Numeric.Integration.TanhSinh:simpson from integration-0.2.1

Percentage Accurate: 100.0% → 100.0%

Time: 1.4s

Alternatives: 2

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

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ x\_s \cdot \left(y\_s \cdot \left(\left(x\_m + x\_m\right) \cdot y\_m\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m)
 :precision binary64
 (* x_s (* y_s (* (+ x_m x_m) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m);
double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * ((x_m + x_m) * y_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = x_s * (y_s * ((x_m + x_m) * y_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m;
public static double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * ((x_m + x_m) * y_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_s, y_s, x_m, y_m):
	return x_s * (y_s * ((x_m + x_m) * y_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m = sort([x_m, y_m])
function code(x_s, y_s, x_m, y_m)
	return Float64(x_s * Float64(y_s * Float64(Float64(x_m + x_m) * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m = num2cell(sort([x_m, y_m])){:}
function tmp = code(x_s, y_s, x_m, y_m)
	tmp = x_s * (y_s * ((x_m + x_m) * y_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_] := N[(x$95$s * N[(y$95$s * N[(N[(x$95$m + x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. distribute-lft-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ x\_s \cdot \left(y\_s \cdot \left(\left(y\_m + y\_m\right) \cdot x\_m\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m)
 :precision binary64
 (* x_s (* y_s (* (+ y_m y_m) x_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m);
double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * ((y_m + y_m) * x_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = x_s * (y_s * ((y_m + y_m) * x_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m;
public static double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * ((y_m + y_m) * x_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_s, y_s, x_m, y_m):
	return x_s * (y_s * ((y_m + y_m) * x_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m = sort([x_m, y_m])
function code(x_s, y_s, x_m, y_m)
	return Float64(x_s * Float64(y_s * Float64(Float64(y_m + y_m) * x_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m = num2cell(sort([x_m, y_m])){:}
function tmp = code(x_s, y_s, x_m, y_m)
	tmp = x_s * (y_s * ((y_m + y_m) * x_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_] := N[(x$95$s * N[(y$95$s * N[(N[(y$95$m + y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Reproduce

herbie shell --seed 2024244 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:simpson  from integration-0.2.1"
  :precision binary64
  (* x (+ y y)))