Average Error: 29.6 → 0.7
Time: 1.0m
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 r9670695 = x;
        double r9670696 = exp(r9670695);
        double r9670697 = 2.0;
        double r9670698 = r9670696 - r9670697;
        double r9670699 = -r9670695;
        double r9670700 = exp(r9670699);
        double r9670701 = r9670698 + r9670700;
        return r9670701;
}
↓
double f(double x) {
        double r9670702 = x;
        double r9670703 = r9670702 * r9670702;
        double r9670704 = r9670703 * r9670703;
        double r9670705 = 0.002777777777777778;
        double r9670706 = r9670704 * r9670705;
        double r9670707 = 0.08333333333333333;
        double r9670708 = fma(r9670704, r9670707, r9670703);
        double r9670709 = fma(r9670706, r9670703, r9670708);
        return r9670709;
}

Error

Bits error versus x

Target

Original29.6
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.6

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

  2. Simplified29.5

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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