Average Error: 30.3 → 0.6
Time: 27.4s
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 r2896942 = x;
        double r2896943 = exp(r2896942);
        double r2896944 = 2.0;
        double r2896945 = r2896943 - r2896944;
        double r2896946 = -r2896942;
        double r2896947 = exp(r2896946);
        double r2896948 = r2896945 + r2896947;
        return r2896948;
}
↓
double f(double x) {
        double r2896949 = x;
        double r2896950 = r2896949 * r2896949;
        double r2896951 = r2896950 * r2896950;
        double r2896952 = r2896950 * r2896951;
        double r2896953 = 0.002777777777777778;
        double r2896954 = r2896952 * r2896953;
        double r2896955 = 0.08333333333333333;
        double r2896956 = r2896951 * r2896955;
        double r2896957 = r2896956 + r2896950;
        double r2896958 = r2896954 + r2896957;
        return r2896958;
}

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

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

  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 2019152 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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