Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Accuracy: 91.3% → 99.9%

Time: 5.3s

Precision: binary64

Cost: 713

Math

\[x \cdot \left(1 + y \cdot y\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+139} \lor \neg \left(y \leq 9 \cdot 10^{+147}\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.4e+139) (not (<= y 9e+147)))
   (* y (* y x))
   (+ x (* x (* y y)))))

C

double code(double x, double y) {
	return x * (1.0 + (y * y));
}

↓

double code(double x, double y) {
	double tmp;
	if ((y <= -2.4e+139) || !(y <= 9e+147)) {
		tmp = y * (y * x);
	} else {
		tmp = x + (x * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.4d+139)) .or. (.not. (y <= 9d+147))) then
        tmp = y * (y * x)
    else
        tmp = x + (x * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (y * y));
}

↓

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.4e+139) || !(y <= 9e+147)) {
		tmp = y * (y * x);
	} else {
		tmp = x + (x * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	return x * (1.0 + (y * y))

↓

def code(x, y):
	tmp = 0
	if (y <= -2.4e+139) or not (y <= 9e+147):
		tmp = y * (y * x)
	else:
		tmp = x + (x * (y * y))
	return tmp

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end

↓

function code(x, y)
	tmp = 0.0
	if ((y <= -2.4e+139) || !(y <= 9e+147))
		tmp = Float64(y * Float64(y * x));
	else
		tmp = Float64(x + Float64(x * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (y * y));
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.4e+139) || ~((y <= 9e+147)))
		tmp = y * (y * x);
	else
		tmp = x + (x * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[Or[LessEqual[y, -2.4e+139], N[Not[LessEqual[y, 9e+147]], $MachinePrecision]], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	91.3%
Target	99.9%
Herbie	99.9%

\[x + \left(x \cdot y\right) \cdot y \]

Derivation

1. Split input into 2 regimes

2. if y < -2.40000000000000008e139 or 9.00000000000000016e147 < y

   1. Initial program 11.8%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around inf 11.8%

      \[\leadsto \color{blue}{{y}^{2} \cdot x} \]

   3. Simplified 99.6%

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

      [Start] 11.8	\[ {y}^{2} \cdot x \]
      *-commutative [=>] 11.8	\[ \color{blue}{x \cdot {y}^{2}} \]
      unpow2 [=>] 11.8	\[ x \cdot \color{blue}{\left(y \cdot y\right)} \]
      associate-*r* [=>] 99.6	\[ \color{blue}{\left(x \cdot y\right) \cdot y} \]

   if -2.40000000000000008e139 < y < 9.00000000000000016e147

   1. Initial program 99.9%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot y\right) + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+139} \lor \neg \left(y \leq 9 \cdot 10^{+147}\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+139} \lor \neg \left(y \leq 2 \cdot 10^{+145}\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \end{array} \]

Alternative 2
Accuracy	98.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	89.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 4
Accuracy	99.9%
Cost	448

\[x + y \cdot \left(y \cdot x\right) \]

Alternative 5
Accuracy	67.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023129 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

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

  (* x (+ 1.0 (* y y))))