Average Error: 22.5 → 0.1
Time: 4.3s
Precision: 64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -340993515449.59576416015625 \lor \neg \left(y \le 3406145480945.9931640625\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \left(\frac{1 - x}{1 \cdot \left(1 - y\right) + {y}^{2}} \cdot \frac{y}{y + 1}\right)\right) - \left(1 \cdot 1 - y \cdot 1\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \le -340993515449.59576416015625 \lor \neg \left(y \le 3406145480945.9931640625\right):\\
\;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\

\mathbf{else}:\\
\;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \left(\frac{1 - x}{1 \cdot \left(1 - y\right) + {y}^{2}} \cdot \frac{y}{y + 1}\right)\right) - \left(1 \cdot 1 - y \cdot 1\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\\

\end{array}
double f(double x, double y) {
        double r667312 = 1.0;
        double r667313 = x;
        double r667314 = r667312 - r667313;
        double r667315 = y;
        double r667316 = r667314 * r667315;
        double r667317 = r667315 + r667312;
        double r667318 = r667316 / r667317;
        double r667319 = r667312 - r667318;
        return r667319;
}
↓
double f(double x, double y) {
        double r667320 = y;
        double r667321 = -340993515449.59576;
        bool r667322 = r667320 <= r667321;
        double r667323 = 3406145480945.993;
        bool r667324 = r667320 <= r667323;
        double r667325 = !r667324;
        bool r667326 = r667322 || r667325;
        double r667327 = 1.0;
        double r667328 = 1.0;
        double r667329 = r667328 / r667320;
        double r667330 = x;
        double r667331 = r667330 / r667320;
        double r667332 = r667329 - r667331;
        double r667333 = r667327 * r667332;
        double r667334 = r667333 + r667330;
        double r667335 = r667320 * r667320;
        double r667336 = r667327 - r667330;
        double r667337 = r667327 - r667320;
        double r667338 = r667327 * r667337;
        double r667339 = 2.0;
        double r667340 = pow(r667320, r667339);
        double r667341 = r667338 + r667340;
        double r667342 = r667336 / r667341;
        double r667343 = r667320 + r667327;
        double r667344 = r667320 / r667343;
        double r667345 = r667342 * r667344;
        double r667346 = r667335 * r667345;
        double r667347 = r667327 - r667346;
        double r667348 = r667327 * r667327;
        double r667349 = r667320 * r667327;
        double r667350 = r667348 - r667349;
        double r667351 = r667336 * r667320;
        double r667352 = 3.0;
        double r667353 = pow(r667320, r667352);
        double r667354 = pow(r667327, r667352);
        double r667355 = r667353 + r667354;
        double r667356 = r667351 / r667355;
        double r667357 = r667350 * r667356;
        double r667358 = r667347 - r667357;
        double r667359 = r667326 ? r667334 : r667358;
        return r667359;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.5
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -340993515449.59576 or 3406145480945.993 < y

    1. Initial program 46.4

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -340993515449.59576 < y < 3406145480945.993

    1. Initial program 0.4

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

    3. Applied flip3-+0.4

      \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}}\]

    4. Applied associate-/r/0.4

      \[\leadsto 1 - \color{blue}{\frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)}\]
    5. Using strategy rm

    6. Applied distribute-rgt-in0.4

      \[\leadsto 1 - \color{blue}{\left(\left(y \cdot y\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}} + \left(1 \cdot 1 - y \cdot 1\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right)}\]

    7. Applied associate--r+0.2

      \[\leadsto \color{blue}{\left(1 - \left(y \cdot y\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\right) - \left(1 \cdot 1 - y \cdot 1\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}}\]
    8. Using strategy rm

    9. Applied sum-cubes0.3

      \[\leadsto \left(1 - \left(y \cdot y\right) \cdot \frac{\left(1 - x\right) \cdot y}{\color{blue}{\left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right) \cdot \left(y + 1\right)}}\right) - \left(1 \cdot 1 - y \cdot 1\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\]

    10. Applied times-frac0.2

      \[\leadsto \left(1 - \left(y \cdot y\right) \cdot \color{blue}{\left(\frac{1 - x}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)} \cdot \frac{y}{y + 1}\right)}\right) - \left(1 \cdot 1 - y \cdot 1\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\]

    11. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -340993515449.59576416015625 \lor \neg \left(y \le 3406145480945.9931640625\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(y \cdot y\right) \cdot \left(\frac{1 - x}{1 \cdot \left(1 - y\right) + {y}^{2}} \cdot \frac{y}{y + 1}\right)\right) - \left(1 \cdot 1 - y \cdot 1\right) \cdot \frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))