Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x - y \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* y z)))

C

double code(double x, double y, double z) {
	return x - (y * z);
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return x - (y * z);
}

Python

def code(x, y, z):
	return x - (y * z)

Julia

function code(x, y, z)
	return Float64(x - Float64(y * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - (y * z);
end

Wolfram

code[x_, y_, z_] := N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - y \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* y z)))

C

double code(double x, double y, double z) {
	return x - (y * z);
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return x - (y * z);
}

Python

def code(x, y, z):
	return x - (y * z)

Julia

function code(x, y, z)
	return Float64(x - Float64(y * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - (y * z);
end

Wolfram

code[x_, y_, z_] := N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, -y, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z (- y) x))

C

double code(double x, double y, double z) {
	return fma(z, -y, x);
}

Julia

function code(x, y, z)
	return fma(z, Float64(-y), x)
end

Wolfram

code[x_, y_, z_] := N[(z * (-y) + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - y \cdot z \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. fma-define 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, -y, x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 68.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+105} \lor \neg \left(y \leq -7.6 \cdot 10^{-30}\right) \land \left(y \leq -8 \cdot 10^{-110} \lor \neg \left(y \leq 5.8 \cdot 10^{-115}\right)\right):\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.1e+105)
         (and (not (<= y -7.6e-30)) (or (<= y -8e-110) (not (<= y 5.8e-115)))))
   (* z (- y))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.1e+105) || (!(y <= -7.6e-30) && ((y <= -8e-110) || !(y <= 5.8e-115)))) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.1d+105)) .or. (.not. (y <= (-7.6d-30))) .and. (y <= (-8d-110)) .or. (.not. (y <= 5.8d-115))) then
        tmp = z * -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.1e+105) || (!(y <= -7.6e-30) && ((y <= -8e-110) || !(y <= 5.8e-115)))) {
		tmp = z * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.1e+105) or (not (y <= -7.6e-30) and ((y <= -8e-110) or not (y <= 5.8e-115))):
		tmp = z * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.1e+105) || (!(y <= -7.6e-30) && ((y <= -8e-110) || !(y <= 5.8e-115))))
		tmp = Float64(z * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.1e+105) || (~((y <= -7.6e-30)) && ((y <= -8e-110) || ~((y <= 5.8e-115)))))
		tmp = z * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.1e+105], And[N[Not[LessEqual[y, -7.6e-30]], $MachinePrecision], Or[LessEqual[y, -8e-110], N[Not[LessEqual[y, 5.8e-115]], $MachinePrecision]]]], N[(z * (-y)), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.1000000000000001e105 or -7.6000000000000006e-30 < y < -8.0000000000000004e-110 or 5.7999999999999996e-115 < y

   1. Initial program 100.0%

      \[x - y \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.2%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 71.2%

         \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot z} \]

      2. neg-mul-1 71.2%

         \[\leadsto \color{blue}{\left(-y\right)} \cdot z \]

      3. *-commutative 71.2%

         \[\leadsto \color{blue}{z \cdot \left(-y\right)} \]

   5. Simplified 71.2%

      \[\leadsto \color{blue}{z \cdot \left(-y\right)} \]

   if -2.1000000000000001e105 < y < -7.6000000000000006e-30 or -8.0000000000000004e-110 < y < 5.7999999999999996e-115

   1. Initial program 100.0%

      \[x - y \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 73.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+105} \lor \neg \left(y \leq -7.6 \cdot 10^{-30}\right) \land \left(y \leq -8 \cdot 10^{-110} \lor \neg \left(y \leq 5.8 \cdot 10^{-115}\right)\right):\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - z \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x (* z y)))

C

double code(double x, double y, double z) {
	return x - (z * y);
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return x - (z * y);
}

Python

def code(x, y, z):
	return x - (z * y)

Julia

function code(x, y, z)
	return Float64(x - Float64(z * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - (z * y);
end

Wolfram

code[x_, y_, z_] := N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x - y \cdot z \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x - z \cdot y \]
4. Add Preprocessing

Alternative 4: 50.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[x - y \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around inf 47.8%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer target: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + y \cdot z\\ \frac{t\_0}{\frac{t\_0}{x - y \cdot z}} \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (* y z)))) (/ t_0 (/ t_0 (- x (* y z))))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y * z);
	return t_0 / (t_0 / (x - (y * z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    t_0 = x + (y * z)
    code = t_0 / (t_0 / (x - (y * z)))
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y * z);
	return t_0 / (t_0 / (x - (y * z)));
}

Python

def code(x, y, z):
	t_0 = x + (y * z)
	return t_0 / (t_0 / (x - (y * z)))

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y * z))
	return Float64(t_0 / Float64(t_0 / Float64(x - Float64(y * z))))
end

MATLAB

function tmp = code(x, y, z)
	t_0 = x + (y * z);
	tmp = t_0 / (t_0 / (x - (y * z)));
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 / N[(t$95$0 / N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024097 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C"
  :precision binary64

  :alt
  (/ (+ x (* y z)) (/ (+ x (* y z)) (- x (* y z))))

  (- x (* y z)))