Average Error: 7.9 → 4.7
Time: 21.5s
Precision: 64
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\log \left(e^{\left(\frac{1}{1 - x1} \cdot x0\right) \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{x0 + \frac{x0}{1 - x1}}\]
double f(double x0, double x1) {
        double r19274243 = x0;
        double r19274244 = 1.0;
        double r19274245 = x1;
        double r19274246 = r19274244 - r19274245;
        double r19274247 = r19274243 / r19274246;
        double r19274248 = r19274247 - r19274243;
        return r19274248;
}
↓
double f(double x0, double x1) {
        double r19274249 = 1.0;
        double r19274250 = x1;
        double r19274251 = r19274249 - r19274250;
        double r19274252 = r19274249 / r19274251;
        double r19274253 = x0;
        double r19274254 = r19274252 * r19274253;
        double r19274255 = r19274253 / r19274251;
        double r19274256 = r19274254 * r19274255;
        double r19274257 = r19274253 * r19274253;
        double r19274258 = r19274256 - r19274257;
        double r19274259 = exp(r19274258);
        double r19274260 = log(r19274259);
        double r19274261 = r19274253 + r19274255;
        double r19274262 = r19274260 / r19274261;
        return r19274262;
}


\frac{x0}{1 - x1} - x0
↓
\frac{\log \left(e^{\left(\frac{1}{1 - x1} \cdot x0\right) \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{x0 + \frac{x0}{1 - x1}}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
Target0.2
Herbie4.7
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

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

  3. Applied flip--7.3

    \[\leadsto \color{blue}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}}\]
  4. Using strategy rm

  5. Applied div-inv5.6

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

  7. Applied add-log-exp4.7

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

  8. Final simplification4.7

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

Reproduce

herbie shell --seed 2019101 
(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 x1))

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