Average Error: 20.8 → 0.0
Time: 20.7s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x - y}} \cdot \log \left(e^{\frac{x + y}{\mathsf{hypot}\left(x, y\right)}}\right)\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x - y}} \cdot \log \left(e^{\frac{x + y}{\mathsf{hypot}\left(x, y\right)}}\right)
double f(double x, double y) {
        double r77120 = x;
        double r77121 = y;
        double r77122 = r77120 - r77121;
        double r77123 = r77120 + r77121;
        double r77124 = r77122 * r77123;
        double r77125 = r77120 * r77120;
        double r77126 = r77121 * r77121;
        double r77127 = r77125 + r77126;
        double r77128 = r77124 / r77127;
        return r77128;
}
↓
double f(double x, double y) {
        double r77129 = 1.0;
        double r77130 = x;
        double r77131 = y;
        double r77132 = hypot(r77130, r77131);
        double r77133 = r77130 - r77131;
        double r77134 = r77132 / r77133;
        double r77135 = r77129 / r77134;
        double r77136 = r77130 + r77131;
        double r77137 = r77136 / r77132;
        double r77138 = exp(r77137);
        double r77139 = log(r77138);
        double r77140 = r77135 * r77139;
        return r77140;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.8
Target0.0
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Initial program 20.8

    \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt20.8

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

  4. Applied times-frac20.8

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

  5. Simplified20.8

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

  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 clear-num0.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x - y}}} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\]
  9. Using strategy rm

  10. Applied add-log-exp0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +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))))