Average Error: 30.2 → 0.6
Time: 4.4s
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 r92137 = x;
        double r92138 = exp(r92137);
        double r92139 = 2.0;
        double r92140 = r92138 - r92139;
        double r92141 = -r92137;
        double r92142 = exp(r92141);
        double r92143 = r92140 + r92142;
        return r92143;
}
↓
double f(double x) {
        double r92144 = x;
        double r92145 = 0.002777777777777778;
        double r92146 = 6.0;
        double r92147 = pow(r92144, r92146);
        double r92148 = 0.08333333333333333;
        double r92149 = 4.0;
        double r92150 = pow(r92144, r92149);
        double r92151 = r92148 * r92150;
        double r92152 = fma(r92145, r92147, r92151);
        double r92153 = fma(r92144, r92144, r92152);
        return r92153;
}

Error

Bits error versus x

Target

Original30.2
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.2

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

  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 2020001 +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))))