Average Error: 29.7 → 0.7
Time: 5.7s
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 r100184 = x;
        double r100185 = exp(r100184);
        double r100186 = 2.0;
        double r100187 = r100185 - r100186;
        double r100188 = -r100184;
        double r100189 = exp(r100188);
        double r100190 = r100187 + r100189;
        return r100190;
}
↓
double f(double x) {
        double r100191 = x;
        double r100192 = 0.002777777777777778;
        double r100193 = 6.0;
        double r100194 = pow(r100191, r100193);
        double r100195 = 0.08333333333333333;
        double r100196 = 4.0;
        double r100197 = pow(r100191, r100196);
        double r100198 = r100195 * r100197;
        double r100199 = fma(r100192, r100194, r100198);
        double r100200 = fma(r100191, r100191, r100199);
        return r100200;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.7

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020027 +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))))