Average Error: 0.0 → 0.0
Time: 3.3s
Precision: 64
\[\frac{x - y}{x + y}\]
↓
\[\log \left(\sqrt{e^{\frac{x - y}{x + y}}}\right) + \frac{1}{2} \cdot \frac{x - y}{x + y}\]
\frac{x - y}{x + y}
↓
\log \left(\sqrt{e^{\frac{x - y}{x + y}}}\right) + \frac{1}{2} \cdot \frac{x - y}{x + y}
double f(double x, double y) {
        double r833131 = x;
        double r833132 = y;
        double r833133 = r833131 - r833132;
        double r833134 = r833131 + r833132;
        double r833135 = r833133 / r833134;
        return r833135;
}
↓
double f(double x, double y) {
        double r833136 = x;
        double r833137 = y;
        double r833138 = r833136 - r833137;
        double r833139 = r833136 + r833137;
        double r833140 = r833138 / r833139;
        double r833141 = exp(r833140);
        double r833142 = sqrt(r833141);
        double r833143 = log(r833142);
        double r833144 = 0.5;
        double r833145 = r833144 * r833140;
        double r833146 = r833143 + r833145;
        return r833146;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

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 add-sqr-sqrt0.0

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

  6. Applied log-prod0.0

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

  8. Applied pow10.0

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

  9. Applied sqrt-pow10.0

    \[\leadsto \log \left(\sqrt{e^{\frac{x - y}{x + y}}}\right) + \log \color{blue}{\left({\left(e^{\frac{x - y}{x + y}}\right)}^{\left(\frac{1}{2}\right)}\right)}\]

  10. Applied log-pow0.0

    \[\leadsto \log \left(\sqrt{e^{\frac{x - y}{x + y}}}\right) + \color{blue}{\frac{1}{2} \cdot \log \left(e^{\frac{x - y}{x + y}}\right)}\]

  11. Simplified0.0

    \[\leadsto \log \left(\sqrt{e^{\frac{x - y}{x + y}}}\right) + \frac{1}{2} \cdot \color{blue}{\frac{x - y}{x + y}}\]

  12. Final simplification0.0

    \[\leadsto \log \left(\sqrt{e^{\frac{x - y}{x + y}}}\right) + \frac{1}{2} \cdot \frac{x - y}{x + y}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"
  :precision binary64

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

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