Average Error: 20.0 → 0.1
Time: 19.4s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\sqrt[3]{{\left(\log \left(e^{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right)\right)}^{3}} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\sqrt[3]{{\left(\log \left(e^{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right)\right)}^{3}} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}
double f(double x, double y) {
        double r64783 = x;
        double r64784 = y;
        double r64785 = r64783 - r64784;
        double r64786 = r64783 + r64784;
        double r64787 = r64785 * r64786;
        double r64788 = r64783 * r64783;
        double r64789 = r64784 * r64784;
        double r64790 = r64788 + r64789;
        double r64791 = r64787 / r64790;
        return r64791;
}
↓
double f(double x, double y) {
        double r64792 = x;
        double r64793 = y;
        double r64794 = r64792 - r64793;
        double r64795 = hypot(r64792, r64793);
        double r64796 = r64794 / r64795;
        double r64797 = exp(r64796);
        double r64798 = log(r64797);
        double r64799 = 3.0;
        double r64800 = pow(r64798, r64799);
        double r64801 = cbrt(r64800);
        double r64802 = r64792 + r64793;
        double r64803 = r64802 / r64795;
        double r64804 = r64801 * r64803;
        return r64804;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt20.0

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

  4. Applied times-frac20.1

    \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]

  5. Simplified20.1

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

  6. Simplified0.0

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

  8. Applied add-cbrt-cube32.1

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

  9. Applied add-cbrt-cube32.0

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

  10. Applied cbrt-undiv32.0

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

  11. Simplified0.0

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

  13. Applied add-log-exp0.1

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

  14. Final simplification0.1

    \[\leadsto \sqrt[3]{{\left(\log \left(e^{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}\right)\right)}^{3}} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\]

Reproduce

herbie shell --seed 2019235 +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))))