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

Percentage Accurate: 95.8% → 97.9%

Time: 9.5s

Alternatives: 10

Speedup: 1.3×

• 42.3% 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: 95.8% 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: 97.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(z * t + N[(x * y + N[(a * b + N[(c * i), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 97.3

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

4. Applied rewrites 97.3%

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

Alternative 2: 43.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.2 \cdot 10^{+95}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.3 \cdot 10^{-298}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 9.5 \cdot 10^{-35}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.35 \cdot 10^{+135}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -4.2e+95)
   (* a b)
   (if (<= (* a b) -2.3e-298)
     (* x y)
     (if (<= (* a b) 9.5e-35)
       (* z t)
       (if (<= (* a b) 1.35e+135) (* x y) (* 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) <= -4.2e+95) {
		tmp = a * b;
	} else if ((a * b) <= -2.3e-298) {
		tmp = x * y;
	} else if ((a * b) <= 9.5e-35) {
		tmp = z * t;
	} else if ((a * b) <= 1.35e+135) {
		tmp = x * y;
	} 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) <= (-4.2d+95)) then
        tmp = a * b
    else if ((a * b) <= (-2.3d-298)) then
        tmp = x * y
    else if ((a * b) <= 9.5d-35) then
        tmp = z * t
    else if ((a * b) <= 1.35d+135) 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 c, double i) {
	double tmp;
	if ((a * b) <= -4.2e+95) {
		tmp = a * b;
	} else if ((a * b) <= -2.3e-298) {
		tmp = x * y;
	} else if ((a * b) <= 9.5e-35) {
		tmp = z * t;
	} else if ((a * b) <= 1.35e+135) {
		tmp = x * y;
	} else {
		tmp = a * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (a * b) <= -4.2e+95:
		tmp = a * b
	elif (a * b) <= -2.3e-298:
		tmp = x * y
	elif (a * b) <= 9.5e-35:
		tmp = z * t
	elif (a * b) <= 1.35e+135:
		tmp = x * y
	else:
		tmp = a * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(a * b) <= -4.2e+95)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -2.3e-298)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 9.5e-35)
		tmp = Float64(z * t);
	elseif (Float64(a * b) <= 1.35e+135)
		tmp = Float64(x * y);
	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) <= -4.2e+95)
		tmp = a * b;
	elseif ((a * b) <= -2.3e-298)
		tmp = x * y;
	elseif ((a * b) <= 9.5e-35)
		tmp = z * t;
	elseif ((a * b) <= 1.35e+135)
		tmp = x * y;
	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], -4.2e+95], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -2.3e-298], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 9.5e-35], N[(z * t), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.35e+135], N[(x * y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -4.2e95 or 1.34999999999999992e135 < (*.f64 a b)

   1. Initial program 92.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -4.2e95 < (*.f64 a b) < -2.3000000000000001e-298 or 9.5000000000000003e-35 < (*.f64 a b) < 1.34999999999999992e135

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 42.2

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

   5. Applied rewrites 42.2%

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

   if -2.3000000000000001e-298 < (*.f64 a b) < 9.5000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 47.3

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

   5. Applied rewrites 47.3%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -4.2 \cdot 10^{+95}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -2.3 \cdot 10^{-298}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 9.5 \cdot 10^{-35}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;a \cdot b \leq 1.35 \cdot 10^{+135}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t, x \cdot y\right)\\ t_2 := x \cdot y + z \cdot t\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma z t (* x y))) (t_2 (+ (* x y) (* z t))))
   (if (<= t_2 -1e+158) t_1 (if (<= t_2 1e+208) (fma i c (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(z, t, (x * y));
	double t_2 = (x * y) + (z * t);
	double tmp;
	if (t_2 <= -1e+158) {
		tmp = t_1;
	} else if (t_2 <= 1e+208) {
		tmp = fma(i, c, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(z, t, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(z * t))
	tmp = 0.0
	if (t_2 <= -1e+158)
		tmp = t_1;
	elseif (t_2 <= 1e+208)
		tmp = fma(i, c, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+158], t$95$1, If[LessEqual[t$95$2, 1e+208], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (*.f64 z t)) < -9.99999999999999953e157 or 9.9999999999999998e207 < (+.f64 (*.f64 x y) (*.f64 z t))

   1. Initial program 92.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 82.0

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

   7. Applied rewrites 82.0%

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

   if -9.99999999999999953e157 < (+.f64 (*.f64 x y) (*.f64 z t)) < 9.9999999999999998e207

   1. Initial program 99.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 71.8

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

   7. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, c, a \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-219}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* a b))))
   (if (<= (* x y) -2e-14)
     t_1
     (if (<= (* x y) 1e-219)
       (fma z t (* a b))
       (if (<= (* x y) 2e+115) (fma z t (* c i)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, y, (a * b));
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = t_1;
	} else if ((x * y) <= 1e-219) {
		tmp = fma(z, t, (a * b));
	} else if ((x * y) <= 2e+115) {
		tmp = fma(z, t, (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(a * b))
	tmp = 0.0
	if (Float64(x * y) <= -2e-14)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-219)
		tmp = fma(z, t, Float64(a * b));
	elseif (Float64(x * y) <= 2e+115)
		tmp = fma(z, t, Float64(c * i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e-14], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-219], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+115], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e-14 or 2e115 < (*.f64 x y)

   1. Initial program 93.0%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(x, y, a \cdot b\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.5%

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

   if -2e-14 < (*.f64 x y) < 1e-219

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if 1e-219 < (*.f64 x y) < 2e115

