Average Error: 20.0 → 0.0
Time: 17.1s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\log \left(e^{\frac{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{y + x}} \cdot \left(x - y\right)}{\mathsf{hypot}\left(x, y\right)}}\right)\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\log \left(e^{\frac{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{y + x}} \cdot \left(x - y\right)}{\mathsf{hypot}\left(x, y\right)}}\right)
double f(double x, double y) {
        double r72488 = x;
        double r72489 = y;
        double r72490 = r72488 - r72489;
        double r72491 = r72488 + r72489;
        double r72492 = r72490 * r72491;
        double r72493 = r72488 * r72488;
        double r72494 = r72489 * r72489;
        double r72495 = r72493 + r72494;
        double r72496 = r72492 / r72495;
        return r72496;
}
↓
double f(double x, double y) {
        double r72497 = 1.0;
        double r72498 = x;
        double r72499 = y;
        double r72500 = hypot(r72498, r72499);
        double r72501 = r72499 + r72498;
        double r72502 = r72500 / r72501;
        double r72503 = r72497 / r72502;
        double r72504 = r72498 - r72499;
        double r72505 = r72503 * r72504;
        double r72506 = r72505 / r72500;
        double r72507 = exp(r72506);
        double r72508 = log(r72507);
        return r72508;
}

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

Derivation

  1. Initial program 20.0

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

  2. Simplified20.0

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

  4. Applied add-sqr-sqrt20.0

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

  5. Applied times-frac20.1

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

  6. Simplified20.1

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

  7. Simplified0.0

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

  9. Applied add-log-exp0.0

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

  10. Simplified0.0

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

  12. Applied pow-exp0.0

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

  13. Simplified0.0

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

  15. Applied clear-num0.0

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

  16. Simplified0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

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