Average Error: 20.0 → 0.1
Time: 3.8s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{\frac{x + y}{\mathsf{hypot}\left(x, y\right)}}}\right)\right)\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{\frac{x + y}{\mathsf{hypot}\left(x, y\right)}}}\right)\right)
double f(double x, double y) {
        double r94080 = x;
        double r94081 = y;
        double r94082 = r94080 - r94081;
        double r94083 = r94080 + r94081;
        double r94084 = r94082 * r94083;
        double r94085 = r94080 * r94080;
        double r94086 = r94081 * r94081;
        double r94087 = r94085 + r94086;
        double r94088 = r94084 / r94087;
        return r94088;
}
↓
double f(double x, double y) {
        double r94089 = x;
        double r94090 = y;
        double r94091 = r94089 - r94090;
        double r94092 = hypot(r94089, r94090);
        double r94093 = 1.0;
        double r94094 = r94089 + r94090;
        double r94095 = r94094 / r94092;
        double r94096 = r94093 / r94095;
        double r94097 = r94092 * r94096;
        double r94098 = r94091 / r94097;
        double r94099 = log1p(r94098);
        double r94100 = expm1(r94099);
        return r94100;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.0
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Initial program 20.0

    \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

  2. Simplified20.1

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

  4. Applied *-un-lft-identity20.1

    \[\leadsto \frac{x - y}{\frac{\mathsf{fma}\left(x, x, y \cdot y\right)}{\color{blue}{1 \cdot \left(x + y\right)}}}\]

  5. Applied add-sqr-sqrt20.1

    \[\leadsto \frac{x - y}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}}}{1 \cdot \left(x + y\right)}}\]

  6. Applied times-frac20.1

    \[\leadsto \frac{x - y}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}}{x + y}}}\]

  7. Simplified20.1

    \[\leadsto \frac{x - y}{\color{blue}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, y \cdot y\right)}}{x + y}}\]

  8. Simplified0.0

    \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}}\]
  9. Using strategy rm

  10. Applied expm1-log1p-u0.1

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

  12. Applied clear-num0.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\frac{1}{\frac{x + y}{\mathsf{hypot}\left(x, y\right)}}}}\right)\right)\]

  13. Final simplification0.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{\frac{x + y}{\mathsf{hypot}\left(x, y\right)}}}\right)\right)\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

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