Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.3 → 0.1

Time: 6.3s

Precision: binary64

Cost: 708

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 20000000000000:\\ \;\;\;\;x + \left(y \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 20000000000000.0) (+ x (* (* y y) x)) (* 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) <= 20000000000000.0) {
		tmp = x + ((y * y) * x);
	} 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) <= 20000000000000.0d0) then
        tmp = x + ((y * y) * x)
    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) <= 20000000000000.0) {
		tmp = x + ((y * y) * x);
	} 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) <= 20000000000000.0:
		tmp = x + ((y * y) * x)
	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) <= 20000000000000.0)
		tmp = Float64(x + Float64(Float64(y * y) * x));
	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) <= 20000000000000.0)
		tmp = x + ((y * y) * x);
	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], 20000000000000.0], N[(x + N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Target

Original	5.3
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2e13

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   if 2e13 < (*.f64 y y)

   1. Initial program 16.4

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

   2. Taylor expanded in y around inf 16.5

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{y \cdot \left(y \cdot x\right)} \]
      Proof
      (*.f64 y (*.f64 y x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y y) x)): 71 points increase in error, 15 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 20000000000000:\\ \;\;\;\;x + \left(y \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	6.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.48329585899359384:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]

Alternative 2
Error	0.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Error	20.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022317 
(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))))