Linear.V4:$cdot from linear-1.19.1.3, C

Percentage Accurate: 95.7% → 97.7%

Time: 7.2s

Alternatives: 12

Speedup: 0.5×

• 41.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 40.0%

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

      1. *-commutative 40.0%

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

      2. fma-def 50.0%

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

      3. *-commutative 50.0%

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in c around 0 50.0%

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

      1. *-commutative 40.0%

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

      2. fma-def 50.0%

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

      3. *-commutative 50.0%

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

   7. Applied egg-rr 60.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a \cdot b + \left(z \cdot t + x \cdot y\right)\right) + c \cdot i \leq \infty:\\ \;\;\;\;\left(a \cdot b + \left(z \cdot t + x \cdot y\right)\right) + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a, z \cdot t\right)\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

   1. +-commutative 96.1%

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

   2. fma-def 96.9%

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

   3. associate-+l+ 96.9%

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

   4. fma-def 98.0%

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

   5. fma-def 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 3: 97.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 40.0%

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

      1. *-commutative 40.0%

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

      2. fma-def 50.0%

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

      3. *-commutative 50.0%

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in c around 0 50.0%

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

4. Final simplification 98.0%

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

Alternative 4: 44.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.6 \cdot 10^{+79}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.5 \cdot 10^{-293}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 3 \cdot 10^{-157}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 7.5 \cdot 10^{-72}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 1.65 \cdot 10^{-49}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 4.3 \cdot 10^{+49}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -1.6e+79)
   (* a b)
   (if (<= (* a b) -2.5e-293)
     (* z t)
     (if (<= (* a b) 3e-157)
       (* x y)
       (if (<= (* a b) 7.5e-72)
         (* c i)
         (if (<= (* a b) 1.65e-49)
           (* x y)
           (if (<= (* a b) 4.3e+49) (* z t) (* a b))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -1.6e+79) {
		tmp = a * b;
	} else if ((a * b) <= -2.5e-293) {
		tmp = z * t;
	} else if ((a * b) <= 3e-157) {
		tmp = x * y;
	} else if ((a * b) <= 7.5e-72) {
		tmp = c * i;
	} else if ((a * b) <= 1.65e-49) {
		tmp = x * y;
	} else if ((a * b) <= 4.3e+49) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-1.6d+79)) then
        tmp = a * b
    else if ((a * b) <= (-2.5d-293)) then
        tmp = z * t
    else if ((a * b) <= 3d-157) then
        tmp = x * y
    else if ((a * b) <= 7.5d-72) then
        tmp = c * i
    else if ((a * b) <= 1.65d-49) then
        tmp = x * y
    else if ((a * b) <= 4.3d+49) then
        tmp = z * t
    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 c, double i) {
	double tmp;
	if ((a * b) <= -1.6e+79) {
		tmp = a * b;
	} else if ((a * b) <= -2.5e-293) {
		tmp = z * t;
	} else if ((a * b) <= 3e-157) {
		tmp = x * y;
	} else if ((a * b) <= 7.5e-72) {
		tmp = c * i;
	} else if ((a * b) <= 1.65e-49) {
		tmp = x * y;
	} else if ((a * b) <= 4.3e+49) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -1.6e+79:
		tmp = a * b
	elif (a * b) <= -2.5e-293:
		tmp = z * t
	elif (a * b) <= 3e-157:
		tmp = x * y
	elif (a * b) <= 7.5e-72:
		tmp = c * i
	elif (a * b) <= 1.65e-49:
		tmp = x * y
	elif (a * b) <= 4.3e+49:
		tmp = z * t
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -1.6e+79)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -2.5e-293)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 3e-157)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 7.5e-72)
		tmp = Float64(c * i);
	elseif (Float64(a * b) <= 1.65e-49)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 4.3e+49)
		tmp = Float64(z * t);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -1.6e+79)
		tmp = a * b;
	elseif ((a * b) <= -2.5e-293)
		tmp = z * t;
	elseif ((a * b) <= 3e-157)
		tmp = x * y;
	elseif ((a * b) <= 7.5e-72)
		tmp = c * i;
	elseif ((a * b) <= 1.65e-49)
		tmp = x * y;
	elseif ((a * b) <= 4.3e+49)
		tmp = z * t;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -1.6e+79], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2.5e-293], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 3e-157], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 7.5e-72], N[(c * i), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.65e-49], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4.3e+49], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 a b) < -1.60000000000000001e79 or 4.2999999999999999e49 < (*.f64 a b)

