Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.3 → 0.1

Time: 6.2s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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
    code = x + (y * (y * x))
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) {
	return x + (y * (y * x));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	5.3
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 5.3

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

2. Simplified 5.3

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

   [Start] 5.3	\[ x \cdot \left(1 + y \cdot y\right) \]
   distribute-lft-in [=>] 5.3	\[ \color{blue}{x \cdot 1 + x \cdot \left(y \cdot y\right)} \]
   +-commutative [=>] 5.3	\[ \color{blue}{x \cdot \left(y \cdot y\right) + x \cdot 1} \]
   *-rgt-identity [=>] 5.3	\[ x \cdot \left(y \cdot y\right) + \color{blue}{x} \]
   fma-def [=>] 5.3	\[ \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]

3. Taylor expanded in x around 0 5.3

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

4. Simplified 0.1

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

   [Start] 5.3	\[ \left(1 + {y}^{2}\right) \cdot x \]
   +-commutative [=>] 5.3	\[ \color{blue}{\left({y}^{2} + 1\right)} \cdot x \]
   distribute-lft1-in [<=] 5.3	\[ \color{blue}{{y}^{2} \cdot x + x} \]
   unpow2 [=>] 5.3	\[ \color{blue}{\left(y \cdot y\right)} \cdot x + x \]
   associate-*l* [=>] 0.1	\[ \color{blue}{y \cdot \left(y \cdot x\right)} + x \]
   *-commutative [<=] 0.1	\[ y \cdot \color{blue}{\left(x \cdot y\right)} + x \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(y, x \cdot y, x\right)} \]
   *-commutative [=>] 0.1	\[ \mathsf{fma}\left(y, \color{blue}{y \cdot x}, x\right) \]

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 200000000000:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 200000000000:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Error	6.2
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
Error	1.0
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 5
Error	20.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023187 
(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))))