Average Error: 5.4 → 0.1
Time: 2.7s
Precision: binary64
Cost: 448
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[x + y \cdot \left(x \cdot y\right)\]
x \cdot \left(1 + y \cdot y\right)
↓
x + y \cdot \left(x \cdot y\right)
(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))
↓
(FPCore (x y) :precision binary64 (+ x (* y (* x y))))
double code(double x, double y) {
	return x * (1.0 + (y * y));
}
↓
double code(double x, double y) {
	return x + (y * (x * y));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.4
Target0.1
Herbie0.1
\[x + \left(x \cdot y\right) \cdot y\]

Alternatives

Alternative 1
Error5.3
Cost448
\[x + x \cdot \left(y \cdot y\right)\]
Alternative 2
Error5.4
Cost448
\[x \cdot \left(y \cdot y + 1\right)\]
Alternative 3
Error6.3
Cost769
\[\begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.993078768153627:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array}\]
Alternative 4
Error20.0
Cost64
\[x\]
Alternative 5
Error60.6
Cost385
\[\begin{array}{l} \mathbf{if}\;x \leq 3.1961021957859 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 6
Error61.8
Cost64
\[1\]

Error

Derivation

  1. Initial program 5.4

    \[x \cdot \left(1 + y \cdot y\right)\]
  2. Using strategy rm

  3. Applied distribute-rgt-in_binary64_136685.3

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

  4. Simplified5.3

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

  5. Simplified5.3

    \[\leadsto x + \color{blue}{x \cdot \left(y \cdot y\right)}\]
  6. Using strategy rm

  7. Applied associate-*r*_binary64_136580.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2021044 
(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))))