Average Error: 29.3 → 0.7
Time: 20.9s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(\frac{1}{12}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{1}{360}, x \cdot x\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(\frac{1}{12}, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right), \frac{1}{360}, x \cdot x\right)\right)
double f(double x) {
        double r2067717 = x;
        double r2067718 = exp(r2067717);
        double r2067719 = 2.0;
        double r2067720 = r2067718 - r2067719;
        double r2067721 = -r2067717;
        double r2067722 = exp(r2067721);
        double r2067723 = r2067720 + r2067722;
        return r2067723;
}
↓
double f(double x) {
        double r2067724 = 0.08333333333333333;
        double r2067725 = x;
        double r2067726 = r2067725 * r2067725;
        double r2067727 = r2067726 * r2067726;
        double r2067728 = r2067726 * r2067725;
        double r2067729 = r2067728 * r2067728;
        double r2067730 = 0.002777777777777778;
        double r2067731 = fma(r2067729, r2067730, r2067726);
        double r2067732 = fma(r2067724, r2067727, r2067731);
        return r2067732;
}

Error

Bits error versus x

Target

Original29.3
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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