   1. Initial program 93.8%

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

   2. Taylor expanded in x around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. fma-def 85.9%

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

      3. *-commutative 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in b around inf 65.4%

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

   if -1.60000000000000001e79 < (*.f64 a b) < -2.5000000000000001e-293 or 1.65e-49 < (*.f64 a b) < 4.2999999999999999e49

   1. Initial program 96.5%

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

   2. Taylor expanded in x around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. fma-def 76.6%

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

      3. *-commutative 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in z around inf 48.4%

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

   if -2.5000000000000001e-293 < (*.f64 a b) < 3e-157 or 7.5000000000000004e-72 < (*.f64 a b) < 1.65e-49

   1. Initial program 98.3%

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

   2. Taylor expanded in x around inf 80.2%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]

   3. Taylor expanded in y around inf 50.4%

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

   if 3e-157 < (*.f64 a b) < 7.5000000000000004e-72

   1. Initial program 100.0%

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

   2. Taylor expanded in c around inf 55.3%

      \[\leadsto \color{blue}{c \cdot i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.6 \cdot 10^{+79}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.5 \cdot 10^{-293}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 3 \cdot 10^{-157}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 7.5 \cdot 10^{-72}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 1.65 \cdot 10^{-49}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 4.3 \cdot 10^{+49}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 5: 63.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot i + x \cdot y\\ t_2 := a \cdot b + z \cdot t\\ \mathbf{if}\;a \cdot b \leq -8.6 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq -3.6 \cdot 10^{-289}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.8 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 2.3 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 4.8 \cdot 10^{+230}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* c i) (* x y))) (t_2 (+ (* a b) (* z t))))
   (if (<= (* a b) -8.6e+56)
     t_2
     (if (<= (* a b) -3.6e-289)
       (+ (* c i) (* z t))
       (if (<= (* a b) 2.8e-49)
         t_1
         (if (<= (* a b) 2.3e+87)
           t_2
           (if (<= (* a b) 4.8e+230) t_1 (* a b))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * i) + (x * y);
	double t_2 = (a * b) + (z * t);
	double tmp;
	if ((a * b) <= -8.6e+56) {
		tmp = t_2;
	} else if ((a * b) <= -3.6e-289) {
		tmp = (c * i) + (z * t);
	} else if ((a * b) <= 2.8e-49) {
		tmp = t_1;
	} else if ((a * b) <= 2.3e+87) {
		tmp = t_2;
	} else if ((a * b) <= 4.8e+230) {
		tmp = t_1;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (c * i) + (x * y)
    t_2 = (a * b) + (z * t)
    if ((a * b) <= (-8.6d+56)) then
        tmp = t_2
    else if ((a * b) <= (-3.6d-289)) then
        tmp = (c * i) + (z * t)
    else if ((a * b) <= 2.8d-49) then
        tmp = t_1
    else if ((a * b) <= 2.3d+87) then
        tmp = t_2
    else if ((a * b) <= 4.8d+230) then
        tmp = t_1
    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 c, double i) {
	double t_1 = (c * i) + (x * y);
	double t_2 = (a * b) + (z * t);
	double tmp;
	if ((a * b) <= -8.6e+56) {
		tmp = t_2;
	} else if ((a * b) <= -3.6e-289) {
		tmp = (c * i) + (z * t);
	} else if ((a * b) <= 2.8e-49) {
		tmp = t_1;
	} else if ((a * b) <= 2.3e+87) {
		tmp = t_2;
	} else if ((a * b) <= 4.8e+230) {
		tmp = t_1;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (c * i) + (x * y)
	t_2 = (a * b) + (z * t)
	tmp = 0
	if (a * b) <= -8.6e+56:
		tmp = t_2
	elif (a * b) <= -3.6e-289:
		tmp = (c * i) + (z * t)
	elif (a * b) <= 2.8e-49:
		tmp = t_1
	elif (a * b) <= 2.3e+87:
		tmp = t_2
	elif (a * b) <= 4.8e+230:
		tmp = t_1
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) + Float64(x * y))
	t_2 = Float64(Float64(a * b) + Float64(z * t))
	tmp = 0.0
	if (Float64(a * b) <= -8.6e+56)
		tmp = t_2;
	elseif (Float64(a * b) <= -3.6e-289)
		tmp = Float64(Float64(c * i) + Float64(z * t));
	elseif (Float64(a * b) <= 2.8e-49)
		tmp = t_1;
	elseif (Float64(a * b) <= 2.3e+87)
		tmp = t_2;
	elseif (Float64(a * b) <= 4.8e+230)
		tmp = t_1;
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (c * i) + (x * y);
	t_2 = (a * b) + (z * t);
	tmp = 0.0;
	if ((a * b) <= -8.6e+56)
		tmp = t_2;
	elseif ((a * b) <= -3.6e-289)
		tmp = (c * i) + (z * t);
	elseif ((a * b) <= 2.8e-49)
		tmp = t_1;
	elseif ((a * b) <= 2.3e+87)
		tmp = t_2;
	elseif ((a * b) <= 4.8e+230)
		tmp = t_1;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * i), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -8.6e+56], t$95$2, If[LessEqual[N[(a * b), $MachinePrecision], -3.6e-289], N[(N[(c * i), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2.8e-49], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 2.3e+87], t$95$2, If[LessEqual[N[(a * b), $MachinePrecision], 4.8e+230], t$95$1, N[(a * b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 a b) < -8.6000000000000007e56 or 2.79999999999999997e-49 < (*.f64 a b) < 2.3000000000000002e87

