], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.0000000000000001e76 or 1.9999999999999999e31 < (*.f64 x y)

   1. Initial program 94.4%

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

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

      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 84.8

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

   5. Applied rewrites 84.8%

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

   if -2.0000000000000001e76 < (*.f64 x y) < 1.9999999999999999e31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-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. lower-*.f64 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+76} \lor \neg \left(x \cdot y \leq 2 \cdot 10^{+31}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.5 \cdot 10^{+153} \lor \neg \left(x \cdot y \leq 7.5 \cdot 10^{+141}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -1.5e+153) (not (<= (* x y) 7.5e+141)))
   (* y x)
   (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) <= -1.5e+153) || !((x * y) <= 7.5e+141)) {
		tmp = y * x;
	} 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) <= -1.5e+153) || !(Float64(x * y) <= 7.5e+141))
		tmp = Float64(y * x);
	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], -1.5e+153], N[Not[LessEqual[N[(x * y), $MachinePrecision], 7.5e+141]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.50000000000000009e153 or 7.49999999999999937e141 < (*.f64 x y)

   1. Initial program 92.4%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 13.6%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 79.4

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

   14. Applied rewrites 79.4%

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

   if -1.50000000000000009e153 < (*.f64 x y) < 7.49999999999999937e141

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-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. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.5 \cdot 10^{+153} \lor \neg \left(x \cdot y \leq 7.5 \cdot 10^{+141}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{elif}\;z \cdot t \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -5e+53)
   (fma t z (* y x))
   (if (<= (* z t) 20000000000.0) (fma b a (* y x)) (fma z t (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -5e+53) {
		tmp = fma(t, z, (y * x));
	} else if ((z * t) <= 20000000000.0) {
		tmp = fma(b, a, (y * x));
	} else {
		tmp = fma(z, t, (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -5e+53)
		tmp = fma(t, z, Float64(y * x));
	elseif (Float64(z * t) <= 20000000000.0)
		tmp = fma(b, a, Float64(y * x));
	else
		tmp = fma(z, t, Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+53], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 20000000000.0], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(z * t + N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e53

   1. Initial program 98.4%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if -5.0000000000000004e53 < (*.f64 z t) < 2e10

   1. Initial program 97.8%

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

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

      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 91.3

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

   5. Applied rewrites 91.3%

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

   if 2e10 < (*.f64 z t)

   1. Initial program 96.6%

      \[\left(x \cdot y + z \cdot t\right) + a \cdot b \]
   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. +-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 98.3

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

      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 98.3

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

   4. Applied rewrites 98.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 87.0

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

   7. Applied rewrites 87.0%

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

Alternative 5: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(t, z, y \cdot x\right)\\ \mathbf{elif}\;z \cdot t \leq 20000000000:\\ \;\;\;\;\mathsf{fma}\left(b, a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -5e+53)
   (fma t z (* y x))
   (if (<= (* z t) 20000000000.0) (fma b a (* y x)) (fma b a (* t z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -5e+53) {
		tmp = fma(t, z, (y * x));
	} else if ((z * t) <= 20000000000.0) {
		tmp = fma(b, a, (y * x));
	} else {
		tmp = fma(b, a, (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -5e+53)
		tmp = fma(t, z, Float64(y * x));
	elseif (Float64(z * t) <= 20000000000.0)
		tmp = fma(b, a, Float64(y * x));
	else
		tmp = fma(b, a, Float64(t * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -5e+53], N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 20000000000.0], N[(b * a + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(t * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e53

   1. Initial program 98.4%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if -5.0000000000000004e53 < (*.f64 z t) < 2e10

   1. Initial program 97.8%

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

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

      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 91.3

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

   5. Applied rewrites 91.3%

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

   if 2e10 < (*.f64 z t)

   1. Initial program 96.6%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

Alternative 6: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+53} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+112}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* z t) -5e+53) (not (<= (* z t) 5e+112))) (* t z) (* b a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((z * t) <= -5e+53) || !((z * t) <= 5e+112)) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	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 (((z * t) <= (-5d+53)) .or. (.not. ((z * t) <= 5d+112))) then
        tmp = t * z
    else
        tmp = b * a
    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 (((z * t) <= -5e+53) || !((z * t) <= 5e+112)) {
		tmp = t * z;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((z * t) <= -5e+53) or not ((z * t) <= 5e+112):
		tmp = t * z
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+53) || !(Float64(z * t) <= 5e+112))
		tmp = Float64(t * z);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((z * t) <= -5e+53) || ~(((z * t) <= 5e+112)))
		tmp = t * z;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+53], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+112]], $MachinePrecision]], N[(t * z), $MachinePrecision], N[(b * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000004e53 or 5e112 < (*.f64 z t)

   1. Initial program 97.1%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.0

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

   8. Applied rewrites 91.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 72.3%

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

   if -5.0000000000000004e53 < (*.f64 z t) < 5e112

   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

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

      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 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.8%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+53} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+112}\right):\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-25}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-109}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+70}:\\ \;\;\;\;t \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.2e-25)
   (* y x)
   (if (<= y 4.2e-109) (* b a) (if (<= y 1.9e+70) (* t z) (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.2e-25) {
		tmp = y * x;
	} else if (y <= 4.2e-109) {
		tmp = b * a;
	} else if (y <= 1.9e+70) {
		tmp = t * z;
	} else {
		tmp = y * x;
	}
	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 (y <= (-2.2d-25)) then
        tmp = y * x
    else if (y <= 4.2d-109) then
        tmp = b * a
    else if (y <= 1.9d+70) then
        tmp = t * z
    else
        tmp = y * x
    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 (y <= -2.2e-25) {
		tmp = y * x;
	} else if (y <= 4.2e-109) {
		tmp = b * a;
	} else if (y <= 1.9e+70) {
		tmp = t * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.2e-25:
		tmp = y * x
	elif y <= 4.2e-109:
		tmp = b * a
	elif y <= 1.9e+70:
		tmp = t * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.2e-25)
		tmp = Float64(y * x);
	elseif (y <= 4.2e-109)
		tmp = Float64(b * a);
	elseif (y <= 1.9e+70)
		tmp = Float64(t * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.2e-25)
		tmp = y * x;
	elseif (y <= 4.2e-109)
		tmp = b * a;
	elseif (y <= 1.9e+70)
		tmp = t * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.2e-25], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.2e-109], N[(b * a), $MachinePrecision], If[LessEqual[y, 1.9e+70], N[(t * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2000000000000002e-25 or 1.8999999999999999e70 < y

   1. Initial program 95.1%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 27.8%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 53.3

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

   14. Applied rewrites 53.3%

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

   if -2.2000000000000002e-25 < y < 4.19999999999999992e-109

   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

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

      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 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.3%

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

   if 4.19999999999999992e-109 < y < 1.8999999999999999e70

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-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. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.5

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.0%

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

Alternative 8: 35.4% 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 97.6%

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

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

   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 65.5

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

5. Applied rewrites 65.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 34.4%

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

Reproduce

herbie shell --seed 2024312 
(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)))