Numeric.Integration.TanhSinh:simpson from integration-0.2.1

Percentage Accurate: 100.0% → 100.0%

Time: 2.4s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(y + y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 2.0 (* x y)))

C

double code(double x, double y) {
	return 2.0 * (x * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 * (x * y)
end function

Java

public static double code(double x, double y) {
	return 2.0 * (x * y);
}

Python

def code(x, y):
	return 2.0 * (x * y)

Julia

function code(x, y)
	return Float64(2.0 * Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 * (x * y);
end

Wolfram

code[x_, y_] := N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x \cdot \left(y + y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto 2 \cdot \left(x \cdot y\right) \]
5. Add Preprocessing

Alternative 2: 4.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ y + y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return Float64(y + y)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(y + y), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x \cdot \left(y + y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 36.0%

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

   2. exp-prod 33.3%

      \[\leadsto \log \color{blue}{\left({\left(e^{x}\right)}^{\left(y + y\right)}\right)} \]

   3. flip-+ 17.4%

      \[\leadsto \log \left({\left(e^{x}\right)}^{\color{blue}{\left(\frac{y \cdot y - y \cdot y}{y - y}\right)}}\right) \]

   4. div-inv 17.4%

      \[\leadsto \log \left({\left(e^{x}\right)}^{\color{blue}{\left(\left(y \cdot y - y \cdot y\right) \cdot \frac{1}{y - y}\right)}}\right) \]

   5. +-inverses 17.4%

      \[\leadsto \log \left({\left(e^{x}\right)}^{\left(\color{blue}{0} \cdot \frac{1}{y - y}\right)}\right) \]

   6. +-inverses 17.4%

      \[\leadsto \log \left({\left(e^{x}\right)}^{\left(\color{blue}{\left(y - y\right)} \cdot \frac{1}{y - y}\right)}\right) \]

   7. pow-unpow 18.7%

      \[\leadsto \log \color{blue}{\left({\left({\left(e^{x}\right)}^{\left(y - y\right)}\right)}^{\left(\frac{1}{y - y}\right)}\right)} \]

   8. +-inverses 18.7%

      \[\leadsto \log \left({\left({\left(e^{x}\right)}^{\color{blue}{0}}\right)}^{\left(\frac{1}{y - y}\right)}\right) \]

   9. metadata-eval 18.7%

      \[\leadsto \log \left({\color{blue}{1}}^{\left(\frac{1}{y - y}\right)}\right) \]

   10. 1-exp 18.7%

       \[\leadsto \log \left({\color{blue}{\left(e^{0}\right)}}^{\left(\frac{1}{y - y}\right)}\right) \]

   11. +-inverses 18.7%

       \[\leadsto \log \left({\left(e^{\color{blue}{y - y}}\right)}^{\left(\frac{1}{y - y}\right)}\right) \]

   12. exp-prod 0.0%

       \[\leadsto \log \color{blue}{\left(e^{\left(y - y\right) \cdot \frac{1}{y - y}}\right)} \]

   13. +-inverses 0.0%

       \[\leadsto \log \left(e^{\color{blue}{0} \cdot \frac{1}{y - y}}\right) \]

   14. +-inverses 0.0%

       \[\leadsto \log \left(e^{\color{blue}{\left(y \cdot y - y \cdot y\right)} \cdot \frac{1}{y - y}}\right) \]

   15. div-inv 0.0%

       \[\leadsto \log \left(e^{\color{blue}{\frac{y \cdot y - y \cdot y}{y - y}}}\right) \]

   16. flip-+ 25.3%

       \[\leadsto \log \left(e^{\color{blue}{y + y}}\right) \]

   17. add-cube-cbrt 25.3%

       \[\leadsto \log \left(e^{\color{blue}{\left(\sqrt[3]{y + y} \cdot \sqrt[3]{y + y}\right) \cdot \sqrt[3]{y + y}}}\right) \]

   18. add-log-exp 4.0%

       \[\leadsto \color{blue}{\left(\sqrt[3]{y + y} \cdot \sqrt[3]{y + y}\right) \cdot \sqrt[3]{y + y}} \]

   19. add-cube-cbrt 4.0%

       \[\leadsto \color{blue}{y + y} \]

4. Applied egg-rr 4.0%

   \[\leadsto \color{blue}{y + y} \]

5. Final simplification 4.0%

   \[\leadsto y + y \]
6. Add Preprocessing

Alternative 3: 3.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -2.0)

