Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Average Error: 5.3 → 0.1

Time: 6.8s

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. Applied egg-rr 5.3

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

3. Taylor expanded in x around 0 5.3

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

4. Simplified 0.1

   \[\leadsto \color{blue}{y \cdot \left(y \cdot x\right)} + x \]
   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)): 65 points increase in error, 23 points decrease in error
   (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	6.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.0361618571361528 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 2
Error	1.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3
Error	20.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(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))))