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

Percentage Accurate: 97.4% → 98.8%

Time: 9.9s

Alternatives: 11

Speedup: 0.4×

• 34.2% 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: 97.4% 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: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b + \frac{x \cdot y}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* a b) (+ (* x y) (* z t)))))
   (if (<= t_1 INFINITY) t_1 (* a (+ b (/ (* x y) a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + ((x * y) + (z * t));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = a * (b + ((x * y) / a));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a * b) + ((x * y) + (z * t));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = a * (b + ((x * y) / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a * b) + ((x * y) + (z * t))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = a * (b + ((x * y) / a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a * b) + Float64(Float64(x * y) + Float64(z * t)))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(a * Float64(b + Float64(Float64(x * y) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a * b) + ((x * y) + (z * t));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = a * (b + ((x * y) / a));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b)) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (+.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 a b))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf 57.1%

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

   4. Taylor expanded in a around inf 71.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b + \left(x \cdot y + z \cdot t\right) \leq \infty:\\ \;\;\;\;a \cdot b + \left(x \cdot y + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b + \frac{x \cdot y}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. associate-+l+ 97.2%

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

   2. fma-define 98.0%

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

   3. fma-define 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 98.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. +-commutative 97.2%

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

   2. fma-define 97.6%

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

   3. fma-define 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 4: 97.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

   1. fma-define 97.6%

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

3. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 5: 53.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+71}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-135}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-151}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-211}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-239}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 10^{+143}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2e+71)
   (* a b)
   (if (<= (* a b) -2e-135)
     (* x y)
     (if (<= (* a b) -2e-151)
       (* z t)
       (if (<= (* a b) -5e-211)
         (* x y)
         (if (<= (* a b) 2e-239)
           (* z t)
           (if (<= (* a b) 1e+143) (* x y) (* a b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2e+71) {
		tmp = a * b;
	} else if ((a * b) <= -2e-135) {
		tmp = x * y;
	} else if ((a * b) <= -2e-151) {
		tmp = z * t;
	} else if ((a * b) <= -5e-211) {
		tmp = x * y;
	} else if ((a * b) <= 2e-239) {
		tmp = z * t;
	} else if ((a * b) <= 1e+143) {
		tmp = x * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * b) <= (-2d+71)) then
        tmp = a * b
    else if ((a * b) <= (-2d-135)) then
        tmp = x * y
    else if ((a * b) <= (-2d-151)) then
        tmp = z * t
    else if ((a * b) <= (-5d-211)) then
        tmp = x * y
    else if ((a * b) <= 2d-239) then
        tmp = z * t
    else if ((a * b) <= 1d+143) then
        tmp = x * y
    else
        tmp = a * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2e+71) {
		tmp = a * b;
	} else if ((a * b) <= -2e-135) {
		tmp = x * y;
	} else if ((a * b) <= -2e-151) {
		tmp = z * t;
	} else if ((a * b) <= -5e-211) {
		tmp = x * y;
	} else if ((a * b) <= 2e-239) {
		tmp = z * t;
	} else if ((a * b) <= 1e+143) {
		tmp = x * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2e+71:
		tmp = a * b
	elif (a * b) <= -2e-135:
		tmp = x * y
	elif (a * b) <= -2e-151:
		tmp = z * t
	elif (a * b) <= -5e-211:
		tmp = x * y
	elif (a * b) <= 2e-239:
		tmp = z * t
	elif (a * b) <= 1e+143:
		tmp = x * y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2e+71)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -2e-135)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= -2e-151)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= -5e-211)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 2e-239)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 1e+143)
		tmp = Float64(x * y);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -2e+71)
		tmp = a * b;
	elseif ((a * b) <= -2e-135)
		tmp = x * y;
	elseif ((a * b) <= -2e-151)
		tmp = z * t;
	elseif ((a * b) <= -5e-211)
		tmp = x * y;
	elseif ((a * b) <= 2e-239)
		tmp = z * t;
	elseif ((a * b) <= 1e+143)
		tmp = x * y;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+71], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2e-135], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2e-151], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -5e-211], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e-239], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1e+143], N[(x * y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.0000000000000001e71 or 1e143 < (*.f64 a b)

   1. Initial program 95.4%

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

   3. Taylor expanded in a around inf 82.9%

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

   if -2.0000000000000001e71 < (*.f64 a b) < -2.0000000000000001e-135 or -1.9999999999999999e-151 < (*.f64 a b) < -5.0000000000000002e-211 or 2.0000000000000002e-239 < (*.f64 a b) < 1e143

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf 70.5%

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

   4. Taylor expanded in a around inf 60.8%

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

   5. Taylor expanded in a around 0 56.4%

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

   if -2.0000000000000001e-135 < (*.f64 a b) < -1.9999999999999999e-151 or -5.0000000000000002e-211 < (*.f64 a b) < 2.0000000000000002e-239

