Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.0% → 99.8%

Time: 6.3s

Precision: binary64

Cost: 7428

• 24.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(1 - {y}^{4}\right) \cdot x}{1 - y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4e+46)
   (/ (* (- 1.0 (pow y 4.0)) x) (- 1.0 (* y y)))
   (* y (* y x))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+46) {
		tmp = ((1.0 - pow(y, 4.0)) * x) / (1.0 - (y * y));
	} else {
		tmp = y * (y * x);
	}
	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 * y) <= 4d+46) then
        tmp = ((1.0d0 - (y ** 4.0d0)) * x) / (1.0d0 - (y * y))
    else
        tmp = y * (y * x)
    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 * y) <= 4e+46) {
		tmp = ((1.0 - Math.pow(y, 4.0)) * x) / (1.0 - (y * y));
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if (y * y) <= 4e+46:
		tmp = ((1.0 - math.pow(y, 4.0)) * x) / (1.0 - (y * y))
	else:
		tmp = y * (y * x)
	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 (Float64(y * y) <= 4e+46)
		tmp = Float64(Float64(Float64(1.0 - (y ^ 4.0)) * x) / Float64(1.0 - Float64(y * y)));
	else
		tmp = Float64(y * Float64(y * x));
	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 * y) <= 4e+46)
		tmp = ((1.0 - (y ^ 4.0)) * x) / (1.0 - (y * y));
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 4e+46], N[(N[(N[(1.0 - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Target

Original	94.0%
Target	99.9%
Herbie	99.8%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4e46

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

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

      [Start] 100.0%	\[ x \cdot \left(1 + y \cdot y\right) \]
      *-commutative [=>] 100.0%	\[ \color{blue}{\left(1 + y \cdot y\right) \cdot x} \]
      flip-+ [=>] 99.9%	\[ \color{blue}{\frac{1 \cdot 1 - \left(y \cdot y\right) \cdot \left(y \cdot y\right)}{1 - y \cdot y}} \cdot x \]
      associate-*l/ [=>] 99.9%	\[ \color{blue}{\frac{\left(1 \cdot 1 - \left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot x}{1 - y \cdot y}} \]
      metadata-eval [=>] 99.9%	\[ \frac{\left(\color{blue}{1} - \left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot x}{1 - y \cdot y} \]
      pow2 [=>] 99.9%	\[ \frac{\left(1 - \color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)\right) \cdot x}{1 - y \cdot y} \]
      pow2 [=>] 99.9%	\[ \frac{\left(1 - {y}^{2} \cdot \color{blue}{{y}^{2}}\right) \cdot x}{1 - y \cdot y} \]
      pow-prod-up [=>] 100.0%	\[ \frac{\left(1 - \color{blue}{{y}^{\left(2 + 2\right)}}\right) \cdot x}{1 - y \cdot y} \]
      metadata-eval [=>] 100.0%	\[ \frac{\left(1 - {y}^{\color{blue}{4}}\right) \cdot x}{1 - y \cdot y} \]

   if 4e46 < (*.f64 y y)

   1. Initial program 91.2%

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

   2. Taylor expanded in y around inf 91.2%

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

   3. Simplified 99.8%

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

      [Start] 91.2%	\[ {y}^{2} \cdot x \]
      unpow2 [=>] 91.2%	\[ \color{blue}{\left(y \cdot y\right)} \cdot x \]
      associate-*r* [<=] 99.8%	\[ \color{blue}{y \cdot \left(y \cdot x\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(1 - {y}^{4}\right) \cdot x}{1 - y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(1 - {y}^{4}\right) \cdot x}{1 - y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+122}:\\ \;\;\;\;\left(1 - {y}^{4}\right) \cdot \frac{x}{1 - y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+122}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4
Accuracy	97.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 5
Accuracy	53.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023272 
(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))))