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

Percentage Accurate: 95.6% → 97.9%

Time: 10.4s

Alternatives: 16

Speedup: 1.3×

• 41.3% 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: 95.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.8%

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i \]
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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.0

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate-+l+ N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 98.4

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 43.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+127}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-204}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-87}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 3.48 \cdot 10^{+154}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* a b) -2e+127)
   (* a b)
   (if (<= (* a b) -1e-204)
     (* x y)
     (if (<= (* a b) 2e-87)
       (* c i)
       (if (<= (* a b) 3.48e+154) (* 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) <= -2e+127) {
		tmp = a * b;
	} else if ((a * b) <= -1e-204) {
		tmp = x * y;
	} else if ((a * b) <= 2e-87) {
		tmp = c * i;
	} else if ((a * b) <= 3.48e+154) {
		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) <= (-2d+127)) then
        tmp = a * b
    else if ((a * b) <= (-1d-204)) then
        tmp = x * y
    else if ((a * b) <= 2d-87) then
        tmp = c * i
    else if ((a * b) <= 3.48d+154) 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) <= -2e+127) {
		tmp = a * b;
	} else if ((a * b) <= -1e-204) {
		tmp = x * y;
	} else if ((a * b) <= 2e-87) {
		tmp = c * i;
	} else if ((a * b) <= 3.48e+154) {
		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) <= -2e+127:
		tmp = a * b
	elif (a * b) <= -1e-204:
		tmp = x * y
	elif (a * b) <= 2e-87:
		tmp = c * i
	elif (a * b) <= 3.48e+154:
		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) <= -2e+127)
		tmp = Float64(a * b);
	elseif (Float64(a * b) <= -1e-204)
		tmp = Float64(x * y);
	elseif (Float64(a * b) <= 2e-87)
		tmp = Float64(c * i);
	elseif (Float64(a * b) <= 3.48e+154)
		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) <= -2e+127)
		tmp = a * b;
	elseif ((a * b) <= -1e-204)
		tmp = x * y;
	elseif ((a * b) <= 2e-87)
		tmp = c * i;
	elseif ((a * b) <= 3.48e+154)
		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], -2e+127], N[(a * b), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], -1e-204], N[(x * y), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 2e-87], N[(c * i), $MachinePrecision], If[LessEqual[N[(a * b), $MachinePrecision], 3.48e+154], N[(x * y), $MachinePrecision], N[(a * b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a b) < -1.99999999999999991e127 or 3.47999999999999987e154 < (*.f64 a b)

   1. Initial program 90.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -1.99999999999999991e127 < (*.f64 a b) < -1e-204 or 2.00000000000000004e-87 < (*.f64 a b) < 3.47999999999999987e154

   1. Initial program 97.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.7

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

   5. Applied rewrites 30.7%

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

   if -1e-204 < (*.f64 a b) < 2.00000000000000004e-87

   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 c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 35.4

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

   5. Applied rewrites 35.4%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot b \leq -2 \cdot 10^{+127}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;a \cdot b \leq -1 \cdot 10^{-204}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;a \cdot b \leq 2 \cdot 10^{-87}:\\ \;\;\;\;c \cdot i\\ \mathbf{elif}\;a \cdot b \leq 3.48 \cdot 10^{+154}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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)))