Average Error: 29.6 → 0.6
Time: 40.8s
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 r5850578 = x;
        double r5850579 = exp(r5850578);
        double r5850580 = 2.0;
        double r5850581 = r5850579 - r5850580;
        double r5850582 = -r5850578;
        double r5850583 = exp(r5850582);
        double r5850584 = r5850581 + r5850583;
        return r5850584;
}
↓
double f(double x) {
        double r5850585 = x;
        double r5850586 = r5850585 * r5850585;
        double r5850587 = 0.08333333333333333;
        double r5850588 = r5850586 * r5850586;
        double r5850589 = r5850587 * r5850588;
        double r5850590 = r5850586 + r5850589;
        double r5850591 = 0.002777777777777778;
        double r5850592 = r5850588 * r5850586;
        double r5850593 = r5850591 * r5850592;
        double r5850594 = r5850590 + r5850593;
        return r5850594;
}


\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

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

Derivation

  1. Initial program 29.6

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

  2. Simplified29.6

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

  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 2019102 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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