Average Error: 29.3 → 0.7
Time: 22.1s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + x \cdot x\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\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) + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + x \cdot x
double f(double x) {
        double r1536357 = x;
        double r1536358 = exp(r1536357);
        double r1536359 = 2.0;
        double r1536360 = r1536358 - r1536359;
        double r1536361 = -r1536357;
        double r1536362 = exp(r1536361);
        double r1536363 = r1536360 + r1536362;
        return r1536363;
}
↓
double f(double x) {
        double r1536364 = x;
        double r1536365 = r1536364 * r1536364;
        double r1536366 = r1536364 * r1536365;
        double r1536367 = 0.002777777777777778;
        double r1536368 = r1536366 * r1536367;
        double r1536369 = r1536368 * r1536366;
        double r1536370 = 0.08333333333333333;
        double r1536371 = r1536365 * r1536365;
        double r1536372 = r1536370 * r1536371;
        double r1536373 = r1536369 + r1536372;
        double r1536374 = r1536373 + r1536365;
        return r1536374;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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