Average Error: 30.0 → 0.6
Time: 47.4s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\]
double f(double x) {
        double r11957947 = x;
        double r11957948 = exp(r11957947);
        double r11957949 = 2.0;
        double r11957950 = r11957948 - r11957949;
        double r11957951 = -r11957947;
        double r11957952 = exp(r11957951);
        double r11957953 = r11957950 + r11957952;
        return r11957953;
}
↓
double f(double x) {
        double r11957954 = x;
        double r11957955 = r11957954 * r11957954;
        double r11957956 = 0.08333333333333333;
        double r11957957 = r11957955 * r11957955;
        double r11957958 = r11957956 * r11957957;
        double r11957959 = r11957955 + r11957958;
        double r11957960 = 0.002777777777777778;
        double r11957961 = r11957957 * r11957955;
        double r11957962 = r11957960 * r11957961;
        double r11957963 = r11957959 + r11957962;
        return r11957963;
}


\left(e^{x} - 2\right) + e^{-x}
↓
\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 30.0

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

  2. Simplified30.0

    \[\leadsto \color{blue}{\left(e^{x} - 2\right) - \frac{-1}{e^{x}}}\]

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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