Average Error: 29.7 → 0.6
Time: 14.0s
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 r89940 = x;
        double r89941 = exp(r89940);
        double r89942 = 2.0;
        double r89943 = r89941 - r89942;
        double r89944 = -r89940;
        double r89945 = exp(r89944);
        double r89946 = r89943 + r89945;
        return r89946;
}
↓
double f(double x) {
        double r89947 = x;
        double r89948 = 0.002777777777777778;
        double r89949 = 6.0;
        double r89950 = pow(r89947, r89949);
        double r89951 = 0.08333333333333333;
        double r89952 = 4.0;
        double r89953 = pow(r89947, r89952);
        double r89954 = r89951 * r89953;
        double r89955 = fma(r89948, r89950, r89954);
        double r89956 = fma(r89947, r89947, r89955);
        return r89956;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.7

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

  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 2019347 +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))))