Average Error: 20.6 → 0.0
Time: 27.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\sqrt[3]{\left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right) \cdot \left(\left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right) \cdot \left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right)\right)}\]

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.6
Target0.0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Initial program 20.6

    \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

  2. Simplified20.6

    \[\leadsto \color{blue}{\frac{\left(x - y\right) \cdot \left(y + x\right)}{(x \cdot x + \left(y \cdot y\right))_*}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt20.6

    \[\leadsto \frac{\left(x - y\right) \cdot \left(y + x\right)}{\color{blue}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*} \cdot \sqrt{(x \cdot x + \left(y \cdot y\right))_*}}}\]

  5. Applied times-frac20.6

    \[\leadsto \color{blue}{\frac{x - y}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*}} \cdot \frac{y + x}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*}}}\]

  6. Simplified20.6

    \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x^2 + y^2}^*}} \cdot \frac{y + x}{\sqrt{(x \cdot x + \left(y \cdot y\right))_*}}\]

  7. Simplified0.0

    \[\leadsto \frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \color{blue}{\frac{x + y}{\sqrt{x^2 + y^2}^*}}\]
  8. Using strategy rm

  9. Applied add-cbrt-cube0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\right) \cdot \left(\frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\right)\right) \cdot \left(\frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\right)}}\]

  10. Final simplification0.0

    \[\leadsto \sqrt[3]{\left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right) \cdot \left(\left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right) \cdot \left(\frac{y + x}{\sqrt{x^2 + y^2}^*} \cdot \frac{x - y}{\sqrt{x^2 + y^2}^*}\right)\right)}\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))