Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.6 → 5.6

Time: 9.2s

Precision: binary64

Cost: 6784

Math

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

↓

\[{y}^{2} \cdot x + x \]

FPCore

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

↓

(FPCore (x y) :precision binary64 (+ (* (pow y 2.0) x) x))

C

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

↓

double code(double x, double y) {
	return (pow(y, 2.0) * x) + 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 = ((y ** 2.0d0) * x) + 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 (Math.pow(y, 2.0) * x) + x;
}

Python

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

↓

def code(x, y):
	return (math.pow(y, 2.0) * x) + x

Julia

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

↓

function code(x, y)
	return Float64(Float64((y ^ 2.0) * x) + x)
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = ((y ^ 2.0) * x) + x;
end

Wolfram

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

↓

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

Target

Original	5.6
Target	0.1
Herbie	5.6

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

Derivation

1. Initial program 5.6

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

2. Simplified 5.6

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

   [Start] 5.6	\[ x \cdot \left(1 + y \cdot y\right) \]
   rational.json-simplify-17 [=>] 5.6	\[ x \cdot \color{blue}{\left(y \cdot y - -1\right)} \]

3. Taylor expanded in y around 0 5.6

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

4. Final simplification 5.6

   \[\leadsto {y}^{2} \cdot x + x \]

Alternatives

Alternative 1
Error	5.6
Cost	448

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

Alternative 2
Error	21.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023077 
(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))))