Average Error: 15.3 → 0.1
Time: 2.5s
Precision: 64
\[\frac{x + y}{\left(x \cdot 2\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.2211876626416346 \cdot 10^{-12}:\\ \;\;\;\;\frac{x + y}{x \cdot 2} \cdot \frac{1}{y}\\ \mathbf{elif}\;x \le 9.43358736000154189 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y}{\frac{x + y}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}\\ \end{array}\]
\frac{x + y}{\left(x \cdot 2\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.2211876626416346 \cdot 10^{-12}:\\
\;\;\;\;\frac{x + y}{x \cdot 2} \cdot \frac{1}{y}\\

\mathbf{elif}\;x \le 9.43358736000154189 \cdot 10^{-18}:\\
\;\;\;\;\frac{\frac{1}{x}}{\frac{y}{\frac{x + y}{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}\\

\end{array}
double f(double x, double y) {
        double r2779 = x;
        double r2780 = y;
        double r2781 = r2779 + r2780;
        double r2782 = 2.0;
        double r2783 = r2779 * r2782;
        double r2784 = r2783 * r2780;
        double r2785 = r2781 / r2784;
        return r2785;
}
↓
double f(double x, double y) {
        double r2786 = x;
        double r2787 = -5.221187662641635e-12;
        bool r2788 = r2786 <= r2787;
        double r2789 = y;
        double r2790 = r2786 + r2789;
        double r2791 = 2.0;
        double r2792 = r2786 * r2791;
        double r2793 = r2790 / r2792;
        double r2794 = 1.0;
        double r2795 = r2794 / r2789;
        double r2796 = r2793 * r2795;
        double r2797 = 9.433587360001542e-18;
        bool r2798 = r2786 <= r2797;
        double r2799 = r2794 / r2786;
        double r2800 = r2790 / r2791;
        double r2801 = r2789 / r2800;
        double r2802 = r2799 / r2801;
        double r2803 = 0.5;
        double r2804 = r2789 / r2786;
        double r2805 = fma(r2803, r2804, r2803);
        double r2806 = r2805 / r2789;
        double r2807 = r2798 ? r2802 : r2806;
        double r2808 = r2788 ? r2796 : r2807;
        return r2808;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.3
Target0.0
Herbie0.1
\[\frac{0.5}{x} + \frac{0.5}{y}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -5.221187662641635e-12

    1. Initial program 15.5

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

    3. Applied associate-/r*0.2

      \[\leadsto \color{blue}{\frac{\frac{x + y}{x \cdot 2}}{y}}\]
    4. Using strategy rm

    5. Applied div-inv0.2

      \[\leadsto \color{blue}{\frac{x + y}{x \cdot 2} \cdot \frac{1}{y}}\]

    if -5.221187662641635e-12 < x < 9.433587360001542e-18

    1. Initial program 15.5

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

    3. Applied associate-/r*16.6

      \[\leadsto \color{blue}{\frac{\frac{x + y}{x \cdot 2}}{y}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity16.6

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

    6. Applied times-frac16.6

      \[\leadsto \frac{\color{blue}{\frac{1}{x} \cdot \frac{x + y}{2}}}{y}\]

    7. Applied associate-/l*0.1

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{y}{\frac{x + y}{2}}}}\]

    if 9.433587360001542e-18 < x

    1. Initial program 14.6

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

    3. Applied associate-/r*0.2

      \[\leadsto \color{blue}{\frac{\frac{x + y}{x \cdot 2}}{y}}\]

    4. Taylor expanded around 0 0.1

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{y}{x} + 0.5}}{y}\]

    5. Simplified0.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}}{y}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.2211876626416346 \cdot 10^{-12}:\\ \;\;\;\;\frac{x + y}{x \cdot 2} \cdot \frac{1}{y}\\ \mathbf{elif}\;x \le 9.43358736000154189 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y}{\frac{x + y}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{x}, 0.5\right)}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

  :herbie-target
  (+ (/ 0.5 x) (/ 0.5 y))

  (/ (+ x y) (* (* x 2) y)))