   1. Initial program 96.0%

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

   2. Taylor expanded in x around 0 86.9%

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

      1. *-commutative 86.9%

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

      2. fma-def 86.9%

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

      3. *-commutative 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in c around 0 78.0%

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

   if -8.6000000000000007e56 < (*.f64 a b) < -3.6e-289

   1. Initial program 95.1%

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

   2. Taylor expanded in z around inf 71.7%

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]

   if -3.6e-289 < (*.f64 a b) < 2.79999999999999997e-49 or 2.3000000000000002e87 < (*.f64 a b) < 4.79999999999999996e230

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 74.5%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]

   if 4.79999999999999996e230 < (*.f64 a b)

   1. Initial program 87.5%

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

   2. Taylor expanded in x around 0 91.7%

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

      1. *-commutative 91.7%

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

      2. fma-def 95.8%

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

      3. *-commutative 95.8%

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in b around inf 95.8%

      \[\leadsto \color{blue}{a \cdot b} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -8.6 \cdot 10^{+56}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq -3.6 \cdot 10^{-289}:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.8 \cdot 10^{-49}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2.3 \cdot 10^{+87}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 4.8 \cdot 10^{+230}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 6: 64.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot i + z \cdot t\\ t_2 := a \cdot b + z \cdot t\\ \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot b \leq 1.12 \cdot 10^{-197}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 4400000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* c i) (* z t))) (t_2 (+ (* a b) (* z t))))
   (if (<= (* a b) -4e+56)
     t_2
     (if (<= (* a b) 0.0)
       t_1
       (if (<= (* a b) 1.12e-197)
         (* x y)
         (if (<= (* a b) 4400000000.0) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * i) + (z * t);
	double t_2 = (a * b) + (z * t);
	double tmp;
	if ((a * b) <= -4e+56) {
		tmp = t_2;
	} else if ((a * b) <= 0.0) {
		tmp = t_1;
	} else if ((a * b) <= 1.12e-197) {
		tmp = x * y;
	} else if ((a * b) <= 4400000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (c * i) + (z * t)
    t_2 = (a * b) + (z * t)
    if ((a * b) <= (-4d+56)) then
        tmp = t_2
    else if ((a * b) <= 0.0d0) then
        tmp = t_1
    else if ((a * b) <= 1.12d-197) then
        tmp = x * y
    else if ((a * b) <= 4400000000.0d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * i) + (z * t);
	double t_2 = (a * b) + (z * t);
	double tmp;
	if ((a * b) <= -4e+56) {
		tmp = t_2;
	} else if ((a * b) <= 0.0) {
		tmp = t_1;
	} else if ((a * b) <= 1.12e-197) {
		tmp = x * y;
	} else if ((a * b) <= 4400000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (c * i) + (z * t)
	t_2 = (a * b) + (z * t)
	tmp = 0
	if (a * b) <= -4e+56:
		tmp = t_2
	elif (a * b) <= 0.0:
		tmp = t_1
	elif (a * b) <= 1.12e-197:
		tmp = x * y
	elif (a * b) <= 4400000000.0:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) + Float64(z * t))
	t_2 = Float64(Float64(a * b) + Float64(z * t))
	tmp = 0.0
	if (Float64(a * b) <= -4e+56)
		tmp = t_2;
	elseif (Float64(a * b) <= 0.0)
		tmp = t_1;
	elseif (Float64(a * b) <= 1.12e-197)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 4400000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (c * i) + (z * t);
	t_2 = (a * b) + (z * t);
	tmp = 0.0;
	if ((a * b) <= -4e+56)
		tmp = t_2;
	elseif ((a * b) <= 0.0)
		tmp = t_1;
	elseif ((a * b) <= 1.12e-197)
		tmp = x * y;
	elseif ((a * b) <= 4400000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * i), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * b), $MachinePrecision], -4e+56], t$95$2, If[LessEqual[N[(a * b), $MachinePrecision], 0.0], t$95$1, If[LessEqual[N[(a * b), $MachinePrecision], 1.12e-197], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 4400000000.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.00000000000000037e56 or 4.4e9 < (*.f64 a b)