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0 65.4%

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

   4. Taylor expanded in b around inf 38.2%

      \[\leadsto \color{blue}{b \cdot \left(a + \frac{t \cdot z}{b}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in b around 0 65.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+71}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-135}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq -2 \cdot 10^{-151}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -5 \cdot 10^{-211}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-239}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 10^{+143}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+71}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+125}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + t \cdot \frac{z}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2e+71)
   (+ (* a b) (* x y))
   (if (<= (* a b) 5e+125) (+ (* x y) (* z t)) (* b (+ a (* t (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2e+71) {
		tmp = (a * b) + (x * y);
	} else if ((a * b) <= 5e+125) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = b * (a + (t * (z / b)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * b) <= (-2d+71)) then
        tmp = (a * b) + (x * y)
    else if ((a * b) <= 5d+125) then
        tmp = (x * y) + (z * t)
    else
        tmp = b * (a + (t * (z / b)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2e+71) {
		tmp = (a * b) + (x * y);
	} else if ((a * b) <= 5e+125) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = b * (a + (t * (z / b)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2e+71:
		tmp = (a * b) + (x * y)
	elif (a * b) <= 5e+125:
		tmp = (x * y) + (z * t)
	else:
		tmp = b * (a + (t * (z / b)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2e+71)
		tmp = Float64(Float64(a * b) + Float64(x * y));
	elseif (Float64(a * b) <= 5e+125)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	else
		tmp = Float64(b * Float64(a + Float64(t * Float64(z / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -2e+71)
		tmp = (a * b) + (x * y);
	elseif ((a * b) <= 5e+125)
		tmp = (x * y) + (z * t);
	else
		tmp = b * (a + (t * (z / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+71], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+125], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(b * N[(a + N[(t * N[(z / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.0000000000000001e71

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf 93.6%

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

   if -2.0000000000000001e71 < (*.f64 a b) < 4.99999999999999962e125

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

      2. fma-define 98.2%

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

      3. fma-define 98.8%

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

   3. Simplified 98.8%

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

      1. fma-define 98.2%

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

      2. +-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in t around inf 87.6%

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

   8. Taylor expanded in x around 0 97.6%

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

   9. Taylor expanded in t around inf 90.2%

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

   if 4.99999999999999962e125 < (*.f64 a b)

   1. Initial program 92.1%

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

   3. Taylor expanded in x around 0 94.7%

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

   4. Taylor expanded in b around inf 94.7%

      \[\leadsto \color{blue}{b \cdot \left(a + \frac{t \cdot z}{b}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 94.8%

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

   6. Simplified 94.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+71}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+125}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a + t \cdot \frac{z}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.8 \cdot 10^{+257} \lor \neg \left(x \cdot y \leq 2.6 \cdot 10^{+110}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* x y) -3.8e+257) (not (<= (* x y) 2.6e+110)))
   (* x y)
   (+ (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * y) <= -3.8e+257) || !((x * y) <= 2.6e+110)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (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 (((x * y) <= (-3.8d+257)) .or. (.not. ((x * y) <= 2.6d+110))) then
        tmp = x * y
    else
        tmp = (a * b) + (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 (((x * y) <= -3.8e+257) || !((x * y) <= 2.6e+110)) {
		tmp = x * y;
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * y) <= -3.8e+257) or not ((x * y) <= 2.6e+110):
		tmp = x * y
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(x * y) <= -3.8e+257) || !(Float64(x * y) <= 2.6e+110))
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * y) <= -3.8e+257) || ~(((x * y) <= 2.6e+110)))
		tmp = x * y;
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -3.8e+257], N[Not[LessEqual[N[(x * y), $MachinePrecision], 2.6e+110]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.79999999999999998e257 or 2.6e110 < (*.f64 x y)

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf 87.5%

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

   4. Taylor expanded in a around inf 74.1%

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

   5. Taylor expanded in a around 0 80.3%

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

   if -3.79999999999999998e257 < (*.f64 x y) < 2.6e110

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 81.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -3.8 \cdot 10^{+257} \lor \neg \left(x \cdot y \leq 2.6 \cdot 10^{+110}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+71} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+125}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -2e+71) (not (<= (* a b) 5e+125)))
   (+ (* a b) (* z t))
   (+ (* x y) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -2e+71) || !((a * b) <= 5e+125)) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = (x * y) + (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 (((a * b) <= (-2d+71)) .or. (.not. ((a * b) <= 5d+125))) then
        tmp = (a * b) + (z * t)
    else
        tmp = (x * y) + (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 (((a * b) <= -2e+71) || !((a * b) <= 5e+125)) {
		tmp = (a * b) + (z * t);
	} else {
		tmp = (x * y) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -2e+71) or not ((a * b) <= 5e+125):
		tmp = (a * b) + (z * t)
	else:
		tmp = (x * y) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -2e+71) || !(Float64(a * b) <= 5e+125))
		tmp = Float64(Float64(a * b) + Float64(z * t));
	else
		tmp = Float64(Float64(x * y) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((a * b) <= -2e+71) || ~(((a * b) <= 5e+125)))
		tmp = (a * b) + (z * t);
	else
		tmp = (x * y) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -2e+71], N[Not[LessEqual[N[(a * b), $MachinePrecision], 5e+125]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.0000000000000001e71 or 4.99999999999999962e125 < (*.f64 a b)

