Average Error: 7.9 → 5.9
Time: 14.0s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\log \left(\sqrt{e^{\frac{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot x0}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}}\right) + \log \left(\sqrt{e^{\frac{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot x0}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}}\right)}{\left(\frac{x0}{1 - x1} \cdot x0 + x0 \cdot x0\right) + \frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\log \left(\sqrt{e^{\frac{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot x0}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}}\right) + \log \left(\sqrt{e^{\frac{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot x0}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}}\right)}{\left(\frac{x0}{1 - x1} \cdot x0 + x0 \cdot x0\right) + \frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}}
double f(double x0, double x1) {
        double r8182163 = x0;
        double r8182164 = 1.0;
        double r8182165 = x1;
        double r8182166 = r8182164 - r8182165;
        double r8182167 = r8182163 / r8182166;
        double r8182168 = r8182167 - r8182163;
        return r8182168;
}
↓
double f(double x0, double x1) {
        double r8182169 = x0;
        double r8182170 = 1.0;
        double r8182171 = x1;
        double r8182172 = r8182170 - r8182171;
        double r8182173 = r8182169 / r8182172;
        double r8182174 = r8182173 * r8182173;
        double r8182175 = r8182174 * r8182169;
        double r8182176 = r8182175 / r8182172;
        double r8182177 = r8182169 * r8182169;
        double r8182178 = r8182169 * r8182177;
        double r8182179 = r8182176 - r8182178;
        double r8182180 = exp(r8182179);
        double r8182181 = sqrt(r8182180);
        double r8182182 = log(r8182181);
        double r8182183 = r8182182 + r8182182;
        double r8182184 = r8182173 * r8182169;
        double r8182185 = r8182184 + r8182177;
        double r8182186 = r8182185 + r8182174;
        double r8182187 = r8182183 / r8182186;
        return r8182187;
}

Error

Bits error versus x0

Bits error versus x1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.9
Target0.3
Herbie5.9
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

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

  3. Applied flip3--7.7

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

  4. Simplified7.3

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

  6. Applied associate-*r/6.1

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

  8. Applied add-log-exp6.0

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

  10. Applied add-sqr-sqrt6.1

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

  11. Applied log-prod5.9

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

  12. Final simplification5.9

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :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))