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

Percentage Accurate: 97.8% → 98.9%

Time: 5.2s

Alternatives: 7

Speedup: 1.2×

• 33.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 97.3

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 98.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 54.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.25 \cdot 10^{+71}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4.6 \cdot 10^{-60}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -1.25e+71)
   (* z t)
   (if (<= (* z t) 4.6e-60)
     (* x y)
     (if (<= (* z t) 5.5e+126) (* a b) (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -1.25e+71) {
		tmp = z * t;
	} else if ((z * t) <= 4.6e-60) {
		tmp = x * y;
	} else if ((z * t) <= 5.5e+126) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	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) <= (-1.25d+71)) then
        tmp = z * t
    else if ((z * t) <= 4.6d-60) then
        tmp = x * y
    else if ((z * t) <= 5.5d+126) then
        tmp = a * b
    else
        tmp = z * t
    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) <= -1.25e+71) {
		tmp = z * t;
	} else if ((z * t) <= 4.6e-60) {
		tmp = x * y;
	} else if ((z * t) <= 5.5e+126) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z * t) <= -1.25e+71:
		tmp = z * t
	elif (z * t) <= 4.6e-60:
		tmp = x * y
	elif (z * t) <= 5.5e+126:
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -1.25e+71)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 4.6e-60)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= 5.5e+126)
		tmp = Float64(a * b);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z * t) <= -1.25e+71)
		tmp = z * t;
	elseif ((z * t) <= 4.6e-60)
		tmp = x * y;
	elseif ((z * t) <= 5.5e+126)
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -1.25e+71], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4.6e-60], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5.5e+126], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.24999999999999993e71 or 5.5000000000000004e126 < (*.f64 z t)

   1. Initial program 91.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if -1.24999999999999993e71 < (*.f64 z t) < 4.6000000000000003e-60

   1. Initial program 97.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   if 4.6000000000000003e-60 < (*.f64 z t) < 5.5000000000000004e126

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.7

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

   5. Applied rewrites 57.7%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.25 \cdot 10^{+71}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 4.6 \cdot 10^{-60}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 5.5 \cdot 10^{+126}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -1e+134)
   (fma b a (* x y))
   (if (<= (* a b) 2e+116) (fma y x (* z t)) (fma b a (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -1e+134) {
		tmp = fma(b, a, (x * y));
	} else if ((a * b) <= 2e+116) {
		tmp = fma(y, x, (z * t));
	} else {
		tmp = fma(b, a, (z * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -1e+134)
		tmp = fma(b, a, Float64(x * y));
	elseif (Float64(a * b) <= 2e+116)
		tmp = fma(y, x, Float64(z * t));
	else
		tmp = fma(b, a, Float64(z * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -1e+134], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e+116], N[(y * x + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -9.99999999999999921e133

   1. Initial program 89.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if -9.99999999999999921e133 < (*.f64 a b) < 2.00000000000000003e116

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if 2.00000000000000003e116 < (*.f64 a b)

   1. Initial program 88.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 90.2

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

   5. Applied rewrites 90.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{if}\;z \cdot t \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b a (* z t))))
   (if (<= (* z t) -1.15e+71)
     t_1
     (if (<= (* z t) 1.45e-5) (fma b a (* x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, a, (z * t));
	double tmp;
	if ((z * t) <= -1.15e+71) {
		tmp = t_1;
	} else if ((z * t) <= 1.45e-5) {
		tmp = fma(b, a, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, a, Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -1.15e+71)
		tmp = t_1;
	elseif (Float64(z * t) <= 1.45e-5)
		tmp = fma(b, a, Float64(x * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * a + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1.15e+71], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1.45e-5], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.1500000000000001e71 or 1.45e-5 < (*.f64 z t)

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -1.1500000000000001e71 < (*.f64 z t) < 1.45e-5

   1. Initial program 97.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \mathbf{elif}\;z \cdot t \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.15 \cdot 10^{+128}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 1.06 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* z t) -3.15e+128)
   (* z t)
   (if (<= (* z t) 1.06e+127) (fma b a (* x y)) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z * t) <= -3.15e+128) {
		tmp = z * t;
	} else if ((z * t) <= 1.06e+127) {
		tmp = fma(b, a, (x * y));
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(z * t) <= -3.15e+128)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 1.06e+127)
		tmp = fma(b, a, Float64(x * y));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(z * t), $MachinePrecision], -3.15e+128], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1.06e+127], N[(b * a + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.1499999999999999e128 or 1.06000000000000006e127 < (*.f64 z t)

   1. Initial program 90.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -3.1499999999999999e128 < (*.f64 z t) < 1.06000000000000006e127

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.15 \cdot 10^{+128}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 1.06 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(b, a, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+27}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+50}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x y) -5e+27) (* x y) (if (<= (* x y) 4e+50) (* a b) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -5e+27) {
		tmp = x * y;
	} else if ((x * y) <= 4e+50) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((x * y) <= (-5d+27)) then
        tmp = x * y
    else if ((x * y) <= 4d+50) then
        tmp = a * b
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * y) <= -5e+27) {
		tmp = x * y;
	} else if ((x * y) <= 4e+50) {
		tmp = a * b;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * y) <= -5e+27:
		tmp = x * y
	elif (x * y) <= 4e+50:
		tmp = a * b
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * y) <= -5e+27)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 4e+50)
		tmp = Float64(a * b);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((x * y) <= -5e+27)
		tmp = x * y;
	elseif ((x * y) <= 4e+50)
		tmp = a * b;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+27], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+50], N[(a * b), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999979e27 or 4.0000000000000003e50 < (*.f64 x y)

   1. Initial program 92.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if -4.99999999999999979e27 < (*.f64 x y) < 4.0000000000000003e50

   1. Initial program 98.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+27}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+50}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 32.0

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

5. Applied rewrites 32.0%

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

6. Final simplification 32.0%

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

Reproduce

herbie shell --seed 2024236 
(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)))