Average Error: 0.0 → 0.0
Time: 10.6s
Precision: 64
\[\frac{-\left(f + n\right)}{f - n}\]
↓
\[\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, \sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, -1\right)\right)\]
\frac{-\left(f + n\right)}{f - n}
↓
\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, \sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, -1\right)\right)
double f(double f, double n) {
        double r26730 = f;
        double r26731 = n;
        double r26732 = r26730 + r26731;
        double r26733 = -r26732;
        double r26734 = r26730 - r26731;
        double r26735 = r26733 / r26734;
        return r26735;
}
↓
double f(double f, double n) {
        double r26736 = f;
        double r26737 = n;
        double r26738 = r26736 + r26737;
        double r26739 = -r26738;
        double r26740 = r26736 - r26737;
        double r26741 = r26739 / r26740;
        double r26742 = exp(r26741);
        double r26743 = sqrt(r26742);
        double r26744 = -1.0;
        double r26745 = fma(r26743, r26743, r26744);
        double r26746 = log1p(r26745);
        return r26746;
}

Error

Bits error versus f

Bits error versus n

Derivation

  1. Initial program 0.0

    \[\frac{-\left(f + n\right)}{f - n}\]
  2. Using strategy rm

  3. Applied log1p-expm1-u0.0

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-\left(f + n\right)}{f - n}\right)\right)}\]
  4. Using strategy rm

  5. Applied expm1-udef0.0

    \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\frac{-\left(f + n\right)}{f - n}} - 1}\right)\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.0

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}} \cdot \sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}} - 1\right)\]

  8. Applied fma-neg0.0

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, \sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, -1\right)}\right)\]

  9. Simplified0.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, \sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, \color{blue}{-1}\right)\right)\]

  10. Final simplification0.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, \sqrt{e^{\frac{-\left(f + n\right)}{f - n}}}, -1\right)\right)\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))