Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Accuracy: 91.7% → 99.9%

Time: 8.1s

Precision: binary64

Cost: 713

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+109} \lor \neg \left(y \leq 2 \cdot 10^{+146}\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.6e+109) (not (<= y 2e+146)))
   (* 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 <= -3.6e+109) || !(y <= 2e+146)) {
		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 <= (-3.6d+109)) .or. (.not. (y <= 2d+146))) 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 <= -3.6e+109) || !(y <= 2e+146)) {
		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 <= -3.6e+109) or not (y <= 2e+146):
		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 <= -3.6e+109) || !(y <= 2e+146))
		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 <= -3.6e+109) || ~((y <= 2e+146)))
		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, -3.6e+109], N[Not[LessEqual[y, 2e+146]], $MachinePrecision]], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	91.7%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if y < -3.6e109 or 1.99999999999999987e146 < y

   1. Initial program 26.7%

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

   2. Taylor expanded in y around inf 26.7%

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

   3. Simplified 99.6%

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

      [Start] 26.7	\[ {y}^{2} \cdot x \]
      unpow2 [=>] 26.7	\[ \color{blue}{\left(y \cdot y\right)} \cdot x \]
      associate-*l* [=>] 99.6	\[ \color{blue}{y \cdot \left(y \cdot x\right)} \]

   if -3.6e109 < y < 1.99999999999999987e146

   1. Initial program 99.9%

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

   2. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ x \cdot \left(1 + y \cdot y\right) \]
      distribute-rgt-in [=>] 99.9	\[ \color{blue}{1 \cdot x + \left(y \cdot y\right) \cdot x} \]
      *-un-lft-identity [<=] 99.9	\[ \color{blue}{x} + \left(y \cdot y\right) \cdot x \]
      +-commutative [=>] 99.9	\[ \color{blue}{\left(y \cdot y\right) \cdot x + x} \]
      *-commutative [=>] 99.9	\[ \color{blue}{x \cdot \left(y \cdot y\right)} + x \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+109} \lor \neg \left(y \leq 2 \cdot 10^{+146}\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	13376

\[\frac{\mathsf{hypot}\left(1, y\right)}{\frac{1}{\mathsf{hypot}\left(1, y\right) \cdot x}} \]

Alternative 2
Accuracy	99.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+109} \lor \neg \left(y \leq 5 \cdot 10^{+152}\right):\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \end{array} \]

Alternative 3
Accuracy	90.3%
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	98.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 5
Accuracy	67.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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))))