Average Error: 0.0 → 0.0
Time: 4.0s
Precision: 64
\[\frac{x + y}{x - y}\]
↓
\[\frac{x + y}{x - y} \cdot \log e\]
\frac{x + y}{x - y}
↓
\frac{x + y}{x - y} \cdot \log e
double f(double x, double y) {
        double r349146 = x;
        double r349147 = y;
        double r349148 = r349146 + r349147;
        double r349149 = r349146 - r349147;
        double r349150 = r349148 / r349149;
        return r349150;
}
↓
double f(double x, double y) {
        double r349151 = x;
        double r349152 = y;
        double r349153 = r349151 + r349152;
        double r349154 = r349151 - r349152;
        double r349155 = r349153 / r349154;
        double r349156 = exp(1.0);
        double r349157 = log(r349156);
        double r349158 = r349155 * r349157;
        return r349158;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.0
Target0.0
Herbie0.0
\[\frac{1}{\frac{x}{x + y} - \frac{y}{x + y}}\]

Derivation

  1. Initial program 0.0

    \[\frac{x + y}{x - y}\]
  2. Using strategy rm

  3. Applied add-log-exp0.0

    \[\leadsto \color{blue}{\log \left(e^{\frac{x + y}{x - y}}\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.0

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

  6. Applied *-un-lft-identity0.0

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

  7. Applied times-frac0.0

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

  8. Applied exp-prod0.0

    \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{1}}\right)}^{\left(\frac{x + y}{x - y}\right)}\right)}\]

  9. Simplified0.0

    \[\leadsto \log \left({\color{blue}{e}}^{\left(\frac{x + y}{x - y}\right)}\right)\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt0.5

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

  12. Applied unpow-prod-down0.0

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

  13. Applied log-prod0.0

    \[\leadsto \color{blue}{\log \left({\left(\sqrt{e}\right)}^{\left(\frac{x + y}{x - y}\right)}\right) + \log \left({\left(\sqrt{e}\right)}^{\left(\frac{x + y}{x - y}\right)}\right)}\]

  14. Simplified0.0

    \[\leadsto \color{blue}{\log \left({e}^{\left(\frac{\frac{x + y}{x - y}}{2}\right)}\right)} + \log \left({\left(\sqrt{e}\right)}^{\left(\frac{x + y}{x - y}\right)}\right)\]

  15. Simplified0.0

    \[\leadsto \log \left({e}^{\left(\frac{\frac{x + y}{x - y}}{2}\right)}\right) + \color{blue}{\log \left({e}^{\left(\frac{\frac{x + y}{x - y}}{2}\right)}\right)}\]

  16. Final simplification0.0

    \[\leadsto \frac{x + y}{x - y} \cdot \log e\]

Reproduce

herbie shell --seed 2019308 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (/ 1 (- (/ x (+ x y)) (/ y (+ x y))))

  (/ (+ x y) (- x y)))