Average Error: 0.0 → 0.0
Time: 3.5s
Precision: 64
\[\frac{x - y}{x + y}\]
↓
\[\frac{\sqrt[3]{{\left({\left(\frac{x}{x + y}\right)}^{3}\right)}^{3}} - {\left(\frac{y}{x + y}\right)}^{3}}{\frac{y}{x + y} \cdot \left(\frac{y}{x + y} + \frac{x}{x + y}\right) + \frac{x}{x + y} \cdot \frac{x}{x + y}}\]
\frac{x - y}{x + y}
↓
\frac{\sqrt[3]{{\left({\left(\frac{x}{x + y}\right)}^{3}\right)}^{3}} - {\left(\frac{y}{x + y}\right)}^{3}}{\frac{y}{x + y} \cdot \left(\frac{y}{x + y} + \frac{x}{x + y}\right) + \frac{x}{x + y} \cdot \frac{x}{x + y}}
double f(double x, double y) {
        double r716095 = x;
        double r716096 = y;
        double r716097 = r716095 - r716096;
        double r716098 = r716095 + r716096;
        double r716099 = r716097 / r716098;
        return r716099;
}
↓
double f(double x, double y) {
        double r716100 = x;
        double r716101 = y;
        double r716102 = r716100 + r716101;
        double r716103 = r716100 / r716102;
        double r716104 = 3.0;
        double r716105 = pow(r716103, r716104);
        double r716106 = pow(r716105, r716104);
        double r716107 = cbrt(r716106);
        double r716108 = r716101 / r716102;
        double r716109 = pow(r716108, r716104);
        double r716110 = r716107 - r716109;
        double r716111 = r716108 + r716103;
        double r716112 = r716108 * r716111;
        double r716113 = r716103 * r716103;
        double r716114 = r716112 + r716113;
        double r716115 = r716110 / r716114;
        return r716115;
}

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 div-sub0.0

    \[\leadsto \color{blue}{\frac{x}{x + y} - \frac{y}{x + y}}\]
  4. Using strategy rm

  5. Applied flip3--0.0

    \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{3} - {\left(\frac{y}{x + y}\right)}^{3}}{\frac{x}{x + y} \cdot \frac{x}{x + y} + \left(\frac{y}{x + y} \cdot \frac{y}{x + y} + \frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}\]

  6. Simplified0.0

    \[\leadsto \frac{{\left(\frac{x}{x + y}\right)}^{3} - {\left(\frac{y}{x + y}\right)}^{3}}{\color{blue}{\frac{y}{x + y} \cdot \left(\frac{y}{x + y} + \frac{x}{x + y}\right) + \frac{x}{x + y} \cdot \frac{x}{x + y}}}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube0.0

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left({\left(\frac{x}{x + y}\right)}^{3} \cdot {\left(\frac{x}{x + y}\right)}^{3}\right) \cdot {\left(\frac{x}{x + y}\right)}^{3}}} - {\left(\frac{y}{x + y}\right)}^{3}}{\frac{y}{x + y} \cdot \left(\frac{y}{x + y} + \frac{x}{x + y}\right) + \frac{x}{x + y} \cdot \frac{x}{x + y}}\]

  9. Simplified0.0

    \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left({\left(\frac{x}{x + y}\right)}^{3}\right)}^{3}}} - {\left(\frac{y}{x + y}\right)}^{3}}{\frac{y}{x + y} \cdot \left(\frac{y}{x + y} + \frac{x}{x + y}\right) + \frac{x}{x + y} \cdot \frac{x}{x + y}}\]

  10. Final simplification0.0

    \[\leadsto \frac{\sqrt[3]{{\left({\left(\frac{x}{x + y}\right)}^{3}\right)}^{3}} - {\left(\frac{y}{x + y}\right)}^{3}}{\frac{y}{x + y} \cdot \left(\frac{y}{x + y} + \frac{x}{x + y}\right) + \frac{x}{x + y} \cdot \frac{x}{x + y}}\]

Reproduce

herbie shell --seed 2020100 
(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)))