Average Error: 29.9 → 0.5
Time: 16.7s
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 r3275547 = x;
        double r3275548 = exp(r3275547);
        double r3275549 = 2.0;
        double r3275550 = r3275548 - r3275549;
        double r3275551 = -r3275547;
        double r3275552 = exp(r3275551);
        double r3275553 = r3275550 + r3275552;
        return r3275553;
}
↓
double f(double x) {
        double r3275554 = x;
        double r3275555 = r3275554 * r3275554;
        double r3275556 = r3275554 * r3275555;
        double r3275557 = 0.002777777777777778;
        double r3275558 = r3275556 * r3275557;
        double r3275559 = r3275558 * r3275556;
        double r3275560 = 0.08333333333333333;
        double r3275561 = r3275555 * r3275555;
        double r3275562 = r3275560 * r3275561;
        double r3275563 = r3275559 + r3275562;
        double r3275564 = r3275563 + r3275555;
        return r3275564;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019132 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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