Average Error: 20.5 → 0.0
Time: 59.4s
Precision: 64
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\sqrt[3]{\left(\frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\right) \cdot \left(\left(\frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\right) \cdot \left(\frac{x - y}{\sqrt{x^2 + y^2}^*} \cdot \frac{x + y}{\sqrt{x^2 + y^2}^*}\right)\right)}\]

Error

Bits error versus x

Bits error versus y

Target

Original20.5
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 20.5

    \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt20.5

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

  4. Applied times-frac20.5

    \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube20.5

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

  7. Applied add-cbrt-cube20.5

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

  8. Applied cbrt-unprod20.5

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019100 +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))))