   1. Initial program 94.5%

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

   2. Taylor expanded in x around 0 84.9%

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

      1. *-commutative 84.9%

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

      2. fma-def 85.8%

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

      3. *-commutative 85.8%

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

   4. Applied egg-rr 85.8%

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

   5. Taylor expanded in c around 0 76.0%

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

   if -4.00000000000000037e56 < (*.f64 a b) < -0.0 or 1.12e-197 < (*.f64 a b) < 4.4e9

   1. Initial program 97.7%

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

   2. Taylor expanded in z around inf 68.2%

      \[\leadsto \color{blue}{t \cdot z} + c \cdot i \]

   if -0.0 < (*.f64 a b) < 1.12e-197

   1. Initial program 92.9%

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

   2. Taylor expanded in x around inf 85.8%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]

   3. Taylor expanded in y around inf 72.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4 \cdot 10^{+56}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 0:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.12 \cdot 10^{-197}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 4400000000:\\ \;\;\;\;c \cdot i + z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 7: 50.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot b + c \cdot i\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+253}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+134}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-176}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* a b) (* c i))))
   (if (<= x -1.6e+253)
     (* x y)
     (if (<= x -8e+177)
       t_1
       (if (<= x -8.5e+134)
         (* x y)
         (if (<= x -7e-142)
           t_1
           (if (<= x -2.3e-176) (* z t) (if (<= x 8e-90) t_1 (* x y)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a * b) + (c * i);
	double tmp;
	if (x <= -1.6e+253) {
		tmp = x * y;
	} else if (x <= -8e+177) {
		tmp = t_1;
	} else if (x <= -8.5e+134) {
		tmp = x * y;
	} else if (x <= -7e-142) {
		tmp = t_1;
	} else if (x <= -2.3e-176) {
		tmp = z * t;
	} else if (x <= 8e-90) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * b) + (c * i)
    if (x <= (-1.6d+253)) then
        tmp = x * y
    else if (x <= (-8d+177)) then
        tmp = t_1
    else if (x <= (-8.5d+134)) then
        tmp = x * y
    else if (x <= (-7d-142)) then
        tmp = t_1
    else if (x <= (-2.3d-176)) then
        tmp = z * t
    else if (x <= 8d-90) then
        tmp = t_1
    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 c, double i) {
	double t_1 = (a * b) + (c * i);
	double tmp;
	if (x <= -1.6e+253) {
		tmp = x * y;
	} else if (x <= -8e+177) {
		tmp = t_1;
	} else if (x <= -8.5e+134) {
		tmp = x * y;
	} else if (x <= -7e-142) {
		tmp = t_1;
	} else if (x <= -2.3e-176) {
		tmp = z * t;
	} else if (x <= 8e-90) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (a * b) + (c * i)
	tmp = 0
	if x <= -1.6e+253:
		tmp = x * y
	elif x <= -8e+177:
		tmp = t_1
	elif x <= -8.5e+134:
		tmp = x * y
	elif x <= -7e-142:
		tmp = t_1
	elif x <= -2.3e-176:
		tmp = z * t
	elif x <= 8e-90:
		tmp = t_1
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a * b) + Float64(c * i))
	tmp = 0.0
	if (x <= -1.6e+253)
		tmp = Float64(x * y);
	elseif (x <= -8e+177)
		tmp = t_1;
	elseif (x <= -8.5e+134)
		tmp = Float64(x * y);
	elseif (x <= -7e-142)
		tmp = t_1;
	elseif (x <= -2.3e-176)
		tmp = Float64(z * t);
	elseif (x <= 8e-90)
		tmp = t_1;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (a * b) + (c * i);
	tmp = 0.0;
	if (x <= -1.6e+253)
		tmp = x * y;
	elseif (x <= -8e+177)
		tmp = t_1;
	elseif (x <= -8.5e+134)
		tmp = x * y;
	elseif (x <= -7e-142)
		tmp = t_1;
	elseif (x <= -2.3e-176)
		tmp = z * t;
	elseif (x <= 8e-90)
		tmp = t_1;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e+253], N[(x * y), $MachinePrecision], If[LessEqual[x, -8e+177], t$95$1, If[LessEqual[x, -8.5e+134], N[(x * y), $MachinePrecision], If[LessEqual[x, -7e-142], t$95$1, If[LessEqual[x, -2.3e-176], N[(z * t), $MachinePrecision], If[LessEqual[x, 8e-90], t$95$1, N[(x * y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000002e253 or -8.0000000000000001e177 < x < -8.50000000000000024e134 or 7.99999999999999996e-90 < x

