Average Error: 7.8 → 6.1
Time: 11.0s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\frac{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot x0}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(\sqrt[3]{\frac{x0}{1 - x1} \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)} \cdot x0 + x0 \cdot x0\right)}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\frac{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot x0}{1 - x1} - x0 \cdot \left(x0 \cdot x0\right)}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(\sqrt[3]{\frac{x0}{1 - x1} \cdot \left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right)} \cdot x0 + x0 \cdot x0\right)}
double f(double x0, double x1) {
        double r7879135 = x0;
        double r7879136 = 1.0;
        double r7879137 = x1;
        double r7879138 = r7879136 - r7879137;
        double r7879139 = r7879135 / r7879138;
        double r7879140 = r7879139 - r7879135;
        return r7879140;
}
↓
double f(double x0, double x1) {
        double r7879141 = x0;
        double r7879142 = 1.0;
        double r7879143 = x1;
        double r7879144 = r7879142 - r7879143;
        double r7879145 = r7879141 / r7879144;
        double r7879146 = r7879145 * r7879145;
        double r7879147 = r7879146 * r7879141;
        double r7879148 = r7879147 / r7879144;
        double r7879149 = r7879141 * r7879141;
        double r7879150 = r7879141 * r7879149;
        double r7879151 = r7879148 - r7879150;
        double r7879152 = r7879145 * r7879146;
        double r7879153 = cbrt(r7879152);
        double r7879154 = r7879153 * r7879141;
        double r7879155 = r7879154 + r7879149;
        double r7879156 = r7879146 + r7879155;
        double r7879157 = r7879151 / r7879156;
        return r7879157;
}

Error

Bits error versus x0

Bits error versus x1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.8
Target0.2
Herbie6.1
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.8

    \[\frac{x0}{1 - x1} - x0\]
  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. Simplified7.2

    \[\leadsto \frac{\color{blue}{\left(\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1}\right) \cdot \frac{x0}{1 - x1} - \left(x0 \cdot x0\right) \cdot x0}}{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} + \left(x0 \cdot x0 + \frac{x0}{1 - x1} \cdot x0\right)}\]
  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-cbrt-cube6.1

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

  9. Applied add-cbrt-cube6.1

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

  10. Applied cbrt-undiv6.1

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

  11. Simplified6.1

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

  12. Final simplification6.1

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

Reproduce

herbie shell --seed 2019169 
(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))