Average Error: 22.7 → 7.8
Time: 16.1s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -11594563728487306:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, 1 \cdot \left(-\frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \le 863065781124297844139753472:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(y - 1\right) \cdot \frac{1}{\left(y - 1\right) \cdot \left(y + 1\right)}\right), x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, 1 \cdot \left(-\frac{x}{y}\right)\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -11594563728487306:\\
\;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, 1 \cdot \left(-\frac{x}{y}\right)\right)\\

\mathbf{elif}\;y \le 863065781124297844139753472:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(y - 1\right) \cdot \frac{1}{\left(y - 1\right) \cdot \left(y + 1\right)}\right), x - 1, 1\right)\\

\mathbf{else}:\\
\;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, 1 \cdot \left(-\frac{x}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r28544069 = 1.0;
        double r28544070 = x;
        double r28544071 = r28544069 - r28544070;
        double r28544072 = y;
        double r28544073 = r28544071 * r28544072;
        double r28544074 = r28544072 + r28544069;
        double r28544075 = r28544073 / r28544074;
        double r28544076 = r28544069 - r28544075;
        return r28544076;
}
↓
double f(double x, double y) {
        double r28544077 = y;
        double r28544078 = -11594563728487306.0;
        bool r28544079 = r28544077 <= r28544078;
        double r28544080 = x;
        double r28544081 = 1.0;
        double r28544082 = r28544081 / r28544077;
        double r28544083 = r28544080 / r28544077;
        double r28544084 = -r28544083;
        double r28544085 = r28544081 * r28544084;
        double r28544086 = fma(r28544082, r28544083, r28544085);
        double r28544087 = r28544080 + r28544086;
        double r28544088 = 8.630657811242978e+26;
        bool r28544089 = r28544077 <= r28544088;
        double r28544090 = r28544077 - r28544081;
        double r28544091 = 1.0;
        double r28544092 = r28544077 + r28544081;
        double r28544093 = r28544090 * r28544092;
        double r28544094 = r28544091 / r28544093;
        double r28544095 = r28544090 * r28544094;
        double r28544096 = r28544077 * r28544095;
        double r28544097 = r28544080 - r28544081;
        double r28544098 = fma(r28544096, r28544097, r28544081);
        double r28544099 = r28544089 ? r28544098 : r28544087;
        double r28544100 = r28544079 ? r28544087 : r28544099;
        return r28544100;
}

Error

Bits error versus x

Bits error versus y

Target

Original22.7
Target0.2
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -11594563728487306.0 or 8.630657811242978e+26 < y

    1. Initial program 47.0

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

    2. Simplified29.6

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

    4. Applied div-inv29.7

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

    5. Taylor expanded around inf 15.3

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

    6. Simplified15.3

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

    if -11594563728487306.0 < y < 8.630657811242978e+26

    1. Initial program 1.4

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

    2. Simplified1.3

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

    4. Applied div-inv1.3

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

    6. Applied flip-+1.3

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

    7. Applied associate-/r/1.3

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

    8. Simplified1.3

      \[\leadsto \mathsf{fma}\left(y \cdot \left(\color{blue}{\frac{1}{\left(y + 1\right) \cdot \left(y - 1\right)}} \cdot \left(y - 1\right)\right), x - 1, 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -11594563728487306:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, 1 \cdot \left(-\frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \le 863065781124297844139753472:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(y - 1\right) \cdot \frac{1}{\left(y - 1\right) \cdot \left(y + 1\right)}\right), x - 1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, 1 \cdot \left(-\frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))