   1. Initial program 92.6%

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

   2. Taylor expanded in x around inf 64.8%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]

   3. Taylor expanded in y around inf 50.3%

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

   if -1.6000000000000002e253 < x < -8.0000000000000001e177 or -8.50000000000000024e134 < x < -7.00000000000000029e-142 or -2.3000000000000001e-176 < x < 7.99999999999999996e-90

   1. Initial program 98.6%

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

   2. Taylor expanded in a around inf 60.5%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   if -7.00000000000000029e-142 < x < -2.3000000000000001e-176

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf 71.9%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+253}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+177}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+134}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-142}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-176}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-90}:\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.6 \cdot 10^{+80}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.05 \cdot 10^{-280}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 3.2 \cdot 10^{+49}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -3.6e+80)
   (* a b)
   (if (<= (* a b) -1.05e-280)
     (* z t)
     (if (<= (* a b) 2.1e-49)
       (* c i)
       (if (<= (* a b) 3.2e+49) (* z t) (* a b))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -3.6e+80) {
		tmp = a * b;
	} else if ((a * b) <= -1.05e-280) {
		tmp = z * t;
	} else if ((a * b) <= 2.1e-49) {
		tmp = c * i;
	} else if ((a * b) <= 3.2e+49) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-3.6d+80)) then
        tmp = a * b
    else if ((a * b) <= (-1.05d-280)) then
        tmp = z * t
    else if ((a * b) <= 2.1d-49) then
        tmp = c * i
    else if ((a * b) <= 3.2d+49) then
        tmp = z * t
    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 c, double i) {
	double tmp;
	if ((a * b) <= -3.6e+80) {
		tmp = a * b;
	} else if ((a * b) <= -1.05e-280) {
		tmp = z * t;
	} else if ((a * b) <= 2.1e-49) {
		tmp = c * i;
	} else if ((a * b) <= 3.2e+49) {
		tmp = z * t;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -3.6e+80:
		tmp = a * b
	elif (a * b) <= -1.05e-280:
		tmp = z * t
	elif (a * b) <= 2.1e-49:
		tmp = c * i
	elif (a * b) <= 3.2e+49:
		tmp = z * t
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -3.6e+80)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -1.05e-280)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 2.1e-49)
		tmp = Float64(c * i);
	elseif (Float64(a * b) <= 3.2e+49)
		tmp = Float64(z * t);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -3.6e+80)
		tmp = a * b;
	elseif ((a * b) <= -1.05e-280)
		tmp = z * t;
	elseif ((a * b) <= 2.1e-49)
		tmp = c * i;
	elseif ((a * b) <= 3.2e+49)
		tmp = z * t;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -3.6e+80], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -1.05e-280], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2.1e-49], N[(c * i), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 3.2e+49], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -3.59999999999999995e80 or 3.20000000000000014e49 < (*.f64 a b)

   1. Initial program 93.8%

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

   2. Taylor expanded in x around 0 84.8%

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

      1. *-commutative 84.8%

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

      2. fma-def 85.9%

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

      3. *-commutative 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in b around inf 65.4%

