Average Error: 29.7 → 0.6
Time: 24.8s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\right)
double f(double x) {
        double r4484324 = x;
        double r4484325 = exp(r4484324);
        double r4484326 = 2.0;
        double r4484327 = r4484325 - r4484326;
        double r4484328 = -r4484324;
        double r4484329 = exp(r4484328);
        double r4484330 = r4484327 + r4484329;
        return r4484330;
}
↓
double f(double x) {
        double r4484331 = x;
        double r4484332 = r4484331 * r4484331;
        double r4484333 = r4484331 * r4484332;
        double r4484334 = 0.002777777777777778;
        double r4484335 = r4484333 * r4484334;
        double r4484336 = r4484335 * r4484333;
        double r4484337 = r4484332 * r4484332;
        double r4484338 = 0.08333333333333333;
        double r4484339 = r4484337 * r4484338;
        double r4484340 = r4484332 + r4484339;
        double r4484341 = r4484336 + r4484340;
        return r4484341;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

    \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\right)\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))