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

Average Error: 7.9 → 2.0

Time: 8.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861321705391922876060562 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r202205 = x0;
        double r202206 = 1.0;
        double r202207 = x1;
        double r202208 = r202206 - r202207;
        double r202209 = r202205 / r202208;
        double r202210 = r202209 - r202205;
        return r202210;
}

↓

double f(double x0, double x1) {
        double r202211 = x1;
        double r202212 = 0.00021208908081054686;
        bool r202213 = r202211 <= r202212;
        double r202214 = x0;
        double r202215 = 3.0;
        double r202216 = pow(r202214, r202215);
        double r202217 = 1.0;
        double r202218 = r202217 - r202211;
        double r202219 = 6.0;
        double r202220 = pow(r202218, r202219);
        double r202221 = r202216 / r202220;
        double r202222 = r202221 - r202216;
        double r202223 = exp(r202222);
        double r202224 = sqrt(r202223);
        double r202225 = log(r202224);
        double r202226 = r202225 + r202225;
        double r202227 = r202214 * r202214;
        double r202228 = r202218 * r202218;
        double r202229 = r202214 / r202228;
        double r202230 = r202229 + r202214;
        double r202231 = r202230 * r202229;
        double r202232 = r202227 + r202231;
        double r202233 = r202226 / r202232;
        double r202234 = r202214 * r202233;
        double r202235 = r202214 / r202218;
        double r202236 = r202235 + r202214;
        double r202237 = r202234 / r202236;
        double r202238 = sqrt(r202217);
        double r202239 = sqrt(r202211);
        double r202240 = r202238 + r202239;
        double r202241 = pow(r202240, r202219);
        double r202242 = r202238 - r202239;
        double r202243 = pow(r202242, r202219);
        double r202244 = r202241 * r202243;
        double r202245 = r202216 / r202244;
        double r202246 = r202245 - r202216;
        double r202247 = exp(r202246);
        double r202248 = log(r202247);
        double r202249 = r202248 / r202232;
        double r202250 = r202214 * r202249;
        double r202251 = r202250 / r202236;
        double r202252 = r202213 ? r202237 : r202251;
        return r202252;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.3
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if x1 < 0.00021208908081054686

   1. Initial program 11.2

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

   3. Applied flip-- 11.4

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

   4. Simplified 8.7

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

   6. Applied flip3-- 6.0

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

   7. Simplified 6.0

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

   9. Applied add-log-exp 6.0

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

   10. Applied add-log-exp 6.0

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

   11. Applied diff-log 5.1

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

   12. Simplified 5.1

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 3.5

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}} \cdot \sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   15. Applied log-prod 2.5

       \[\leadsto \frac{x0 \cdot \frac{\color{blue}{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   if 0.00021208908081054686 < x1

   1. Initial program 4.5

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

   3. Applied flip-- 3.2

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

   4. Simplified 3.9

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

   6. Applied flip3-- 3.9

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

   7. Simplified 3.9

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

   9. Applied add-log-exp 3.9

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

   10. Applied add-log-exp 3.9

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

   11. Applied diff-log 4.0

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

   12. Simplified 3.9

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 3.9

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(1 - \color{blue}{\sqrt{x1} \cdot \sqrt{x1}}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   15. Applied add-sqr-sqrt 3.9

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \sqrt{x1} \cdot \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   16. Applied difference-of-squares 3.9

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\color{blue}{\left(\left(\sqrt{1} + \sqrt{x1}\right) \cdot \left(\sqrt{1} - \sqrt{x1}\right)\right)}}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   17. Applied unpow-prod-down 1.6

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{\color{blue}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861321705391922876060562 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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 x1))

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