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

Percentage Accurate: 95.9% → 98.1%

Time: 9.3s

Alternatives: 11

Speedup: 1.3×

• 41.6% 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: 95.9% 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: 98.1% 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 97.3%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
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 98.8

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

4. Applied rewrites 98.8%

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

Alternative 2: 42.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+103}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-227}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-259}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 10^{-106}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+78}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* z t) -1e+103)
   (* z t)
   (if (<= (* z t) -1e-227)
     (* c i)
     (if (<= (* z t) 4e-259)
       (* x y)
       (if (<= (* z t) 1e-106)
         (* a b)
         (if (<= (* z t) 5e+78) (* c i) (* 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) <= -1e+103) {
		tmp = z * t;
	} else if ((z * t) <= -1e-227) {
		tmp = c * i;
	} else if ((z * t) <= 4e-259) {
		tmp = x * y;
	} else if ((z * t) <= 1e-106) {
		tmp = a * b;
	} else if ((z * t) <= 5e+78) {
		tmp = c * i;
	} else {
		tmp = 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 ((z * t) <= (-1d+103)) then
        tmp = z * t
    else if ((z * t) <= (-1d-227)) then
        tmp = c * i
    else if ((z * t) <= 4d-259) then
        tmp = x * y
    else if ((z * t) <= 1d-106) then
        tmp = a * b
    else if ((z * t) <= 5d+78) then
        tmp = c * i
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((z * t) <= -1e+103) {
		tmp = z * t;
	} else if ((z * t) <= -1e-227) {
		tmp = c * i;
	} else if ((z * t) <= 4e-259) {
		tmp = x * y;
	} else if ((z * t) <= 1e-106) {
		tmp = a * b;
	} else if ((z * t) <= 5e+78) {
		tmp = c * i;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (z * t) <= -1e+103:
		tmp = z * t
	elif (z * t) <= -1e-227:
		tmp = c * i
	elif (z * t) <= 4e-259:
		tmp = x * y
	elif (z * t) <= 1e-106:
		tmp = a * b
	elif (z * t) <= 5e+78:
		tmp = c * i
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(z * t) <= -1e+103)
		tmp = Float64(z * t);
	elseif (Float64(z * t) <= -1e-227)
		tmp = Float64(c * i);
	elseif (Float64(z * t) <= 4e-259)
		tmp = Float64(x * y);
	elseif (Float64(z * t) <= 1e-106)
		tmp = Float64(a * b);
	elseif (Float64(z * t) <= 5e+78)
		tmp = Float64(c * i);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((z * t) <= -1e+103)
		tmp = z * t;
	elseif ((z * t) <= -1e-227)
		tmp = c * i;
	elseif ((z * t) <= 4e-259)
		tmp = x * y;
	elseif ((z * t) <= 1e-106)
		tmp = a * b;
	elseif ((z * t) <= 5e+78)
		tmp = c * i;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+103], N[(z * t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-227], N[(c * i), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e-259], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e-106], N[(a * b), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+78], N[(c * i), $MachinePrecision], N[(z * t), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -1e103 or 4.99999999999999984e78 < (*.f64 z t)

   1. Initial program 92.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if -1e103 < (*.f64 z t) < -9.99999999999999945e-228 or 9.99999999999999941e-107 < (*.f64 z t) < 4.99999999999999984e78

   1. Initial program 97.8%

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

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

   if -9.99999999999999945e-228 < (*.f64 z t) < 4.0000000000000003e-259

   1. Initial program 97.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 35.7

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

   5. Applied rewrites 35.7%

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

   if 4.0000000000000003e-259 < (*.f64 z t) < 9.99999999999999941e-107

   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 a around inf

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

      1. lower-*.f64 32.3

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

   5. Applied rewrites 32.3%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+103}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-227}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-259}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot t \leq 10^{-106}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+78}:\\ \;\;\;\;c \cdot i\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(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)))