Average Error: 29.9 → 0.6
Time: 3.4s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)
double f(double x) {
        double r68962 = x;
        double r68963 = exp(r68962);
        double r68964 = 2.0;
        double r68965 = r68963 - r68964;
        double r68966 = -r68962;
        double r68967 = exp(r68966);
        double r68968 = r68965 + r68967;
        return r68968;
}
↓
double f(double x) {
        double r68969 = x;
        double r68970 = 0.002777777777777778;
        double r68971 = 6.0;
        double r68972 = pow(r68969, r68971);
        double r68973 = 0.08333333333333333;
        double r68974 = 4.0;
        double r68975 = pow(r68969, r68974);
        double r68976 = r68973 * r68975;
        double r68977 = fma(r68970, r68972, r68976);
        double r68978 = fma(r68969, r68969, r68977);
        return r68978;
}

Error

Bits error versus x

Target

Original29.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\frac{1}{360}, {x}^{6}, \frac{1}{12} \cdot {x}^{4}\right)\right)\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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