C

double code(double x, double y) {
	return -2.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -2.0d0
end function

Java

public static double code(double x, double y) {
	return -2.0;
}

Python

def code(x, y):
	return -2.0

Julia

function code(x, y)
	return -2.0
end

MATLAB

function tmp = code(x, y)
	tmp = -2.0;
end

Wolfram

code[x_, y_] := -2.0

Derivation

1. Initial program 99.6%

   \[x \cdot \left(y + y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 36.0%

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

   2. exp-prod 33.3%

      \[\leadsto \log \color{blue}{\left({\left(e^{x}\right)}^{\left(y + y\right)}\right)} \]

   3. flip-+ 17.4%

      \[\leadsto \log \left({\left(e^{x}\right)}^{\color{blue}{\left(\frac{y \cdot y - y \cdot y}{y - y}\right)}}\right) \]

   4. div-inv 17.4%

      \[\leadsto \log \left({\left(e^{x}\right)}^{\color{blue}{\left(\left(y \cdot y - y \cdot y\right) \cdot \frac{1}{y - y}\right)}}\right) \]

   5. +-inverses 17.4%

      \[\leadsto \log \left({\left(e^{x}\right)}^{\left(\color{blue}{0} \cdot \frac{1}{y - y}\right)}\right) \]

   6. +-inverses 17.4%

      \[\leadsto \log \left({\left(e^{x}\right)}^{\left(\color{blue}{\left(y - y\right)} \cdot \frac{1}{y - y}\right)}\right) \]

   7. pow-unpow 18.7%

      \[\leadsto \log \color{blue}{\left({\left({\left(e^{x}\right)}^{\left(y - y\right)}\right)}^{\left(\frac{1}{y - y}\right)}\right)} \]

   8. +-inverses 18.7%

      \[\leadsto \log \left({\left({\left(e^{x}\right)}^{\color{blue}{0}}\right)}^{\left(\frac{1}{y - y}\right)}\right) \]

   9. metadata-eval 18.7%

      \[\leadsto \log \left({\color{blue}{1}}^{\left(\frac{1}{y - y}\right)}\right) \]

   10. metadata-eval 18.7%

       \[\leadsto \log \left({\color{blue}{\left({\left(e^{1}\right)}^{0}\right)}}^{\left(\frac{1}{y - y}\right)}\right) \]

   11. +-inverses 18.7%

       \[\leadsto \log \left({\left({\left(e^{1}\right)}^{\color{blue}{\left(y - y\right)}}\right)}^{\left(\frac{1}{y - y}\right)}\right) \]

   12. pow-unpow 0.0%

       \[\leadsto \log \color{blue}{\left({\left(e^{1}\right)}^{\left(\left(y - y\right) \cdot \frac{1}{y - y}\right)}\right)} \]

   13. +-inverses 0.0%

       \[\leadsto \log \left({\left(e^{1}\right)}^{\left(\color{blue}{0} \cdot \frac{1}{y - y}\right)}\right) \]

   14. +-inverses 0.0%

       \[\leadsto \log \left({\left(e^{1}\right)}^{\left(\color{blue}{\left(y \cdot y - y \cdot y\right)} \cdot \frac{1}{y - y}\right)}\right) \]

   15. div-inv 0.0%

       \[\leadsto \log \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{y \cdot y - y \cdot y}{y - y}\right)}}\right) \]

   16. flip-+ 25.3%

       \[\leadsto \log \left({\left(e^{1}\right)}^{\color{blue}{\left(y + y\right)}}\right) \]

   17. exp-prod 25.3%

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

   18. *-un-lft-identity 25.3%

       \[\leadsto \log \left(e^{\color{blue}{y + y}}\right) \]

   19. *-un-lft-identity 25.3%

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

   20. log-prod 25.3%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{y + y}\right)} \]

   21. metadata-eval 25.3%

       \[\leadsto \color{blue}{0} + \log \left(e^{y + y}\right) \]

   22. add-log-exp 4.0%

       \[\leadsto 0 + \color{blue}{\left(y + y\right)} \]

   23. flip-+ 0.0%

       \[\leadsto 0 + \color{blue}{\frac{y \cdot y - y \cdot y}{y - y}} \]

4. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{0 + \frac{0}{0}} \]

6. Simplified 3.2%

   \[\leadsto \color{blue}{-2} \]

7. Final simplification 3.2%

   \[\leadsto -2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:simpson  from integration-0.2.1"
  :precision binary64
  (* x (+ y y)))