Average Error: 30.1 → 0.7
Time: 11.9s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + x \cdot x\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \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 r3081716 = x;
        double r3081717 = exp(r3081716);
        double r3081718 = 2.0;
        double r3081719 = r3081717 - r3081718;
        double r3081720 = -r3081716;
        double r3081721 = exp(r3081720);
        double r3081722 = r3081719 + r3081721;
        return r3081722;
}
↓
double f(double x) {
        double r3081723 = x;
        double r3081724 = r3081723 * r3081723;
        double r3081725 = r3081723 * r3081724;
        double r3081726 = r3081725 * r3081725;
        double r3081727 = 0.002777777777777778;
        double r3081728 = r3081726 * r3081727;
        double r3081729 = 0.08333333333333333;
        double r3081730 = r3081724 * r3081724;
        double r3081731 = r3081729 * r3081730;
        double r3081732 = r3081728 + r3081731;
        double r3081733 = r3081732 + r3081724;
        return r3081733;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019129 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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