Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.7 → 0.1

Time: 8.0s

Precision: binary64

Cost: 1344

Math

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

↓

\[x + \left(\frac{y \cdot \left(y \cdot \left(x \cdot 1.5\right)\right)}{2} + \frac{y \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{2}\right) \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (+ x (+ (/ (* y (* y (* x 1.5))) 2.0) (/ (* y (* y (* x 0.5))) 2.0))))

C

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

↓

double code(double x, double y) {
	return x + (((y * (y * (x * 1.5))) / 2.0) + ((y * (y * (x * 0.5))) / 2.0));
}

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 * 1.5d0))) / 2.0d0) + ((y * (y * (x * 0.5d0))) / 2.0d0))
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 * 1.5))) / 2.0) + ((y * (y * (x * 0.5))) / 2.0));
}

Python

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

↓

def code(x, y):
	return x + (((y * (y * (x * 1.5))) / 2.0) + ((y * (y * (x * 0.5))) / 2.0))

Julia

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

↓

function code(x, y)
	return Float64(x + Float64(Float64(Float64(y * Float64(y * Float64(x * 1.5))) / 2.0) + Float64(Float64(y * Float64(y * Float64(x * 0.5))) / 2.0)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = x + (((y * (y * (x * 1.5))) / 2.0) + ((y * (y * (x * 0.5))) / 2.0));
end

Wolfram

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

↓

code[x_, y_] := N[(x + N[(N[(N[(y * N[(y * N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] + N[(N[(y * N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	5.7
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 5.7

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

2. Simplified 5.7

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

   [Start] 5.7	\[ x \cdot \left(1 + y \cdot y\right) \]
   rational_best-simplify-74 [<=] 5.7	\[ \color{blue}{1 \cdot x + \left(y \cdot y\right) \cdot x} \]
   rational_best-simplify-1 [=>] 5.7	\[ \color{blue}{x \cdot 1} + \left(y \cdot y\right) \cdot x \]
   rational_best-simplify-7 [=>] 5.7	\[ \color{blue}{x} + \left(y \cdot y\right) \cdot x \]
   rational_best-simplify-1 [=>] 5.7	\[ x + \color{blue}{x \cdot \left(y \cdot y\right)} \]

3. Applied egg-rr 5.8

   \[\leadsto x + \color{blue}{\left(\frac{x \cdot \left(y \cdot y\right)}{-2} - \left(y \cdot y\right) \cdot \left(x \cdot -1.5\right)\right)} \]

4. Applied egg-rr 0.1

   \[\leadsto x + \color{blue}{\left(\frac{y \cdot \left(y \cdot \left(x \cdot 1.5\right)\right)}{2} + \frac{y \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{2}\right)} \]

5. Final simplification 0.1

   \[\leadsto x + \left(\frac{y \cdot \left(y \cdot \left(x \cdot 1.5\right)\right)}{2} + \frac{y \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{2}\right) \]

Alternatives

Alternative 1
Error	5.7
Cost	448

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

Alternative 2
Error	5.7
Cost	448

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

Alternative 3
Error	0.1
Cost	448

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

Reproduce

herbie shell --seed 2023101 
(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))))