Average Error: 29.0 → 0.7
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 r97309 = x;
        double r97310 = exp(r97309);
        double r97311 = 2.0;
        double r97312 = r97310 - r97311;
        double r97313 = -r97309;
        double r97314 = exp(r97313);
        double r97315 = r97312 + r97314;
        return r97315;
}
↓
double f(double x) {
        double r97316 = x;
        double r97317 = 0.002777777777777778;
        double r97318 = 6.0;
        double r97319 = pow(r97316, r97318);
        double r97320 = 0.08333333333333333;
        double r97321 = 4.0;
        double r97322 = pow(r97316, r97321);
        double r97323 = r97320 * r97322;
        double r97324 = fma(r97317, r97319, r97323);
        double r97325 = fma(r97316, r97316, r97324);
        return r97325;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 29.0

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

  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 2020060 +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))))