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

Percentage Accurate: 100.0% → 100.0%

Time: 1.6s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-146} \lor \neg \left(z \leq 2.25 \cdot 10^{-108} \lor \neg \left(z \leq 8.8 \cdot 10^{-36}\right) \land z \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.1e-146)
         (not (or (<= z 2.25e-108) (and (not (<= z 8.8e-36)) (<= z 3.4e-16)))))
   (- (* y z))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e-146) || !((z <= 2.25e-108) || (!(z <= 8.8e-36) && (z <= 3.4e-16)))) {
		tmp = -(y * z);
	} 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 ((z <= (-2.1d-146)) .or. (.not. (z <= 2.25d-108) .or. (.not. (z <= 8.8d-36)) .and. (z <= 3.4d-16))) then
        tmp = -(y * z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e-146) || !((z <= 2.25e-108) || (!(z <= 8.8e-36) && (z <= 3.4e-16)))) {
		tmp = -(y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.1e-146) or not ((z <= 2.25e-108) or (not (z <= 8.8e-36) and (z <= 3.4e-16))):
		tmp = -(y * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.1e-146) || !((z <= 2.25e-108) || (!(z <= 8.8e-36) && (z <= 3.4e-16))))
		tmp = Float64(-Float64(y * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.1e-146) || ~(((z <= 2.25e-108) || (~((z <= 8.8e-36)) && (z <= 3.4e-16)))))
		tmp = -(y * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.1e-146], N[Not[Or[LessEqual[z, 2.25e-108], And[N[Not[LessEqual[z, 8.8e-36]], $MachinePrecision], LessEqual[z, 3.4e-16]]]], $MachinePrecision]], (-N[(y * z), $MachinePrecision]), x]

Derivation

1. Split input into 2 regimes

2. if z < -2.0999999999999999e-146 or 2.24999999999999985e-108 < z < 8.7999999999999997e-36 or 3.4e-16 < z

   1. Initial program 100.0%

      \[x - y \cdot z \]

   2. Taylor expanded in x around 0 74.1%

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

      1. mul-1-neg 74.1%

         \[\leadsto \color{blue}{-y \cdot z} \]

      2. distribute-rgt-neg-out 74.1%

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

   4. Simplified 74.1%

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

   if -2.0999999999999999e-146 < z < 2.24999999999999985e-108 or 8.7999999999999997e-36 < z < 3.4e-16

   1. Initial program 100.0%

      \[x - y \cdot z \]

   2. Taylor expanded in x around inf 74.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-146} \lor \neg \left(z \leq 2.25 \cdot 10^{-108} \lor \neg \left(z \leq 8.8 \cdot 10^{-36}\right) \land z \leq 3.4 \cdot 10^{-16}\right):\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 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]

Derivation

1. Initial program 100.0%

   \[x - y \cdot z \]

2. Final simplification 100.0%

   \[\leadsto x - y \cdot z \]

Alternative 4: 50.4% 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. Taylor expanded in x around inf 41.7%

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

3. Final simplification 41.7%

   \[\leadsto x \]

Developer target: 87.6% 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 2023229 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C"
  :precision binary64

  :herbie-target
  (/ (+ x (* y z)) (/ (+ x (* y z)) (- x (* y z))))

  (- x (* y z)))