Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.6 → 0.1

Time: 5.9s

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	5.6
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 5.6

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

2. Simplified 5.6

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

   [Start] 5.6	\[ x \cdot \left(1 + y \cdot y\right) \]
   rational_best_oopsla_all_46_json_45_simplify-37 [=>] 5.6	\[ \color{blue}{1 \cdot x + x \cdot \left(y \cdot y\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 5.6	\[ \color{blue}{x \cdot 1} + x \cdot \left(y \cdot y\right) \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 5.6	\[ \color{blue}{x} + x \cdot \left(y \cdot y\right) \]

3. Applied egg-rr 5.6

   \[\leadsto x + \color{blue}{\left(x \cdot \left(y \cdot y\right) + 0\right)} \]

4. Simplified 0.1

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

   [Start] 5.6	\[ x + \left(x \cdot \left(y \cdot y\right) + 0\right) \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 5.6	\[ x + \color{blue}{x \cdot \left(y \cdot y\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.1	\[ x + \color{blue}{y \cdot \left(x \cdot y\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ x + y \cdot \color{blue}{\left(y \cdot x\right)} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	5.6
Cost	448

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

Alternative 2
Error	5.6
Cost	448

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

Alternative 3
Error	21.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))))