Average Error: 29.5 → 0.3
Time: 46.7s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031090765763622602:\\ \;\;\;\;\frac{e^{x} \cdot \left(-4 + e^{x} \cdot e^{x}\right) - \left(-2 - e^{x}\right)}{\left(e^{x} + 2\right) \cdot e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(\frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + x \cdot x\right)\\ \end{array}\]
\left(e^{x} - 2\right) + e^{-x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031090765763622602:\\
\;\;\;\;\frac{e^{x} \cdot \left(-4 + e^{x} \cdot e^{x}\right) - \left(-2 - e^{x}\right)}{\left(e^{x} + 2\right) \cdot e^{x}}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(\frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + x \cdot x\right)\\

\end{array}
double f(double x) {
        double r7123392 = x;
        double r7123393 = exp(r7123392);
        double r7123394 = 2.0;
        double r7123395 = r7123393 - r7123394;
        double r7123396 = -r7123392;
        double r7123397 = exp(r7123396);
        double r7123398 = r7123395 + r7123397;
        return r7123398;
}
↓
double f(double x) {
        double r7123399 = x;
        double r7123400 = -0.031090765763622602;
        bool r7123401 = r7123399 <= r7123400;
        double r7123402 = exp(r7123399);
        double r7123403 = -4.0;
        double r7123404 = r7123402 * r7123402;
        double r7123405 = r7123403 + r7123404;
        double r7123406 = r7123402 * r7123405;
        double r7123407 = -2.0;
        double r7123408 = r7123407 - r7123402;
        double r7123409 = r7123406 - r7123408;
        double r7123410 = 2.0;
        double r7123411 = r7123402 + r7123410;
        double r7123412 = r7123411 * r7123402;
        double r7123413 = r7123409 / r7123412;
        double r7123414 = r7123399 * r7123399;
        double r7123415 = r7123414 * r7123414;
        double r7123416 = r7123414 * r7123415;
        double r7123417 = 0.002777777777777778;
        double r7123418 = r7123416 * r7123417;
        double r7123419 = 0.08333333333333333;
        double r7123420 = r7123419 * r7123415;
        double r7123421 = r7123420 + r7123414;
        double r7123422 = r7123418 + r7123421;
        double r7123423 = r7123401 ? r7123413 : r7123422;
        return r7123423;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.3
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.031090765763622602

    1. Initial program 1.5

      \[\left(e^{x} - 2\right) + e^{-x}\]

    2. Simplified1.6

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

    4. Applied flip--1.6

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

    5. Applied frac-sub1.4

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

    6. Simplified1.4

      \[\leadsto \frac{\color{blue}{e^{x} \cdot \left(e^{x} \cdot e^{x} + -4\right) - \left(-2 - e^{x}\right)}}{\left(e^{x} + 2\right) \cdot e^{x}}\]

    if -0.031090765763622602 < x

    1. Initial program 29.8

      \[\left(e^{x} - 2\right) + e^{-x}\]

    2. Simplified29.8

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

    3. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]

    4. Simplified0.2

      \[\leadsto \color{blue}{\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031090765763622602:\\ \;\;\;\;\frac{e^{x} \cdot \left(-4 + e^{x} \cdot e^{x}\right) - \left(-2 - e^{x}\right)}{\left(e^{x} + 2\right) \cdot e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(\frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + x \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))