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0 90.3%

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

   if -2.0000000000000001e71 < (*.f64 a b) < 4.99999999999999962e125

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

      2. fma-define 98.2%

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

      3. fma-define 98.8%

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

   3. Simplified 98.8%

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

      1. fma-define 98.2%

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

      2. +-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in t around inf 87.6%

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

   8. Taylor expanded in x around 0 97.6%

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

   9. Taylor expanded in t around inf 90.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+71} \lor \neg \left(a \cdot b \leq 5 \cdot 10^{+125}\right):\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+71}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+125}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* a b) -2e+71)
   (+ (* a b) (* x y))
   (if (<= (* a b) 5e+125) (+ (* x y) (* z t)) (+ (* a b) (* z t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a * b) <= -2e+71) {
		tmp = (a * b) + (x * y);
	} else if ((a * b) <= 5e+125) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = (a * b) + (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 ((a * b) <= (-2d+71)) then
        tmp = (a * b) + (x * y)
    else if ((a * b) <= 5d+125) then
        tmp = (x * y) + (z * t)
    else
        tmp = (a * b) + (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 ((a * b) <= -2e+71) {
		tmp = (a * b) + (x * y);
	} else if ((a * b) <= 5e+125) {
		tmp = (x * y) + (z * t);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a * b) <= -2e+71:
		tmp = (a * b) + (x * y)
	elif (a * b) <= 5e+125:
		tmp = (x * y) + (z * t)
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a * b) <= -2e+71)
		tmp = Float64(Float64(a * b) + Float64(x * y));
	elseif (Float64(a * b) <= 5e+125)
		tmp = Float64(Float64(x * y) + Float64(z * t));
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a * b) <= -2e+71)
		tmp = (a * b) + (x * y);
	elseif ((a * b) <= 5e+125)
		tmp = (x * y) + (z * t);
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+71], N[(N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 5e+125], N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -2.0000000000000001e71

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf 93.6%

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

   if -2.0000000000000001e71 < (*.f64 a b) < 4.99999999999999962e125

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

      2. fma-define 98.2%

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

      3. fma-define 98.8%

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

   3. Simplified 98.8%

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

      1. fma-define 98.2%

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

      2. +-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in t around inf 87.6%

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

   8. Taylor expanded in x around 0 97.6%

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

   9. Taylor expanded in t around inf 90.2%

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

   if 4.99999999999999962e125 < (*.f64 a b)

   1. Initial program 92.1%

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

   3. Taylor expanded in x around 0 94.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+71}:\\ \;\;\;\;a \cdot b + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 5 \cdot 10^{+125}:\\ \;\;\;\;x \cdot y + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.2 \cdot 10^{+71} \lor \neg \left(a \cdot b \leq 6.8 \cdot 10^{+75}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* a b) -1.2e+71) (not (<= (* a b) 6.8e+75))) (* a b) (* z t)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a * b) <= -1.2e+71) || !((a * b) <= 6.8e+75)) {
		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 (((a * b) <= (-1.2d+71)) .or. (.not. ((a * b) <= 6.8d+75))) 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 (((a * b) <= -1.2e+71) || !((a * b) <= 6.8e+75)) {
		tmp = a * b;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((a * b) <= -1.2e+71) or not ((a * b) <= 6.8e+75):
		tmp = a * b
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a * b) <= -1.2e+71) || !(Float64(a * b) <= 6.8e+75))
		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 (((a * b) <= -1.2e+71) || ~(((a * b) <= 6.8e+75)))
		tmp = a * b;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a * b), $MachinePrecision], -1.2e+71], N[Not[LessEqual[N[(a * b), $MachinePrecision], 6.8e+75]], $MachinePrecision]], N[(a * b), $MachinePrecision], N[(z * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.1999999999999999e71 or 6.80000000000000022e75 < (*.f64 a b)

   1. Initial program 95.8%

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

   3. Taylor expanded in a around inf 78.3%

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

   if -1.1999999999999999e71 < (*.f64 a b) < 6.80000000000000022e75

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 53.6%

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

   4. Taylor expanded in b around inf 40.0%

      \[\leadsto \color{blue}{b \cdot \left(a + \frac{t \cdot z}{b}\right)} \]
   5. Step-by-step derivation

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in b around 0 46.7%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.2 \cdot 10^{+71} \lor \neg \left(a \cdot b \leq 6.8 \cdot 10^{+75}\right):\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in a around inf 35.2%

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

4. Final simplification 35.2%

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

Reproduce

herbie shell --seed 2024071 
(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)))