Average Error: 29.8 → 0.6
Time: 3.2s
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 r114934 = x;
        double r114935 = exp(r114934);
        double r114936 = 2.0;
        double r114937 = r114935 - r114936;
        double r114938 = -r114934;
        double r114939 = exp(r114938);
        double r114940 = r114937 + r114939;
        return r114940;
}
↓
double f(double x) {
        double r114941 = x;
        double r114942 = 0.002777777777777778;
        double r114943 = 6.0;
        double r114944 = pow(r114941, r114943);
        double r114945 = 0.08333333333333333;
        double r114946 = 4.0;
        double r114947 = pow(r114941, r114946);
        double r114948 = r114945 * r114947;
        double r114949 = fma(r114942, r114944, r114948);
        double r114950 = fma(r114941, r114941, r114949);
        return r114950;
}

Error

Bits error versus x

Target

Original29.8
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. 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 2020025 +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))))