Average Error: 30.5 → 0.6
Time: 21.7s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + x \cdot x\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\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(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + x \cdot x\right)
double f(double x) {
        double r3519510 = x;
        double r3519511 = exp(r3519510);
        double r3519512 = 2.0;
        double r3519513 = r3519511 - r3519512;
        double r3519514 = -r3519510;
        double r3519515 = exp(r3519514);
        double r3519516 = r3519513 + r3519515;
        return r3519516;
}
↓
double f(double x) {
        double r3519517 = x;
        double r3519518 = r3519517 * r3519517;
        double r3519519 = r3519518 * r3519518;
        double r3519520 = r3519518 * r3519519;
        double r3519521 = 0.002777777777777778;
        double r3519522 = r3519520 * r3519521;
        double r3519523 = 0.08333333333333333;
        double r3519524 = r3519519 * r3519523;
        double r3519525 = r3519524 + r3519518;
        double r3519526 = r3519522 + r3519525;
        return r3519526;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.5

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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