Average Error: 29.2 → 0.6
Time: 19.0s
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 r1872200 = x;
        double r1872201 = exp(r1872200);
        double r1872202 = 2.0;
        double r1872203 = r1872201 - r1872202;
        double r1872204 = -r1872200;
        double r1872205 = exp(r1872204);
        double r1872206 = r1872203 + r1872205;
        return r1872206;
}
↓
double f(double x) {
        double r1872207 = x;
        double r1872208 = r1872207 * r1872207;
        double r1872209 = r1872207 * r1872208;
        double r1872210 = 0.002777777777777778;
        double r1872211 = r1872209 * r1872210;
        double r1872212 = r1872211 * r1872209;
        double r1872213 = 0.08333333333333333;
        double r1872214 = r1872208 * r1872208;
        double r1872215 = r1872213 * r1872214;
        double r1872216 = r1872212 + r1872215;
        double r1872217 = r1872216 + r1872208;
        return r1872217;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.2
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.2

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

  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 2019153 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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