Average Error: 29.8 → 0.5
Time: 1.1m
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*\]
\left(e^{x} - 2\right) + e^{-x}
↓
(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot x\right) + \left((\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12} + \left(x \cdot x\right))_*\right))_*
double f(double x) {
        double r16934607 = x;
        double r16934608 = exp(r16934607);
        double r16934609 = 2.0;
        double r16934610 = r16934608 - r16934609;
        double r16934611 = -r16934607;
        double r16934612 = exp(r16934611);
        double r16934613 = r16934610 + r16934612;
        return r16934613;
}
↓
double f(double x) {
        double r16934614 = x;
        double r16934615 = r16934614 * r16934614;
        double r16934616 = r16934615 * r16934615;
        double r16934617 = 0.002777777777777778;
        double r16934618 = r16934616 * r16934617;
        double r16934619 = 0.08333333333333333;
        double r16934620 = fma(r16934616, r16934619, r16934615);
        double r16934621 = fma(r16934618, r16934615, r16934620);
        return r16934621;
}

Error

Bits error versus x

Target

Original29.8
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. Simplified29.8

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

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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