Average Error: 29.2 → 0.1
Time: 21.9s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.006169443383165685093616481537992513040081:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1, \frac{\left({\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{2} - 1 \cdot 1\right) \cdot 2}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)}\\ \mathbf{elif}\;-2 \cdot x \le 1.858478937487164482871759426069017961947 \cdot 10^{-9}:\\ \;\;\;\;1 \cdot x - \mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}\right)}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.006169443383165685093616481537992513040081:\\
\;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1, \frac{\left({\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{2} - 1 \cdot 1\right) \cdot 2}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)}\\

\mathbf{elif}\;-2 \cdot x \le 1.858478937487164482871759426069017961947 \cdot 10^{-9}:\\
\;\;\;\;1 \cdot x - \mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(e^{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}\right)}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)\right)}\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r48392 = 2.0;
        double r48393 = 1.0;
        double r48394 = -2.0;
        double r48395 = x;
        double r48396 = r48394 * r48395;
        double r48397 = exp(r48396);
        double r48398 = r48393 + r48397;
        double r48399 = r48392 / r48398;
        double r48400 = r48399 - r48393;
        return r48400;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r48401 = -2.0;
        double r48402 = x;
        double r48403 = r48401 * r48402;
        double r48404 = -0.006169443383165685;
        bool r48405 = r48403 <= r48404;
        double r48406 = 2.0;
        double r48407 = 1.0;
        double r48408 = exp(r48403);
        double r48409 = r48407 + r48408;
        double r48410 = r48406 / r48409;
        double r48411 = 3.0;
        double r48412 = pow(r48410, r48411);
        double r48413 = pow(r48407, r48411);
        double r48414 = r48412 - r48413;
        double r48415 = r48408 + r48407;
        double r48416 = r48406 / r48415;
        double r48417 = 2.0;
        double r48418 = pow(r48416, r48417);
        double r48419 = r48407 * r48407;
        double r48420 = r48418 - r48419;
        double r48421 = r48420 * r48406;
        double r48422 = r48410 - r48407;
        double r48423 = r48409 * r48422;
        double r48424 = r48421 / r48423;
        double r48425 = fma(r48407, r48407, r48424);
        double r48426 = r48414 / r48425;
        double r48427 = 1.8584789374871645e-09;
        bool r48428 = r48403 <= r48427;
        double r48429 = r48407 * r48402;
        double r48430 = 5.551115123125783e-17;
        double r48431 = 4.0;
        double r48432 = pow(r48402, r48431);
        double r48433 = 0.33333333333333337;
        double r48434 = pow(r48402, r48411);
        double r48435 = r48433 * r48434;
        double r48436 = fma(r48430, r48432, r48435);
        double r48437 = r48429 - r48436;
        double r48438 = exp(r48414);
        double r48439 = log(r48438);
        double r48440 = r48410 + r48407;
        double r48441 = r48410 * r48440;
        double r48442 = fma(r48407, r48407, r48441);
        double r48443 = r48439 / r48442;
        double r48444 = r48428 ? r48437 : r48443;
        double r48445 = r48405 ? r48426 : r48444;
        return r48445;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (* -2.0 x) < -0.006169443383165685

    1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied flip3--0.0

      \[\leadsto \color{blue}{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} + \left(1 \cdot 1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 1\right)}}\]

    4. Simplified0.0

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\color{blue}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)\right)}}\]
    5. Using strategy rm

    6. Applied flip-+0.0

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} - 1}}\right)}\]

    7. Applied frac-times0.0

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1, \color{blue}{\frac{2 \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1\right)}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}}\right)}\]

    8. Simplified0.0

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1, \frac{\color{blue}{\left({\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{2} - 1 \cdot 1\right) \cdot 2}}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)}\]

    if -0.006169443383165685 < (* -2.0 x) < 1.8584789374871645e-09

    1. Initial program 59.4

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{1 \cdot x - \mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)}\]

    if 1.8584789374871645e-09 < (* -2.0 x)

    1. Initial program 0.4

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied flip3--0.4

      \[\leadsto \color{blue}{\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} + \left(1 \cdot 1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 1\right)}}\]

    4. Simplified0.4

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\color{blue}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)\right)}}\]
    5. Using strategy rm

    6. Applied add-log-exp0.4

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - \color{blue}{\log \left(e^{{1}^{3}}\right)}}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)\right)}\]

    7. Applied add-log-exp0.4

      \[\leadsto \frac{\color{blue}{\log \left(e^{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}\right)} - \log \left(e^{{1}^{3}}\right)}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)\right)}\]

    8. Applied diff-log0.4

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}}}{e^{{1}^{3}}}\right)}}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)\right)}\]

    9. Simplified0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.006169443383165685093616481537992513040081:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, 1, \frac{\left({\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{2} - 1 \cdot 1\right) \cdot 2}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)}\\ \mathbf{elif}\;-2 \cdot x \le 1.858478937487164482871759426069017961947 \cdot 10^{-9}:\\ \;\;\;\;1 \cdot x - \mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}\right)}{\mathsf{fma}\left(1, 1, \frac{2}{1 + e^{-2 \cdot x}} \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))