Average Error: 0.0 → 0.0
Time: 17.4s
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 r33926448 = x;
        double r33926449 = y;
        double r33926450 = r33926448 - r33926449;
        double r33926451 = r33926448 + r33926449;
        double r33926452 = r33926450 / r33926451;
        return r33926452;
}
↓
double f(double x, double y) {
        double r33926453 = x;
        double r33926454 = y;
        double r33926455 = r33926454 + r33926453;
        double r33926456 = r33926453 / r33926455;
        double r33926457 = exp(r33926456);
        double r33926458 = 1.0;
        double r33926459 = r33926454 / r33926455;
        double r33926460 = expm1(r33926459);
        double r33926461 = r33926458 + r33926460;
        double r33926462 = r33926457 / r33926461;
        double r33926463 = log(r33926462);
        return r33926463;
}

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)))