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

   if -3.59999999999999995e80 < (*.f64 a b) < -1.05e-280 or 2.0999999999999999e-49 < (*.f64 a b) < 3.20000000000000014e49

   1. Initial program 96.5%

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

   2. Taylor expanded in x around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. fma-def 76.3%

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

      3. *-commutative 76.3%

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

   4. Applied egg-rr 76.3%

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

   5. Taylor expanded in z around inf 48.9%

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

   if -1.05e-280 < (*.f64 a b) < 2.0999999999999999e-49

   1. Initial program 98.6%

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

   2. Taylor expanded in c around inf 37.4%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -3.6 \cdot 10^{+80}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1.05 \cdot 10^{-280}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 3.2 \cdot 10^{+49}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 9: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot i + \left(a \cdot b + z \cdot t\right)\\ t_2 := c \cdot i + x \cdot y\\ \mathbf{if}\;y \leq -1.06 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* c i) (+ (* a b) (* z t)))) (t_2 (+ (* c i) (* x y))))
   (if (<= y -1.06e-44)
     t_2
     (if (<= y 7.5e+59)
       t_1
       (if (<= y 8e+153) t_2 (if (<= y 7.2e+228) t_1 (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * i) + ((a * b) + (z * t));
	double t_2 = (c * i) + (x * y);
	double tmp;
	if (y <= -1.06e-44) {
		tmp = t_2;
	} else if (y <= 7.5e+59) {
		tmp = t_1;
	} else if (y <= 8e+153) {
		tmp = t_2;
	} else if (y <= 7.2e+228) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (c * i) + ((a * b) + (z * t))
    t_2 = (c * i) + (x * y)
    if (y <= (-1.06d-44)) then
        tmp = t_2
    else if (y <= 7.5d+59) then
        tmp = t_1
    else if (y <= 8d+153) then
        tmp = t_2
    else if (y <= 7.2d+228) then
        tmp = t_1
    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 c, double i) {
	double t_1 = (c * i) + ((a * b) + (z * t));
	double t_2 = (c * i) + (x * y);
	double tmp;
	if (y <= -1.06e-44) {
		tmp = t_2;
	} else if (y <= 7.5e+59) {
		tmp = t_1;
	} else if (y <= 8e+153) {
		tmp = t_2;
	} else if (y <= 7.2e+228) {
		tmp = t_1;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (c * i) + ((a * b) + (z * t))
	t_2 = (c * i) + (x * y)
	tmp = 0
	if y <= -1.06e-44:
		tmp = t_2
	elif y <= 7.5e+59:
		tmp = t_1
	elif y <= 8e+153:
		tmp = t_2
	elif y <= 7.2e+228:
		tmp = t_1
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * i) + Float64(Float64(a * b) + Float64(z * t)))
	t_2 = Float64(Float64(c * i) + Float64(x * y))
	tmp = 0.0
	if (y <= -1.06e-44)
		tmp = t_2;
	elseif (y <= 7.5e+59)
		tmp = t_1;
	elseif (y <= 8e+153)
		tmp = t_2;
	elseif (y <= 7.2e+228)
		tmp = t_1;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (c * i) + ((a * b) + (z * t));
	t_2 = (c * i) + (x * y);
	tmp = 0.0;
	if (y <= -1.06e-44)
		tmp = t_2;
	elseif (y <= 7.5e+59)
		tmp = t_1;
	elseif (y <= 8e+153)
		tmp = t_2;
	elseif (y <= 7.2e+228)
		tmp = t_1;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * i), $MachinePrecision] + N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * i), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.06e-44], t$95$2, If[LessEqual[y, 7.5e+59], t$95$1, If[LessEqual[y, 8e+153], t$95$2, If[LessEqual[y, 7.2e+228], t$95$1, N[(x * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.0599999999999999e-44 or 7.4999999999999996e59 < y < 8e153

   1. Initial program 94.5%

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

   2. Taylor expanded in x around inf 62.7%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]

   if -1.0599999999999999e-44 < y < 7.4999999999999996e59 or 8e153 < y < 7.2e228

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0 86.1%

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

   if 7.2e228 < y

   1. Initial program 83.3%

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

   2. Taylor expanded in x around inf 78.7%

      \[\leadsto \color{blue}{y \cdot x} + c \cdot i \]

