Average Error: 7.9 → 5.9
Time: 13.7s
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 r7369125 = x0;
        double r7369126 = 1.0;
        double r7369127 = x1;
        double r7369128 = r7369126 - r7369127;
        double r7369129 = r7369125 / r7369128;
        double r7369130 = r7369129 - r7369125;
        return r7369130;
}
↓
double f(double x0, double x1) {
        double r7369131 = x0;
        double r7369132 = 1.0;
        double r7369133 = x1;
        double r7369134 = r7369132 - r7369133;
        double r7369135 = r7369131 / r7369134;
        double r7369136 = r7369135 * r7369135;
        double r7369137 = r7369136 * r7369131;
        double r7369138 = r7369137 / r7369134;
        double r7369139 = r7369131 * r7369131;
        double r7369140 = r7369131 * r7369139;
        double r7369141 = r7369138 - r7369140;
        double r7369142 = exp(r7369141);
        double r7369143 = sqrt(r7369142);
        double r7369144 = log(r7369143);
        double r7369145 = r7369144 + r7369144;
        double r7369146 = r7369135 * r7369131;
        double r7369147 = r7369146 + r7369139;
        double r7369148 = r7369147 + r7369136;
        double r7369149 = r7369145 / r7369148;
        return r7369149;
}

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))