Average Error: 19.8 → 4.9
Time: 2.5s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.1473874126766077 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.8337771258008988 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{{x}^{2} + {y}^{2}}}}\\ \mathbf{elif}\;y \le 5.7798438136698065 \cdot 10^{-177}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{{x}^{2} + {y}^{2}}}}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.1473874126766077 \cdot 10^{151}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -1.8337771258008988 \cdot 10^{-161}:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{{x}^{2} + {y}^{2}}}}\\

\mathbf{elif}\;y \le 5.7798438136698065 \cdot 10^{-177}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{{x}^{2} + {y}^{2}}}}\\

\end{array}
double f(double x, double y) {
        double r84014 = x;
        double r84015 = y;
        double r84016 = r84014 - r84015;
        double r84017 = r84014 + r84015;
        double r84018 = r84016 * r84017;
        double r84019 = r84014 * r84014;
        double r84020 = r84015 * r84015;
        double r84021 = r84019 + r84020;
        double r84022 = r84018 / r84021;
        return r84022;
}
↓
double f(double x, double y) {
        double r84023 = y;
        double r84024 = -1.1473874126766077e+151;
        bool r84025 = r84023 <= r84024;
        double r84026 = -1.0;
        double r84027 = -1.833777125800899e-161;
        bool r84028 = r84023 <= r84027;
        double r84029 = 1.0;
        double r84030 = x;
        double r84031 = r84030 - r84023;
        double r84032 = r84030 + r84023;
        double r84033 = r84031 * r84032;
        double r84034 = 2.0;
        double r84035 = pow(r84030, r84034);
        double r84036 = pow(r84023, r84034);
        double r84037 = r84035 + r84036;
        double r84038 = r84033 / r84037;
        double r84039 = r84029 / r84038;
        double r84040 = r84029 / r84039;
        double r84041 = 5.7798438136698065e-177;
        bool r84042 = r84023 <= r84041;
        double r84043 = r84042 ? r84029 : r84040;
        double r84044 = r84028 ? r84040 : r84043;
        double r84045 = r84025 ? r84026 : r84044;
        return r84045;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.8
Target0.1
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.1473874126766077e+151

    1. Initial program 62.7

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

    2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{-1}\]

    if -1.1473874126766077e+151 < y < -1.833777125800899e-161 or 5.7798438136698065e-177 < y

    1. Initial program 0.9

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

    3. Applied clear-num0.9

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

    5. Applied clear-num0.9

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

    6. Simplified0.9

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{{x}^{2} + {y}^{2}}}}}\]

    if -1.833777125800899e-161 < y < 5.7798438136698065e-177

    1. Initial program 29.5

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

    2. Taylor expanded around inf 15.0

      \[\leadsto \color{blue}{1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.1473874126766077 \cdot 10^{151}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -1.8337771258008988 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{{x}^{2} + {y}^{2}}}}\\ \mathbf{elif}\;y \le 5.7798438136698065 \cdot 10^{-177}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\left(x - y\right) \cdot \left(x + y\right)}{{x}^{2} + {y}^{2}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))