Average Error: 0.0 → 0.0
Time: 11.1s
Precision: 64
\[\frac{x - y}{x + y}\]
↓
\[\log \left(\frac{e^{\frac{x}{y + x}}}{1 + \mathsf{expm1}\left(\frac{y}{y + x}\right)}\right)\]
\frac{x - y}{x + y}
↓
\log \left(\frac{e^{\frac{x}{y + x}}}{1 + \mathsf{expm1}\left(\frac{y}{y + x}\right)}\right)
double f(double x, double y) {
        double r24150097 = x;
        double r24150098 = y;
        double r24150099 = r24150097 - r24150098;
        double r24150100 = r24150097 + r24150098;
        double r24150101 = r24150099 / r24150100;
        return r24150101;
}
↓
double f(double x, double y) {
        double r24150102 = x;
        double r24150103 = y;
        double r24150104 = r24150103 + r24150102;
        double r24150105 = r24150102 / r24150104;
        double r24150106 = exp(r24150105);
        double r24150107 = 1.0;
        double r24150108 = r24150103 / r24150104;
        double r24150109 = expm1(r24150108);
        double r24150110 = r24150107 + r24150109;
        double r24150111 = r24150106 / r24150110;
        double r24150112 = log(r24150111);
        return r24150112;
}

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 log1p-expm1-u0.0

    \[\leadsto \frac{x}{x + y} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{x + y}\right)\right)}\]
  6. Using strategy rm

  7. Applied log1p-udef0.0

    \[\leadsto \frac{x}{x + y} - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{y}{x + y}\right)\right)}\]

  8. Applied add-log-exp0.0

    \[\leadsto \color{blue}{\log \left(e^{\frac{x}{x + y}}\right)} - \log \left(1 + \mathsf{expm1}\left(\frac{y}{x + y}\right)\right)\]

  9. Applied diff-log0.0

    \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{x}{x + y}}}{1 + \mathsf{expm1}\left(\frac{y}{x + y}\right)}\right)}\]

  10. Final simplification0.0

    \[\leadsto \log \left(\frac{e^{\frac{x}{y + x}}}{1 + \mathsf{expm1}\left(\frac{y}{y + x}\right)}\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, D"

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

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