(- (/ x0 (- 1 x1)) x0)

Average Error: 8.3 → 5.1

Time: 5.3s

Precision: binary64

\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]

Math

\[\frac{x0}{1 - x1} - x0\]

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 0.0182045976562499982:\\ \;\;\;\;\frac{\frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}} \cdot \left(\frac{x0}{1 - x1} + \left(-x0\right)\right)}{\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}}}\right)\\ \end{array}\]

Error

Bits error versus x0

Bits error versus x1

Target

Original	8.3
Target	0.5
Herbie	5.1

\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

1. Split input into 2 regimes

2. if x1 < 0.0182045976562499982

   1. Initial program 11.3

      \[\frac{x0}{1 - x1} - x0\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 11.3

      \[\leadsto \frac{x0}{1 - x1} - \color{blue}{\sqrt{x0} \cdot \sqrt{x0}}\]

   4. Applied add-sqr-sqrt 10.7

      \[\leadsto \color{blue}{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}}} - \sqrt{x0} \cdot \sqrt{x0}\]

   5. Applied difference-of-squares 10.7

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]
   6. Using strategy rm

   7. Applied flip3-- 12.1

      \[\leadsto \left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \color{blue}{\frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}} + \left(\sqrt{x0} \cdot \sqrt{x0} + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right)}}\]

   8. Simplified 12.1

      \[\leadsto \left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\color{blue}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}}}\]
   9. Using strategy rm

   10. Applied flip-+ 12.1

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}} - \sqrt{x0} \cdot \sqrt{x0}}{\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}}} \cdot \frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}}\]

   11. Applied associate-*l/ 12.1

       \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}} - \sqrt{x0} \cdot \sqrt{x0}\right) \cdot \frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}}}{\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}}}\]

   12. Simplified 8.7

       \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}} \cdot \left(\frac{x0}{1 - x1} + \left(-x0\right)\right)}}{\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}}\]

   if 0.0182045976562499982 < x1

   1. Initial program 5.5

      \[\frac{x0}{1 - x1} - x0\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 5.5

      \[\leadsto \frac{x0}{1 - x1} - \color{blue}{\sqrt{x0} \cdot \sqrt{x0}}\]

   4. Applied add-sqr-sqrt 4.4

      \[\leadsto \color{blue}{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}}} - \sqrt{x0} \cdot \sqrt{x0}\]

   5. Applied difference-of-squares 4.6

      \[\leadsto \color{blue}{\left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]
   6. Using strategy rm

   7. Applied flip3-- 3.5

      \[\leadsto \left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \color{blue}{\frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}} + \left(\sqrt{x0} \cdot \sqrt{x0} + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right)}}\]

   8. Simplified 3.2

      \[\leadsto \left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\color{blue}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}}}\]
   9. Using strategy rm

   10. Applied add-log-exp 1.6

       \[\leadsto \color{blue}{\log \left(e^{\left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}}}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 0.0182045976562499982:\\ \;\;\;\;\frac{\frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}} \cdot \left(\frac{x0}{1 - x1} + \left(-x0\right)\right)}{\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\sqrt{\frac{x0}{1 - x1}} + \sqrt{x0}\right) \cdot \frac{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - {\left(\sqrt{x0}\right)}^{3}}{\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :pre (or (and (== x0 1.855) (== x1 0.000209)) (and (== x0 2.985) (== x1 0.0186)))

  :herbie-target
  (/ (* x0 x1) (- 1.0 x1))

  (- (/ x0 (- 1.0 x1)) x0))