Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.4 → 0.1

Time: 53.3s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;y \leq -5 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 24000000000:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= y -5e+43) t_0 (if (<= y 24000000000.0) (* x (+ 1.0 (* y y))) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (y <= -5e+43) {
		tmp = t_0;
	} else if (y <= 24000000000.0) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (y * x) * y
    if (y <= (-5d+43)) then
        tmp = t_0
    else if (y <= 24000000000.0d0) then
        tmp = x * (1.0d0 + (y * y))
    else
        tmp = t_0
    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 t_0 = (y * x) * y;
	double tmp;
	if (y <= -5e+43) {
		tmp = t_0;
	} else if (y <= 24000000000.0) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = (y * x) * y
	tmp = 0
	if y <= -5e+43:
		tmp = t_0
	elif y <= 24000000000.0:
		tmp = x * (1.0 + (y * y))
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (y <= -5e+43)
		tmp = t_0;
	elseif (y <= 24000000000.0)
		tmp = Float64(x * Float64(1.0 + Float64(y * y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	tmp = 0.0;
	if (y <= -5e+43)
		tmp = t_0;
	elseif (y <= 24000000000.0)
		tmp = x * (1.0 + (y * y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -5e+43], t$95$0, If[LessEqual[y, 24000000000.0], N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	5.4
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000004e43 or 2.4e10 < y

   1. Initial program 19.2

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

   2. Taylor expanded in y around inf 19.4

      \[\leadsto \color{blue}{{\left(\frac{1}{y}\right)}^{-2} \cdot x} \]

   3. Taylor expanded in y around 0 19.2

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

   4. Simplified 19.2

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

   5. Taylor expanded in y around 0 19.2

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

   6. Simplified 0.3

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

   if -5.0000000000000004e43 < y < 2.4e10

   1. Initial program 0.0

      \[x \cdot \left(1 + y \cdot y\right) \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	1.0
Cost	584

\[\begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

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

Alternative 3
Error	20.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(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))))