   3. Taylor expanded in y around inf 84.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{-44}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+59}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+153}:\\ \;\;\;\;c \cdot i + x \cdot y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+228}:\\ \;\;\;\;c \cdot i + \left(a \cdot b + z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10: 61.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{+117} \lor \neg \left(c \leq 3.4 \cdot 10^{-41}\right):\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= c -3.5e+117) (not (<= c 3.4e-41)))
   (+ (* a b) (* c i))
   (+ (* a b) (* z t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c <= -3.5e+117) || !(c <= 3.4e-41)) {
		tmp = (a * b) + (c * i);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((c <= (-3.5d+117)) .or. (.not. (c <= 3.4d-41))) then
        tmp = (a * b) + (c * i)
    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 c, double i) {
	double tmp;
	if ((c <= -3.5e+117) || !(c <= 3.4e-41)) {
		tmp = (a * b) + (c * i);
	} else {
		tmp = (a * b) + (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (c <= -3.5e+117) or not (c <= 3.4e-41):
		tmp = (a * b) + (c * i)
	else:
		tmp = (a * b) + (z * t)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((c <= -3.5e+117) || !(c <= 3.4e-41))
		tmp = Float64(Float64(a * b) + Float64(c * i));
	else
		tmp = Float64(Float64(a * b) + Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((c <= -3.5e+117) || ~((c <= 3.4e-41)))
		tmp = (a * b) + (c * i);
	else
		tmp = (a * b) + (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[c, -3.5e+117], N[Not[LessEqual[c, 3.4e-41]], $MachinePrecision]], N[(N[(a * b), $MachinePrecision] + N[(c * i), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.49999999999999983e117 or 3.3999999999999998e-41 < c

   1. Initial program 94.9%

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

   2. Taylor expanded in a around inf 61.3%

      \[\leadsto \color{blue}{a \cdot b} + c \cdot i \]

   if -3.49999999999999983e117 < c < 3.3999999999999998e-41

   1. Initial program 96.8%

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

   2. Taylor expanded in x around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. fma-def 72.0%

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

      3. *-commutative 72.0%

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

   4. Applied egg-rr 72.0%

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

   5. Taylor expanded in c around 0 62.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{+117} \lor \neg \left(c \leq 3.4 \cdot 10^{-41}\right):\\ \;\;\;\;a \cdot b + c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b + z \cdot t\\ \end{array} \]

Alternative 11: 44.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+57}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 9.6 \cdot 10^{+24}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2e+57) (* a b) (if (<= (* a b) 9.6e+24) (* c i) (* a b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((a * b) <= -2e+57) {
		tmp = a * b;
	} else if ((a * b) <= 9.6e+24) {
		tmp = c * i;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    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), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((a * b) <= (-2d+57)) then
        tmp = a * b
    else if ((a * b) <= 9.6d+24) then
        tmp = c * i
    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 c, double i) {
	double tmp;
	if ((a * b) <= -2e+57) {
		tmp = a * b;
	} else if ((a * b) <= 9.6e+24) {
		tmp = c * i;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -2e+57:
		tmp = a * b
	elif (a * b) <= 9.6e+24:
		tmp = c * i
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -2e+57)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 9.6e+24)
		tmp = Float64(c * i);
	else
		tmp = Float64(a * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((a * b) <= -2e+57)
		tmp = a * b;
	elseif ((a * b) <= 9.6e+24)
		tmp = c * i;
	else
		tmp = a * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(a * b), $MachinePrecision], -2e+57], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 9.6e+24], N[(c * i), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.0000000000000001e57 or 9.6000000000000003e24 < (*.f64 a b)

   1. Initial program 94.4%

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

   2. Taylor expanded in x around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. fma-def 85.4%

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

      3. *-commutative 85.4%

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

   4. Applied egg-rr 85.4%

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

   5. Taylor expanded in b around inf 61.5%

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

   if -2.0000000000000001e57 < (*.f64 a b) < 9.6000000000000003e24

   1. Initial program 97.3%

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

   2. Taylor expanded in c around inf 32.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+57}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 9.6 \cdot 10^{+24}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 12: 28.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

2. Taylor expanded in x around 0 74.5%

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

   1. *-commutative 74.5%

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

   2. fma-def 74.9%

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

   3. *-commutative 74.9%

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

4. Applied egg-rr 74.9%

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

5. Taylor expanded in b around inf 28.2%

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

6. Final simplification 28.2%

   \[\leadsto a \cdot b \]

Reproduce

herbie shell --seed 2023252 
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3, C"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))