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

Alternative 5: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* c i) -5e+187)
   (fma z t (* c i))
   (if (<= (* c i) 2e+285) (fma x y (fma a b (* z t))) (fma i c (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((c * i) <= -5e+187) {
		tmp = fma(z, t, (c * i));
	} else if ((c * i) <= 2e+285) {
		tmp = fma(x, y, fma(a, b, (z * t)));
	} else {
		tmp = fma(i, c, (a * b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(c * i) <= -5e+187)
		tmp = fma(z, t, Float64(c * i));
	elseif (Float64(c * i) <= 2e+285)
		tmp = fma(x, y, fma(a, b, Float64(z * t)));
	else
		tmp = fma(i, c, Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(c * i), $MachinePrecision], -5e+187], N[(z * t + N[(c * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * i), $MachinePrecision], 2e+285], N[(x * y + N[(a * b + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * c + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c i) < -5.0000000000000001e187

   1. Initial program 83.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

   if -5.0000000000000001e187 < (*.f64 c i) < 2e285

   1. Initial program 98.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if 2e285 < (*.f64 c i)

   1. Initial program 82.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 94.1

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

   5. Applied rewrites 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 94.1

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

   7. Applied rewrites 94.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot i \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(z, t, c \cdot i\right)\\ \mathbf{elif}\;c \cdot i \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\mathsf{fma}\left(x, y, \mathsf{fma}\left(a, b, z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, c, a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(z, t, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma x y (* a b))))
   (if (<= (* x y) -2e-14)
     t_1
     (if (<= (* x y) 5e+171) (fma z t (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(x, y, (a * b));
	double tmp;
	if ((x * y) <= -2e-14) {
		tmp = t_1;
	} else if ((x * y) <= 5e+171) {
		tmp = fma(z, t, (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(x, y, Float64(a * b))
	tmp = 0.0
	if (Float64(x * y) <= -2e-14)
		tmp = t_1;
	elseif (Float64(x * y) <= 5e+171)
		tmp = fma(z, t, Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e-14], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 5e+171], N[(z * t + N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e-14 or 5.0000000000000004e171 < (*.f64 x y)

   1. Initial program 92.5%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(x, y, a \cdot b\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.4%

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

   if -2e-14 < (*.f64 x y) < 5.0000000000000004e171

   1. Initial program 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 70.4

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

   7. Applied rewrites 70.4%

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

Alternative 7: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+153}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -2e+153)
   (* z t)
   (if (<= (* z t) 2e+239) (fma x y (* 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 ((z * t) <= -2e+153) {
		tmp = z * t;
	} else if ((z * t) <= 2e+239) {
		tmp = fma(x, y, (a * b));
	} else {
		tmp = z * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -2e+153)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= 2e+239)
		tmp = fma(x, y, Float64(a * b));
	else
		tmp = Float64(z * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+153], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+239], N[(x * y + N[(a * b), $MachinePrecision]), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e153 or 1.99999999999999998e239 < (*.f64 z t)

   1. Initial program 98.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -2e153 < (*.f64 z t) < 1.99999999999999998e239

   1. Initial program 95.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(x, y, a \cdot b\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 66.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+153}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+239}:\\ \;\;\;\;\mathsf{fma}\left(x, y, a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.8 \cdot 10^{+61}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.6 \cdot 10^{+95}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2.8e+61) (* a b) (if (<= (* a b) 3.6e+95) (* 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) <= -2.8e+61) {
		tmp = a * b;
	} else if ((a * b) <= 3.6e+95) {
		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) <= (-2.8d+61)) then
        tmp = a * b
    else if ((a * b) <= 3.6d+95) 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) <= -2.8e+61) {
		tmp = a * b;
	} else if ((a * b) <= 3.6e+95) {
		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) <= -2.8e+61:
		tmp = a * b
	elif (a * b) <= 3.6e+95:
		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) <= -2.8e+61)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 3.6e+95)
		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) <= -2.8e+61)
		tmp = a * b;
	elseif ((a * b) <= 3.6e+95)
		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], -2.8e+61], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 3.6e+95], N[(z * t), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -2.8000000000000001e61 or 3.59999999999999978e95 < (*.f64 a b)

   1. Initial program 92.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if -2.8000000000000001e61 < (*.f64 a b) < 3.59999999999999978e95

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 35.1

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

   5. Applied rewrites 35.1%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2.8 \cdot 10^{+61}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 3.6 \cdot 10^{+95}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -1.45 \cdot 10^{+52}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq 1.75 \cdot 10^{+29}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -1.45e+52)
   (* a b)
   (if (<= (* a b) 1.75e+29) (* 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) <= -1.45e+52) {
		tmp = a * b;
	} else if ((a * b) <= 1.75e+29) {
		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) <= (-1.45d+52)) then
        tmp = a * b
    else if ((a * b) <= 1.75d+29) 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) <= -1.45e+52) {
		tmp = a * b;
	} else if ((a * b) <= 1.75e+29) {
		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) <= -1.45e+52:
		tmp = a * b
	elif (a * b) <= 1.75e+29:
		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) <= -1.45e+52)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= 1.75e+29)
		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) <= -1.45e+52)
		tmp = a * b;
	elseif ((a * b) <= 1.75e+29)
		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], -1.45e+52], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 1.75e+29], N[(c * i), $MachinePrecision], N[(a * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a b) < -1.45e52 or 1.74999999999999989e29 < (*.f64 a b)

   1. Initial program 93.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   if -1.45e52 < (*.f64 a b) < 1.74999999999999989e29

   1. Initial program 97.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 31.2

         \[\leadsto \color{blue}{c \cdot i} \]

   5. Applied rewrites 31.2%

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

Alternative 10: 26.9% 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. Add Preprocessing

3. Taylor expanded in a around inf

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

   1. lower-*.f64 29.1

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

5. Applied rewrites 29.1%

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

Reproduce

herbie shell --seed 2024221 
(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)))