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

Percentage Accurate: 100.0% → 100.0%

Time: 2.7s

Alternatives: 5

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, 1.0× 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. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-neg-out N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+90} \lor \neg \left(y \cdot z \leq 50\right):\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y z) -5e+90) (not (<= (* y z) 50.0))) (* (- y) z) (fma z y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -5e+90) || !((y * z) <= 50.0)) {
		tmp = -y * z;
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * z) <= -5e+90) || !(Float64(y * z) <= 50.0))
		tmp = Float64(Float64(-y) * z);
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -5e+90], N[Not[LessEqual[N[(y * z), $MachinePrecision], 50.0]], $MachinePrecision]], N[((-y) * z), $MachinePrecision], N[(z * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5.0000000000000004e90 or 50 < (*.f64 y z)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 80.3

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

   5. Applied rewrites 80.3%

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

   if -5.0000000000000004e90 < (*.f64 y z) < 50

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-neg-out N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-z, y, x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

         \[\leadsto {\left(\left(-z\right) \cdot y\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + x \]

      4. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(\left(-z\right) \cdot y\right)}^{2}}} + x \]

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. *-commutative N/A

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

      10. neg-fabs N/A

          \[\leadsto \color{blue}{\left|y \cdot z\right|} + x \]

      11. rem-sqrt-square-rev N/A

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

      12. pow2 N/A

          \[\leadsto \sqrt{\color{blue}{{\left(y \cdot z\right)}^{2}}} + x \]

      13. sqrt-pow1 N/A

          \[\leadsto \color{blue}{{\left(y \cdot z\right)}^{\left(\frac{2}{2}\right)}} + x \]

      14. metadata-eval N/A

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

      15. unpow1 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 76.3

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

   6. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+90} \lor \neg \left(y \cdot z \leq 50\right):\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing
3. Add Preprocessing

Alternative 4: 49.8% accurate, 1.3× 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, y, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-neg-out N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-z, y, x\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

      \[\leadsto {\left(\left(-z\right) \cdot y\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + x \]

   4. sqrt-pow1 N/A

      \[\leadsto \color{blue}{\sqrt{{\left(\left(-z\right) \cdot y\right)}^{2}}} + x \]

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. *-commutative N/A

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

   10. neg-fabs N/A

       \[\leadsto \color{blue}{\left|y \cdot z\right|} + x \]

   11. rem-sqrt-square-rev N/A

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

   12. pow2 N/A

       \[\leadsto \sqrt{\color{blue}{{\left(y \cdot z\right)}^{2}}} + x \]

   13. sqrt-pow1 N/A

       \[\leadsto \color{blue}{{\left(y \cdot z\right)}^{\left(\frac{2}{2}\right)}} + x \]

   14. metadata-eval N/A

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

   15. unpow1 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 52.7

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

6. Applied rewrites 52.7%

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

Alternative 5: 2.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = z * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 47.3

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

5. Applied rewrites 47.3%

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

7. Applied rewrites 2.3%

   \[\leadsto \color{blue}{z \cdot y} \]
8. Add Preprocessing

Developer Target 1: 87.2% accurate, 0.2× 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 2024332 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, C"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (+ x (* y z)) (/ (+ x (* y z)) (- x (* y z)))))

  (- x (* y z)))