Average Error: 30.3 → 0.6
Time: 23.5s
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 r1464408 = x;
        double r1464409 = exp(r1464408);
        double r1464410 = 2.0;
        double r1464411 = r1464409 - r1464410;
        double r1464412 = -r1464408;
        double r1464413 = exp(r1464412);
        double r1464414 = r1464411 + r1464413;
        return r1464414;
}
↓
double f(double x) {
        double r1464415 = x;
        double r1464416 = r1464415 * r1464415;
        double r1464417 = r1464415 * r1464416;
        double r1464418 = 0.002777777777777778;
        double r1464419 = r1464417 * r1464418;
        double r1464420 = r1464419 * r1464417;
        double r1464421 = 0.08333333333333333;
        double r1464422 = r1464416 * r1464416;
        double r1464423 = r1464421 * r1464422;
        double r1464424 = r1464420 + r1464423;
        double r1464425 = r1464424 + r1464416;
        return r1464425;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.3

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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