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

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

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(Float64(-z), 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. 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. *-commutative 100.0%

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

   4. distribute-lft-neg-in 100.0%

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

   5. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 69.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+123} \lor \neg \left(y \leq -9.5 \cdot 10^{+105}\right) \land \left(y \leq -2 \cdot 10^{+77} \lor \neg \left(y \leq -1.1 \cdot 10^{+43}\right) \land \left(y \leq -1.75 \cdot 10^{-24} \lor \neg \left(y \leq 2.4 \cdot 10^{-91}\right)\right)\right):\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8.6e+123)
         (and (not (<= y -9.5e+105))
              (or (<= y -2e+77)
                  (and (not (<= y -1.1e+43))
                       (or (<= y -1.75e-24) (not (<= y 2.4e-91)))))))
   (* (- z) y)
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.6e+123) || (!(y <= -9.5e+105) && ((y <= -2e+77) || (!(y <= -1.1e+43) && ((y <= -1.75e-24) || !(y <= 2.4e-91)))))) {
		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 <= (-8.6d+123)) .or. (.not. (y <= (-9.5d+105))) .and. (y <= (-2d+77)) .or. (.not. (y <= (-1.1d+43))) .and. (y <= (-1.75d-24)) .or. (.not. (y <= 2.4d-91))) 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 <= -8.6e+123) || (!(y <= -9.5e+105) && ((y <= -2e+77) || (!(y <= -1.1e+43) && ((y <= -1.75e-24) || !(y <= 2.4e-91)))))) {
		tmp = -z * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8.6e+123) or (not (y <= -9.5e+105) and ((y <= -2e+77) or (not (y <= -1.1e+43) and ((y <= -1.75e-24) or not (y <= 2.4e-91))))):
		tmp = -z * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8.6e+123) || (!(y <= -9.5e+105) && ((y <= -2e+77) || (!(y <= -1.1e+43) && ((y <= -1.75e-24) || !(y <= 2.4e-91))))))
		tmp = Float64(Float64(-z) * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8.6e+123) || (~((y <= -9.5e+105)) && ((y <= -2e+77) || (~((y <= -1.1e+43)) && ((y <= -1.75e-24) || ~((y <= 2.4e-91)))))))
		tmp = -z * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8.6e+123], And[N[Not[LessEqual[y, -9.5e+105]], $MachinePrecision], Or[LessEqual[y, -2e+77], And[N[Not[LessEqual[y, -1.1e+43]], $MachinePrecision], Or[LessEqual[y, -1.75e-24], N[Not[LessEqual[y, 2.4e-91]], $MachinePrecision]]]]]], N[((-z) * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -8.59999999999999972e123 or -9.4999999999999995e105 < y < -1.99999999999999997e77 or -1.1e43 < y < -1.7499999999999998e-24 or 2.40000000000000011e-91 < y

   1. Initial program 100.0%

      \[x - y \cdot z \]

   2. Taylor expanded in x around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      2. *-commutative 70.8%

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

      3. distribute-rgt-neg-in 70.8%

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

   4. Simplified 70.8%

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

   if -8.59999999999999972e123 < y < -9.4999999999999995e105 or -1.99999999999999997e77 < y < -1.1e43 or -1.7499999999999998e-24 < y < 2.40000000000000011e-91

   1. Initial program 100.0%

      \[x - y \cdot z \]

   2. Taylor expanded in x around inf 75.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+123} \lor \neg \left(y \leq -9.5 \cdot 10^{+105}\right) \land \left(y \leq -2 \cdot 10^{+77} \lor \neg \left(y \leq -1.1 \cdot 10^{+43}\right) \land \left(y \leq -1.75 \cdot 10^{-24} \lor \neg \left(y \leq 2.4 \cdot 10^{-91}\right)\right)\right):\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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. Final simplification 100.0%

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

Alternative 4: 51.3% 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 51.1%

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

3. Final simplification 51.1%

   \[\leadsto x \]

Developer target: 88.2% 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 2024024 
(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)))