Average Error: 22.6 → 0.2
Time: 16.9s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -140476787.3433772623538970947265625:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \mathbf{elif}\;y \le 108523950.227932274341583251953125:\\ \;\;\;\;1 - \left(y - 1\right) \cdot \left(\frac{1 - x}{1 + y} \cdot \frac{y}{y - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -140476787.3433772623538970947265625:\\
\;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\

\mathbf{elif}\;y \le 108523950.227932274341583251953125:\\
\;\;\;\;1 - \left(y - 1\right) \cdot \left(\frac{1 - x}{1 + y} \cdot \frac{y}{y - 1}\right)\\

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

\end{array}
double f(double x, double y) {
        double r29451282 = 1.0;
        double r29451283 = x;
        double r29451284 = r29451282 - r29451283;
        double r29451285 = y;
        double r29451286 = r29451284 * r29451285;
        double r29451287 = r29451285 + r29451282;
        double r29451288 = r29451286 / r29451287;
        double r29451289 = r29451282 - r29451288;
        return r29451289;
}
↓
double f(double x, double y) {
        double r29451290 = y;
        double r29451291 = -140476787.34337726;
        bool r29451292 = r29451290 <= r29451291;
        double r29451293 = 1.0;
        double r29451294 = r29451293 / r29451290;
        double r29451295 = x;
        double r29451296 = r29451290 / r29451295;
        double r29451297 = r29451293 / r29451296;
        double r29451298 = r29451297 - r29451295;
        double r29451299 = r29451294 - r29451298;
        double r29451300 = 108523950.22793227;
        bool r29451301 = r29451290 <= r29451300;
        double r29451302 = r29451290 - r29451293;
        double r29451303 = r29451293 - r29451295;
        double r29451304 = r29451293 + r29451290;
        double r29451305 = r29451303 / r29451304;
        double r29451306 = r29451290 / r29451302;
        double r29451307 = r29451305 * r29451306;
        double r29451308 = r29451302 * r29451307;
        double r29451309 = r29451293 - r29451308;
        double r29451310 = r29451301 ? r29451309 : r29451299;
        double r29451311 = r29451292 ? r29451299 : r29451310;
        return r29451311;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.6
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -140476787.34337726 or 108523950.22793227 < y

    1. Initial program 45.5

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

    3. Applied *-un-lft-identity45.5

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

    4. Applied times-frac29.1

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

    5. Simplified29.1

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

    6. Taylor expanded around inf 0.2

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

    7. Simplified0.2

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

    if -140476787.34337726 < y < 108523950.22793227

    1. Initial program 0.2

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

    3. Applied *-un-lft-identity0.2

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

    4. Applied times-frac0.2

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

    5. Simplified0.2

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

    7. Applied flip-+0.2

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

    8. Applied associate-/r/0.2

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

    9. Applied associate-*r*0.2

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

    10. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -140476787.3433772623538970947265625:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \mathbf{elif}\;y \le 108523950.227932274341583251953125:\\ \;\;\;\;1 - \left(y - 1\right) \cdot \left(\frac{1 - x}{1 + y} \cdot \frac{y}{y - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{1}{\frac{y}{x}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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))))