Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.2 → 0.1

Time: 3.2s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -1.96 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+138}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y x))))
   (if (<= y -1.96e+96) t_0 (if (<= y 6.2e+138) (+ x (* x (* 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 * (y * x);
	double tmp;
	if (y <= -1.96e+96) {
		tmp = t_0;
	} else if (y <= 6.2e+138) {
		tmp = x + (x * (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 * (y * x)
    if (y <= (-1.96d+96)) then
        tmp = t_0
    else if (y <= 6.2d+138) then
        tmp = x + (x * (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 * (y * x);
	double tmp;
	if (y <= -1.96e+96) {
		tmp = t_0;
	} else if (y <= 6.2e+138) {
		tmp = x + (x * (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 * (y * x)
	tmp = 0
	if y <= -1.96e+96:
		tmp = t_0
	elif y <= 6.2e+138:
		tmp = x + (x * (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(y * Float64(y * x))
	tmp = 0.0
	if (y <= -1.96e+96)
		tmp = t_0;
	elseif (y <= 6.2e+138)
		tmp = Float64(x + Float64(x * 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 * (y * x);
	tmp = 0.0;
	if (y <= -1.96e+96)
		tmp = t_0;
	elseif (y <= 6.2e+138)
		tmp = x + (x * (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[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.96e+96], t$95$0, If[LessEqual[y, 6.2e+138], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	5.2
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -1.96e96 or 6.1999999999999995e138 < y

   1. Initial program 41.6

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

   2. Simplified 41.6

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, y, 1\right)} \]
      Proof
      (*.f64 x (fma.f64 y y 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y y) 1))): 1 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 41.6

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

   4. Simplified 0.2

      \[\leadsto \color{blue}{y \cdot \left(y \cdot x\right)} \]
      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)): 64 points increase in error, 27 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x): 0 points increase in error, 0 points decrease in error

   if -1.96e96 < y < 6.1999999999999995e138

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, y, 1\right)} \]
      Proof
      (*.f64 x (fma.f64 y y 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y y) 1))): 1 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 y y)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.96 \cdot 10^{+96}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+138}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	712

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.1
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 3
Error	1.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4
Error	20.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(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))))